{"id":240,"date":"2016-12-22T19:44:27","date_gmt":"2016-12-22T19:44:27","guid":{"rendered":"http:\/\/rentabilitet.dk\/opgaver\/?page_id=240"},"modified":"2016-12-22T19:44:27","modified_gmt":"2016-12-22T19:44:27","slug":"vektorregning","status":"publish","type":"page","link":"https:\/\/rentabilitet.dk\/opgaver\/matematik\/vektorregning\/","title":{"rendered":"Vektorregning"},"content":{"rendered":"<html xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:default=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n<style>\n.docxpressoPlugin {\n\tfont-size: 83.35%;\n}\n\n.h5p_layout {\n    box-sizing: content-box !important;\n}\n\n.h5p_layout *{\n    float: none;\n    text-align: initial;\n\tbox-sizing: border-box;\n}\n\/*to make sure that floating elements are always inside the document bounds*\/\n.h5p_layout:after {\n    content: \" \";\n    display: block;\n    height: 0;\n    clear: both;\n    visibility: hidden;\n}\n\/* a default layout for documents without maste styles 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{counter-reset: ulLFO4_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li:before {counter-increment: ulLFO4_3;content: counter(ulLFO4_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO4 ul ul ul ul {counter-reset: ulLFO4_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li:before {counter-increment: ulLFO4_4;content: counter(ulLFO4_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO4 ul ul ul ul ul {counter-reset: ulLFO4_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li ul li:before {counter-increment: ulLFO4_5;content: counter(ulLFO4_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO4 ul ul ul ul ul ul {counter-reset: ulLFO4_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO4_6;content: counter(ulLFO4_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO4 ul ul ul ul ul ul ul {counter-reset: ulLFO4_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO4_7;content: counter(ulLFO4_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO4 ul ul ul ul ul ul ul ul {counter-reset: ulLFO4_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO4_8;content: counter(ulLFO4_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO4 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO4_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO4 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO4_9;content: counter(ulLFO4_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO5 {counter-reset: ulLFO5_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 ul {counter-reset: ulLFO5_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li:before {counter-increment: ulLFO5_1;content: counter(ulLFO5_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO5 ul ul {counter-reset: ulLFO5_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li:before {counter-increment: ulLFO5_2;content: counter(ulLFO5_2, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO5 ul ul ul {counter-reset: ulLFO5_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li:before {counter-increment: ulLFO5_3;content: counter(ulLFO5_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO5 ul ul ul ul {counter-reset: ulLFO5_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li:before {counter-increment: ulLFO5_4;content: counter(ulLFO5_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO5 ul ul ul ul ul {counter-reset: ulLFO5_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li ul li:before {counter-increment: ulLFO5_5;content: counter(ulLFO5_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO5 ul ul ul ul ul ul {counter-reset: ulLFO5_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO5_6;content: counter(ulLFO5_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO5 ul ul ul ul ul ul ul {counter-reset: ulLFO5_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO5_7;content: counter(ulLFO5_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO5 ul ul ul ul ul ul ul ul {counter-reset: ulLFO5_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO5_8;content: counter(ulLFO5_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO5 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO5_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO5 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO5_9;content: counter(ulLFO5_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO6 {counter-reset: ulLFO6_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 ul {counter-reset: ulLFO6_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li:before {counter-increment: ulLFO6_1;content: counter(ulLFO6_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO6 ul ul {counter-reset: ulLFO6_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li:before {counter-increment: ulLFO6_2;content: counter(ulLFO6_2, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO6 ul ul ul {counter-reset: ulLFO6_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li:before {counter-increment: ulLFO6_3;content: counter(ulLFO6_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO6 ul ul ul ul {counter-reset: ulLFO6_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li:before {counter-increment: ulLFO6_4;content: counter(ulLFO6_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO6 ul ul ul ul ul {counter-reset: ulLFO6_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li ul li:before {counter-increment: ulLFO6_5;content: counter(ulLFO6_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO6 ul ul ul ul ul ul {counter-reset: ulLFO6_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO6_6;content: counter(ulLFO6_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO6 ul ul ul ul ul ul ul {counter-reset: ulLFO6_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO6_7;content: counter(ulLFO6_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO6 ul ul ul ul ul ul ul ul {counter-reset: ulLFO6_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO6_8;content: counter(ulLFO6_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO6 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO6_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO6 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO6_9;content: counter(ulLFO6_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO7 {counter-reset: ulLFO7_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 ul {counter-reset: ulLFO7_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li:before {counter-increment: ulLFO7_1;content: counter(ulLFO7_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO7 ul ul {counter-reset: ulLFO7_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li:before {counter-increment: ulLFO7_2;content: counter(ulLFO7_2, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO7 ul ul ul {counter-reset: ulLFO7_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li:before {counter-increment: ulLFO7_3;content: counter(ulLFO7_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO7 ul ul ul ul {counter-reset: ulLFO7_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li:before {counter-increment: ulLFO7_4;content: counter(ulLFO7_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO7 ul ul ul ul ul {counter-reset: ulLFO7_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li ul li:before {counter-increment: ulLFO7_5;content: counter(ulLFO7_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO7 ul ul ul ul ul ul {counter-reset: ulLFO7_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO7_6;content: counter(ulLFO7_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO7 ul ul ul ul ul ul ul {counter-reset: ulLFO7_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO7_7;content: counter(ulLFO7_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO7 ul ul ul ul ul ul ul ul {counter-reset: ulLFO7_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO7_8;content: counter(ulLFO7_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO7 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO7_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO7 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO7_9;content: counter(ulLFO7_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO8 {counter-reset: ulLFO8_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 ul {counter-reset: ulLFO8_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li:before {counter-increment: ulLFO8_1;content: counter(ulLFO8_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO8 ul ul {counter-reset: ulLFO8_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li:before {counter-increment: ulLFO8_2;content: counter(ulLFO8_2, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO8 ul ul ul {counter-reset: ulLFO8_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li:before {counter-increment: ulLFO8_3;content: counter(ulLFO8_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO8 ul ul ul ul {counter-reset: ulLFO8_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li:before {counter-increment: ulLFO8_4;content: counter(ulLFO8_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO8 ul ul ul ul ul {counter-reset: ulLFO8_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li ul li:before {counter-increment: ulLFO8_5;content: counter(ulLFO8_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO8 ul ul ul ul ul ul {counter-reset: ulLFO8_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO8_6;content: counter(ulLFO8_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO8 ul ul ul ul ul ul ul {counter-reset: ulLFO8_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO8_7;content: counter(ulLFO8_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO8 ul ul ul ul ul ul ul ul {counter-reset: ulLFO8_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO8_8;content: counter(ulLFO8_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO8 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO8_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO8 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO8_9;content: counter(ulLFO8_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO9 {counter-reset: ulLFO9_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 ul {counter-reset: ulLFO9_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li:before {counter-increment: ulLFO9_1;content: counter(ulLFO9_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO9 ul ul {counter-reset: ulLFO9_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li:before {counter-increment: ulLFO9_2;content: counter(ulLFO9_2, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO9 ul ul ul {counter-reset: ulLFO9_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li:before {counter-increment: ulLFO9_3;content: counter(ulLFO9_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO9 ul ul ul ul {counter-reset: ulLFO9_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li:before {counter-increment: ulLFO9_4;content: counter(ulLFO9_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO9 ul ul ul ul ul {counter-reset: ulLFO9_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li ul li:before {counter-increment: ulLFO9_5;content: counter(ulLFO9_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO9 ul ul ul ul ul ul {counter-reset: ulLFO9_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO9_6;content: counter(ulLFO9_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO9 ul ul ul ul ul ul ul {counter-reset: ulLFO9_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO9_7;content: counter(ulLFO9_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO9 ul ul ul ul ul ul ul ul {counter-reset: ulLFO9_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO9_8;content: counter(ulLFO9_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO9 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO9_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO9 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO9_9;content: counter(ulLFO9_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO10 {counter-reset: ulLFO10_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 ul {counter-reset: ulLFO10_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li:before {counter-increment: ulLFO10_1;content: counter(ulLFO10_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO10 ul ul {counter-reset: ulLFO10_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li:before {counter-increment: ulLFO10_2;content: counter(ulLFO10_2, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO10 ul ul ul {counter-reset: ulLFO10_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li:before {counter-increment: ulLFO10_3;content: counter(ulLFO10_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO10 ul ul ul ul {counter-reset: ulLFO10_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li:before {counter-increment: ulLFO10_4;content: counter(ulLFO10_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO10 ul ul ul ul ul {counter-reset: ulLFO10_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li ul li:before {counter-increment: ulLFO10_5;content: counter(ulLFO10_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO10 ul ul ul ul ul ul {counter-reset: ulLFO10_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO10_6;content: counter(ulLFO10_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO10 ul ul ul ul ul ul ul {counter-reset: ulLFO10_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO10_7;content: counter(ulLFO10_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO10 ul ul ul ul ul ul ul ul {counter-reset: ulLFO10_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO10_8;content: counter(ulLFO10_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO10 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO10_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO10 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO10_9;content: counter(ulLFO10_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO11 {counter-reset: ulLFO11_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 ul {counter-reset: ulLFO11_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li:before {counter-increment: ulLFO11_1;content: counter(ulLFO11_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO11 ul ul {counter-reset: ulLFO11_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li:before {counter-increment: ulLFO11_2;content: counter(ulLFO11_2, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO11 ul ul ul {counter-reset: ulLFO11_4;margin-left: -1.375in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li {text-indent: -0.125in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li:before {counter-increment: ulLFO11_3;content: counter(ulLFO11_3, lower-roman)  \". \";margin-left: -0.125in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO11 ul ul ul ul {counter-reset: ulLFO11_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li:before {counter-increment: ulLFO11_4;content: counter(ulLFO11_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO11 ul ul ul ul ul {counter-reset: ulLFO11_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li ul li:before {counter-increment: ulLFO11_5;content: counter(ulLFO11_5, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO11 ul ul ul ul ul ul {counter-reset: ulLFO11_7;margin-left: -2.875in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li ul li ul li {text-indent: -0.125in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO11_6;content: counter(ulLFO11_6, lower-roman)  \". \";margin-left: -0.125in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO11 ul ul ul ul ul ul ul {counter-reset: ulLFO11_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO11_7;content: counter(ulLFO11_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO11 ul ul ul ul ul ul ul ul {counter-reset: ulLFO11_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO11_8;content: counter(ulLFO11_8, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO11 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO11_10;margin-left: -4.375in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.125in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO11 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO11_9;content: counter(ulLFO11_9, lower-roman)  \". \";margin-left: -0.125in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO12 {counter-reset: ulLFO12_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 ul {counter-reset: ulLFO12_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li:before {counter-increment: ulLFO12_1;content: counter(ulLFO12_1, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO12 ul ul {counter-reset: ulLFO12_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li:before {counter-increment: ulLFO12_2;content: counter(ulLFO12_2, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO12 ul ul ul {counter-reset: ulLFO12_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li:before {counter-increment: ulLFO12_3;content: counter(ulLFO12_3, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO12 ul ul ul ul {counter-reset: ulLFO12_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li:before {counter-increment: ulLFO12_4;content: counter(ulLFO12_4, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO12 ul ul ul ul ul {counter-reset: ulLFO12_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li ul li:before {counter-increment: ulLFO12_5;content: counter(ulLFO12_5, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO12 ul ul ul ul ul ul {counter-reset: ulLFO12_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO12_6;content: counter(ulLFO12_6, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO12 ul ul ul ul ul ul ul {counter-reset: ulLFO12_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO12_7;content: counter(ulLFO12_7, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO12 ul ul ul ul ul ul ul ul {counter-reset: ulLFO12_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO12_8;content: counter(ulLFO12_8, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO12 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO12_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO12 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO12_9;content: counter(ulLFO12_9, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO13 {counter-reset: ulLFO13_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 ul {counter-reset: ulLFO13_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li:before {counter-increment: ulLFO13_1;content: counter(ulLFO13_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO13 ul ul {counter-reset: ulLFO13_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li:before {counter-increment: ulLFO13_2;content: counter(ulLFO13_2, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO13 ul ul ul {counter-reset: ulLFO13_4;margin-left: -1.375in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li {text-indent: -0.125in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li:before {counter-increment: ulLFO13_3;content: counter(ulLFO13_3, lower-roman)  \". \";margin-left: -0.125in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO13 ul ul ul ul {counter-reset: ulLFO13_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li:before {counter-increment: ulLFO13_4;content: counter(ulLFO13_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO13 ul ul ul ul ul {counter-reset: ulLFO13_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li ul li:before {counter-increment: ulLFO13_5;content: counter(ulLFO13_5, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO13 ul ul ul ul ul ul {counter-reset: ulLFO13_7;margin-left: -2.875in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li ul li ul li {text-indent: -0.125in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO13_6;content: counter(ulLFO13_6, lower-roman)  \". \";margin-left: -0.125in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO13 ul ul ul ul ul ul ul {counter-reset: ulLFO13_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO13_7;content: counter(ulLFO13_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO13 ul ul ul ul ul ul ul ul {counter-reset: ulLFO13_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO13_8;content: counter(ulLFO13_8, lower-latin)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO13 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO13_10;margin-left: -4.375in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.125in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO13 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO13_9;content: counter(ulLFO13_9, lower-roman)  \". \";margin-left: -0.125in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO14 {counter-reset: ulLFO14_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 ul {counter-reset: ulLFO14_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li:before {counter-increment: ulLFO14_1;content: counter(ulLFO14_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO14 ul ul {counter-reset: ulLFO14_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li:before {counter-increment: ulLFO14_2;content: counter(ulLFO14_2, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO14 ul ul ul {counter-reset: ulLFO14_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li:before {counter-increment: ulLFO14_3;content: counter(ulLFO14_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO14 ul ul ul ul {counter-reset: ulLFO14_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li:before {counter-increment: ulLFO14_4;content: counter(ulLFO14_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO14 ul ul ul ul ul {counter-reset: ulLFO14_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li ul li:before {counter-increment: ulLFO14_5;content: counter(ulLFO14_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO14 ul ul ul ul ul ul {counter-reset: ulLFO14_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO14_6;content: counter(ulLFO14_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO14 ul ul ul ul ul ul ul {counter-reset: ulLFO14_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO14_7;content: counter(ulLFO14_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO14 ul ul ul ul ul ul ul ul {counter-reset: ulLFO14_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO14_8;content: counter(ulLFO14_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO14 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO14_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO14 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO14_9;content: counter(ulLFO14_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO15 {counter-reset: ulLFO15_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 ul {counter-reset: ulLFO15_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li:before {counter-increment: ulLFO15_1;content: counter(ulLFO15_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO15 ul ul {counter-reset: ulLFO15_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li:before {counter-increment: ulLFO15_2;content: counter(ulLFO15_2, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO15 ul ul ul {counter-reset: ulLFO15_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li:before {counter-increment: ulLFO15_3;content: counter(ulLFO15_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO15 ul ul ul ul {counter-reset: ulLFO15_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li:before {counter-increment: ulLFO15_4;content: counter(ulLFO15_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO15 ul ul ul ul ul {counter-reset: ulLFO15_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li ul li:before {counter-increment: ulLFO15_5;content: counter(ulLFO15_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO15 ul ul ul ul ul ul {counter-reset: ulLFO15_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO15_6;content: counter(ulLFO15_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO15 ul ul ul ul ul ul ul {counter-reset: ulLFO15_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO15_7;content: counter(ulLFO15_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO15 ul ul ul ul ul ul ul ul {counter-reset: ulLFO15_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO15_8;content: counter(ulLFO15_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO15 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO15_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO15 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO15_9;content: counter(ulLFO15_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO16 {counter-reset: ulLFO16_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 ul {counter-reset: ulLFO16_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li:before {counter-increment: ulLFO16_1;content: counter(ulLFO16_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO16 ul ul {counter-reset: ulLFO16_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li:before {counter-increment: ulLFO16_2;content: counter(ulLFO16_2, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO16 ul ul ul {counter-reset: ulLFO16_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li:before {counter-increment: ulLFO16_3;content: counter(ulLFO16_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO16 ul ul ul ul {counter-reset: ulLFO16_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li:before {counter-increment: ulLFO16_4;content: counter(ulLFO16_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO16 ul ul ul ul ul {counter-reset: ulLFO16_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li ul li:before {counter-increment: ulLFO16_5;content: counter(ulLFO16_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO16 ul ul ul ul ul ul {counter-reset: ulLFO16_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO16_6;content: counter(ulLFO16_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO16 ul ul ul ul ul ul ul {counter-reset: ulLFO16_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO16_7;content: counter(ulLFO16_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO16 ul ul ul ul ul ul ul ul {counter-reset: ulLFO16_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO16_8;content: counter(ulLFO16_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO16 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO16_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO16 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO16_9;content: counter(ulLFO16_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO17 {counter-reset: ulLFO17_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 ul {counter-reset: ulLFO17_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li:before {counter-increment: ulLFO17_1;content: counter(ulLFO17_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO17 ul ul {counter-reset: ulLFO17_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li:before {counter-increment: ulLFO17_2;content: counter(ulLFO17_2, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO17 ul ul ul {counter-reset: ulLFO17_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li:before {counter-increment: ulLFO17_3;content: counter(ulLFO17_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO17 ul ul ul ul {counter-reset: ulLFO17_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li:before {counter-increment: ulLFO17_4;content: counter(ulLFO17_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO17 ul ul ul ul ul {counter-reset: ulLFO17_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li ul li:before {counter-increment: ulLFO17_5;content: counter(ulLFO17_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO17 ul ul ul ul ul ul {counter-reset: ulLFO17_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO17_6;content: counter(ulLFO17_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO17 ul ul ul ul ul ul ul {counter-reset: ulLFO17_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO17_7;content: counter(ulLFO17_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO17 ul ul ul ul ul ul ul ul {counter-reset: ulLFO17_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO17_8;content: counter(ulLFO17_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO17 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO17_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO17 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO17_9;content: counter(ulLFO17_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO18 {counter-reset: ulLFO18_1;margin-left: 0;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 ul {counter-reset: ulLFO18_2;margin-left: -0.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li {text-indent: -0.25in;margin-left: 36pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li:before {counter-increment: ulLFO18_1;content: counter(ulLFO18_1, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO18 ul ul {counter-reset: ulLFO18_3;margin-left: -0.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li {text-indent: -0.25in;margin-left: 72pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li:before {counter-increment: ulLFO18_2;content: counter(ulLFO18_2, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO18 ul ul ul {counter-reset: ulLFO18_4;margin-left: -1.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li {text-indent: -0.25in;margin-left: 108pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li:before {counter-increment: ulLFO18_3;content: counter(ulLFO18_3, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO18 ul ul ul ul {counter-reset: ulLFO18_5;margin-left: -1.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li {text-indent: -0.25in;margin-left: 144pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li:before {counter-increment: ulLFO18_4;content: counter(ulLFO18_4, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO18 ul ul ul ul ul {counter-reset: ulLFO18_6;margin-left: -2.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 180pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li ul li:before {counter-increment: ulLFO18_5;content: counter(ulLFO18_5, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO18 ul ul ul ul ul ul {counter-reset: ulLFO18_7;margin-left: -2.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 216pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO18_6;content: counter(ulLFO18_6, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO18 ul ul ul ul ul ul ul {counter-reset: ulLFO18_8;margin-left: -3.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 252pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO18_7;content: counter(ulLFO18_7, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO18 ul ul ul ul ul ul ul ul {counter-reset: ulLFO18_9;margin-left: -3.75in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 288pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO18_8;content: counter(ulLFO18_8, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO18 ul ul ul ul ul ul ul ul ul {counter-reset: ulLFO18_10;margin-left: -4.25in;padding: 0; list-style-type: none;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li ul li ul li ul li ul li ul li {text-indent: -0.25in;margin-left: 324pt;font-size: 11pt;}\n div#h5p_6a487136d9786 ul.LFO18 li ul li ul li ul li ul li ul li ul li ul li ul li:before {counter-increment: ulLFO18_9;content: counter(ulLFO18_9, decimal)  \". \";margin-left: -0.25in;display: table-cell;}\n div#h5p_6a487136d9786 ul.LFO19 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MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}\n div#h5p_6a487136d9786  section.h5p_page p span.doxMathFormula .MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}\n div#h5p_6a487136d9786  section.h5p_page p span.doxMathFormula .MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}\n div#h5p_6a487136d9786  section.h5p_page p span.doxMathFormula .MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}\n div#h5p_6a487136d9786  section.h5p_page p span.doxMathFormula .MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}\n div#h5p_6a487136d9786  section.h5p_page p span.doxMathFormula .mjx-sub span.mjx-char {font-size: 70.7%}\n div#h5p_6a487136d9786  section.h5p_page p span.doxMathFormula .mjx-sup span.mjx-char {font-size: 70.7%}\n div#h5p_6a487136d9786  section.h5p_page p span.doxMathFormula .mjx-under span.mjx-char {font-size: 70.7%}\n div#h5p_6a487136d9786  section.h5p_page p span.doxMathFormula .mjx-over span.mjx-char {font-size: 70.7%}\n div#h5p_6a487136d9786  section.h5p_page p span.doxMathFormula .mjx-root span.mjx-char {font-size: 70.7%;}\n div#h5p_6a487136d9786  section.h5p_page p span.doxMathFormula .mjx-root {font-size: auto !important ; width: 0.5em !important; vertical-align: 1.2em !important}\n<\/style><script src=\"https:\/\/cdn.mathjax.org\/mathjax\/2.6-latest\/MathJax.js?config=MML_CHTML\" type=\"text\/javascript\"> <\/script>\n<div id=\"h5p_6a487136d9786\"><section xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"section_0\" class=\"h5p_layout h5p_page h5p_page_PL0\"><div class=\"PL0\"><p class=\"P1\"><span id=\"_GoBack\"\/><\/p><div style=\"left: 0;height: 750.80016pt;width: 578.90016pt;\" class=\"a14\"><div style=\"left: 0;height: 750.80016pt;position: absolute;width: 578.90016pt;\" class=\"\"><div style=\"position: absolute; float: none !important;height: 10.42778in;width: 8.04028in;margin-left: 0.0125in;margin-top: 0in;\" class=\"a1\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a1\" height=\"10.42778in\" width=\"8.04028in\" viewBox=\"0 0 772 1001\"><g><path d=\"M 0 0 L 772 0 772 1001 0 1001 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 10.42778in;width: 5.88339in;margin-left: 2.16939in;margin-top: 0in;\" class=\"a2\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a2\" height=\"10.42778in\" width=\"5.88339in\" viewBox=\"0 0 565 1001\"><g><path d=\"M 0 0 L 565 0 565 1001 0 1001 Z\"\/><\/g><\/svg><p class=\"Ingenafstand\"><span class=\"T6\">Vektorregning<\/span><\/p><p class=\"Ingenafstand\"><span class=\"T7\">&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P8\">&nbsp;<\/p><p class=\"P9\">&nbsp;<\/p><p class=\"P10\">&nbsp;<\/p><\/div><div style=\"left: 0;height: 453.978pt;position: absolute;width: 78.12288pt;\" class=\"\"><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 1.08504in;margin-top: 4.19957in;\" class=\"a3\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a3\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 1.08504in;margin-top: 3.14672in;\" class=\"a4\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a4\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 0in;margin-top: 3.14672in;\" class=\"a5\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a5\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 0in;margin-top: 2.09388in;\" class=\"a6\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a6\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 0in;margin-top: 4.19957in;\" class=\"a7\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a7\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 1.05284in;width: 1.08504in;margin-left: 1.08504in;margin-top: 5.25241in;\" class=\"a8\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a8\" height=\"1.05284in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><\/div><\/div><div style=\"position: absolute; float: none !important;height: 1.05354in;width: 1.08504in;margin-left: 1.64457in;margin-top: 0in;\" class=\"a9\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a9\" height=\"1.05354in\" width=\"1.08504in\" viewBox=\"0 0 104 101\"><g><path d=\"M 0 0 L 104 0 104 101 0 101 Z\"\/><\/g><\/svg><p class=\"P11\">&nbsp;<\/p><\/div><\/div><div style=\"left: 0;height: 736.11216pt;position: absolute;width: 354.88296pt;\" class=\"\"><div style=\"left: 0;height: 736.11144pt;position: absolute;width: 19.53pt;\" class=\"\"><div style=\"position: absolute; float: none !important;height: 0.26561in;width: 0.27125in;margin-left: 7.29592in;margin-top: 9.95816in;\" class=\"a10\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a10\" height=\"0.26561in\" width=\"0.27125in\" viewBox=\"0 0 26 25\"><g><path d=\"M 0 0 L 26 0 26 25 0 25 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 0.26561in;width: 0.27125in;margin-left: 7.29592in;margin-top: 9.69642in;\" class=\"a11\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a11\" height=\"0.26561in\" width=\"0.27125in\" viewBox=\"0 0 26 25\"><g><path d=\"M 0 0 L 26 0 26 25 0 25 Z\"\/><\/g><\/svg><\/div><div style=\"position: absolute; float: none !important;height: 0.26561in;width: 0.27125in;margin-left: 7.56717in;margin-top: 9.69642in;\" class=\"a12\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a12\" height=\"0.26561in\" width=\"0.27125in\" viewBox=\"0 0 26 25\"><g><path d=\"M 0 0 L 26 0 26 25 0 25 Z\"\/><\/g><\/svg><\/div><\/div><div style=\"position: absolute; float: none !important;height: 0.95896in;width: 4.92893in;margin-left: 2.17136in;margin-top: 9.26482in;\" class=\"a13\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a13\" height=\"0.95896in\" width=\"4.92893in\" viewBox=\"0 0 473 92\"><g><path d=\"M 0 0 L 473 0 473 92 0 92 Z\"\/><\/g><\/svg><p class=\"P12\"><span class=\"T13\">Frederik Hovmark Pedersen<\/span><\/p><p class=\"P14\"><span class=\"T15\">&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><\/p><\/div><\/div><\/div><p class=\"Normal\"\/><div style=\"position: relative; float: none !important;height: 0.28403in;width: 0.26042in;margin-left: 6.51047in;margin-top: 8.70968in;\" class=\"a15\"><svg style=\"position: absolute; top: 0; left: 0;padding: 0 !important; border: none !important\" preserveAspectRatio=\"none\" fill-rule=\"evenodd\" class=\"a15\" height=\"0.28403in\" width=\"0.26042in\" viewBox=\"0 0 25 27\"><g><path d=\"M 0 0 L 25 0 25 27 0 27 Z\"\/><\/g><\/svg><\/div><p class=\"Normal\"\/><p class=\"P17\">&nbsp;<\/p><p class=\"Overskrift\"><span class=\"T18\">Indholdsfortegnelse<\/span><\/p><div><p class=\"P19\"><a href=\"#_Toc433991954\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Indledning<\/span>&emsp;<span>2<\/span><\/a><\/p><p class=\"P20\"><a href=\"#_Toc433991955\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Teori<\/span>&emsp;<span>2<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991956\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Generelt om vektorer<\/span>&emsp;<span>2<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991957\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Addition af vektorer<\/span>&emsp;<span>2<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991958\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Multiplikation af vektor med tal<\/span>&emsp;<span>3<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991959\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Vektorers koordinater<\/span>&emsp;<span>4<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991960\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Stedvektorer<\/span>&emsp;<span>5<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991961\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">L\u00e6ngden af vektorer<\/span>&emsp;<span>6<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991962\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Skalarprodukt(prikprodukt)<\/span>&emsp;<span>6<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991963\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Regneregler for skalarprodukt<\/span>&emsp;<span>9<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991964\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Tv\u00e6rvektor<\/span>&emsp;<span>10<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991965\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Vektorprojektioner<\/span>&emsp;<span>11<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991966\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Areal af parallelogram<\/span>&emsp;<span>11<\/span><\/a><\/p><p class=\"P21\"><a href=\"#_Toc433991967\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Opgaver<\/span>&emsp;<span>13<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991968\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Opgave 1<\/span>&emsp;<span>13<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991969\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Opgave 2<\/span>&emsp;<span>14<\/span><\/a><\/p><p class=\"Indholdsfortegnelse2\"><a href=\"#_Toc433991970\" target=\"_top\" title=\"\"><span class=\"Hyperlink\">Opgave 3<\/span>&emsp;<span>16<\/span><\/a><\/p><\/div><p class=\"Normal\">&nbsp;<\/p><p class=\"P22\">&nbsp;<\/p><h1 class=\"P23\"><span id=\"_Toc433991954\"\/><span>Indledning<\/span><\/h1><p class=\"P24\"><span>I denne emneopgave vil jeg<\/span>&#x200b;&#x200b;&nbsp;<span>gennemg\u00e5 emnet vektorregning.<\/span>&#x200b;&#x200b;&nbsp;<span>Dette vil jeg g\u00f8re p\u00e5 den m\u00e5de, at jeg f\u00f8rst gennemg\u00e5r en lang r\u00e6kke teori samtidig med at jeg kommer med nogle eksempler p\u00e5 brugen af denne teori samt nogle enkelte beviser. Dern\u00e6st vil jeg s\u00e5 benytte teorien til at l\u00f8se de i opgaveformuleringen givne opgaver.<\/span><\/p><h1 class=\"P25\"><span id=\"_Toc433991955\"\/><span>Teori<\/span><\/h1><h2 class=\"P26\"><span id=\"_Toc433991956\"\/><span>Generelt om vektorer<\/span><\/h2><p class=\"P27\"><span>En vektor er defineret som m\u00e6ngden af alle pile, der peger i samme retning og har samme l\u00e6ngde.<\/span>&#x200b;&#x200b;&nbsp;<span>Det vil sige at to pile godt kan repr\u00e6sentere den samme vektor, blot de har samme l\u00e6ngde og retning.<\/span>&#x200b;&#x200b;&nbsp;<\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P28\"><span class=\"T29\">En vektor betegnes som et bogstav med en pil over. Et eksempel kunne v\u00e6re vektor a, som skrives som&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T30\">.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T31\">Et eksempel p\u00e5 en repr\u00e6sentant for&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T32\">&#x200b;&#x200b;&nbsp;kunne s\u00e5 v\u00e6re en pil der begynder i punktet A og slutter&#x200b;&#x200b;&nbsp;<\/span><span class=\"T33\">i punktet B. Denne pil vil s\u00e5 he<\/span><span class=\"T34\">dde&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T35\">.&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P36\"><span class=\"T37\">Skal man beskrive l\u00e6ngden af en vektor skrives dette som&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T38\">, som s\u00e5 betyder l\u00e6ngden af vektor a.<\/span><span class=\"T39\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P40\"><span class=\"T41\">En speciel vektor er vektoren med l\u00e6ngden 0. Denne kaldes for nulvektoren og betegnes&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T42\">.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T43\">Modsat s\u00e5 kaldes alle vektorer<\/span><span class=\"T44\">&#x200b;&#x200b;&nbsp;med en l\u00e6ngde, alts\u00e5 alle<\/span><span class=\"T45\">&#x200b;&#x200b;&nbsp;undtagen nulvektoren<\/span><span class=\"T46\">,<\/span><span class=\"T47\">&#x200b;&#x200b;&nbsp;for egentlige vektorer.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P48\"><span class=\"T49\">En modsatrettet vektor er en vektor der vender den anden vej. Denne skrives fx som&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>-<\/mml:mo><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T50\">.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T51\">Alts\u00e5 s\u00e6tter man minus foran en vektor, skal man blot vende den om<\/span><\/p><p class=\"P52\">&nbsp;<\/p><h2 class=\"P53\"><span id=\"_Toc433991957\"\/><span>Addition af vektorer<\/span><\/h2><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P54\"><span class=\"T55\">Addition af vektorer handler, som det fremg\u00e5r af navnet, om at l\u00e6gge vektorer sammen. Det vil sige at hvis man har to vektorer, fx&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T56\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T57\">, s\u00e5 bliver sumvektoren den vektor der fremkommer, n\u00e5r de to vektorer placeres efter hinanden. Denne vektor starter s\u00e5 hvor&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T58\">&#x200b;&#x200b;&nbsp;starter, og slutter hvor&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T59\">&#x200b;&#x200b;&nbsp;slutter. S<\/span><span class=\"T60\">\u00e5ledes dannes der en trekant.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T61\">I denne trekant kan man s\u00e5 sige at man har punkterne A, B og C. I s\u00e5 fald vil&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T62\">&#x200b;&#x200b;&nbsp;repr\u00e6sentere&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T63\">,&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">B<\/mml:mi><mml:mi mathvariant=\"normal\">C<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T64\">&#x200b;&#x200b;&nbsp;vil repr\u00e6sentere&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T65\">, og s\u00e5 g\u00e6lder det at&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mo>+<\/mml:mo><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi>\u00a0<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mo>+<\/mml:mo><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">B<\/mml:mi><mml:mi mathvariant=\"normal\">C<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mo>=<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">C<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T66\">.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T67\">Sumvektoren skrives&#x200b;&#x200b;&nbsp;<\/span><span class=\"T68\">s\u00e5&#x200b;&#x200b;&nbsp;<\/span><span class=\"T69\">som&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T70\">.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P71\"><span class=\"T72\">Noget der er vigtigt at vide i forbindelse med addition af vektorer, er at r\u00e6kkef\u00f8lgen er ligegyldig. Alts\u00e5 at&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T73\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T74\">.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T75\">Og dermed er det ligegyldigt om starter med&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T76\">&#x200b;&#x200b;&nbsp;og derefter finder&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T77\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T78\">eller om man g\u00f8r det omvendt.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T79\">Dette kaldes<\/span><span class=\"T80\">&#x200b;&#x200b;&nbsp;for<\/span><span class=\"T81\">&#x200b;&#x200b;&nbsp;den kommutative lov.&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P82\"><span class=\"T83\">En anden lov er den a<\/span><span class=\"T84\">ssociative lov<\/span><span class=\"T85\">. Denne siger at hvis man har tre vilk\u00e5rlige vektorer, fx&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T86\">,<\/span><span class=\"T87\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T88\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T89\">, og man ved addition af disse s\u00e6tter parenteser omkring, s\u00e5 har det ingen betydning hvor parenteserne er placeret. Alts\u00e5<\/span><span class=\"T90\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T91\">(<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T92\">&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T93\">) +&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T94\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T95\">&#x200b;&#x200b;&nbsp;+ (<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T96\">&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T97\">)<\/span><span class=\"T98\">.&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P99\">&nbsp;<\/p><h2 class=\"P100\"><span id=\"_Toc433991958\"\/><span>Multiplikation af<\/span>&#x200b;&#x200b;&nbsp;<span>vektor med tal<\/span><\/h2><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P101\"><span class=\"T102\">Dette handler om at gange en vektor med et tal. Hvis jeg lader<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T103\">&#x200b;&#x200b;&nbsp;v\u00e6re en egentlig vektor, og lader t st\u00e5 for et vilk\u00e5rligt tal, s\u00e5 g\u00e6lder f\u00f8lgende.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P104\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>t<\/mml:mi><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T105\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi>t<\/mml:mi><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T106\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"{\" close=\"\" separators=\"|\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mrow><mml:maligngroup\/><mml:mi>t<\/mml:mi><mml:mi>*<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi>H<\/mml:mi><mml:mi>v<\/mml:mi><mml:mi>i<\/mml:mi><mml:mi>s<\/mml:mi><mml:mi>\u00a0<\/mml:mi><mml:mi>t<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:mtd><\/mml:mtr><mml:mtr><mml:mtd><mml:mrow><mml:maligngroup\/><mml:mo>-<\/mml:mo><mml:mi>t<\/mml:mi><mml:mi>\u00a0<\/mml:mi><mml:mi>*<\/mml:mi><mml:mi>\u00a0<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi>\u00a0<\/mml:mi><mml:mi>H<\/mml:mi><mml:mi>v<\/mml:mi><mml:mi>i<\/mml:mi><mml:mi>s<\/mml:mi><mml:mi>\u00a0<\/mml:mi><mml:mi>t<\/mml:mi><mml:mo>&lt;<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:mtd><\/mml:mtr><\/mml:mtable><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P107\"><span class=\"T108\">Dette skal forst\u00e5s s\u00e5ledes<\/span><span class=\"T109\">,<\/span><span class=\"T110\">&#x200b;&#x200b;&nbsp;at hvis t er positiv, s\u00e5 er t *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T111\">&#x200b;&#x200b;&nbsp;ensrettet med&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T112\">. Men er t derimod negativ, s\u00e5 er t *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T113\">&#x200b;&#x200b;&nbsp;modsatrettet af&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T114\">.&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P115\"><span class=\"T116\">Ved&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi mathvariant=\"normal\">t<\/mml:mi><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T117\">&#x200b;&#x200b;&nbsp;forst\u00e5s<\/span><span class=\"T118\">&#x200b;&#x200b;&nbsp;l\u00e6ngden<\/span><span class=\"T119\">. Ved&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T120\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T121\">forst\u00e5s l<\/span><span class=\"T122\">\u00e6ngden af&#x200b;&#x200b;&nbsp;<\/span><span class=\"T123\">en&#x200b;&#x200b;&nbsp;<\/span><span class=\"T124\">vektor<\/span><span class=\"T125\">. Og ved&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi mathvariant=\"normal\">t<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T126\">&#x200b;&#x200b;&nbsp;forst\u00e5s den&#x200b;&#x200b;&nbsp;<\/span><span class=\"T127\">numerisk<\/span><span class=\"T128\">e<\/span><span class=\"T129\">&#x200b;&#x200b;&nbsp;v\u00e6rdi<\/span><span class=\"T130\">. Numerisk v\u00e6rdi betyder at tallet altid skal v\u00e6re positivt eller nul,&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P131\"><span>En ting der ligeledes g\u00e6lder er f\u00f8lgende.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P132\"><span class=\"T133\">t *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T134\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mn>0<\/mml:mn><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T135\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P136\"><span>Her ses det s\u00e5 at t gange nulvektoren er lig nulvektoren.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P137\"><span class=\"T138\">Et eksempel p<\/span><span class=\"T139\">\u00e5 at gange en vektor med et tal<\/span><span class=\"T140\">&#x200b;&#x200b;&nbsp;kunne v\u00e6re 2 *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T141\">. Dette bliver s\u00e5 til 2*<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T142\">.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T143\">Grafisk ville dette se ud som at man nu fik 2 *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T144\">&#x200b;&#x200b;&nbsp;i forl\u00e6ngelse af hinanden.&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P145\"><span>Ved multiplikation af vektorer med tal g\u00e6lder s\u00e5 den distributive lov. Det betyder at f\u00f8lgende regler g\u00f8r sig g\u00e6ldende.<\/span><\/p><ul xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"LFO18\" style=\"counter-reset: ulLFO18_1 0;\"><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;text-align: justify !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P146\"><span class=\"T147\">s * (t *<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T148\">) = t * (s*<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T149\">) = (s * t) *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><\/p><\/li><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;text-align: justify !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P150\"><span class=\"T151\">s *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T152\">&#x200b;&#x200b;&nbsp;+ t *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T153\">&#x200b;&#x200b;&nbsp;= (s + t) *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><\/p><\/li><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;hyphenate: false;hyphenate: false;mleft: 0.5in !important;text-align: justify !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P154\"><span class=\"T155\">t *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T156\">&#x200b;&#x200b;&nbsp;+ t *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T157\">&#x200b;&#x200b;&nbsp;= t * (<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T158\">&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T159\">)<\/span><\/p><\/li><\/ul><h2 class=\"P160\"><span id=\"_Toc433991959\"\/><span>Vektorers koordinater<\/span><\/h2><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P161\"><span class=\"T162\">For at kunne tegne en vektor, er det n\u00f8dvendigt at kende vektorens koordinater. Og for at kunne finde disse, er det n\u00f8dvendigt at indf\u00f8re de to basisvektorer&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">I<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T163\">, som g\u00e5r i x-aksens retning, og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">J<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T164\">&#x200b;&#x200b;&nbsp;som g\u00e5r i y-aksens retning.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T165\">Det specielle ved disse to er s\u00e5, at de begge har l\u00e6ngden 1. Alts\u00e5&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">I<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T166\">&#x200b;&#x200b;&nbsp;= &#x200b;&#x200b;&nbsp;1 og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">J<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T167\">&#x200b;&#x200b;&nbsp;= 1<\/span><span class=\"T168\">.&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P169\"><span class=\"T170\">N\u00e5r man ved dette kan man splitte en vilk\u00e5rlig vektor, fx&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T171\">, op i to dele.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T172\">En der h\u00f8rer til x-aksens retning(a<\/span><span class=\"T173\">1<\/span><span class=\"T174\">), og en der h\u00f8rer til y-aksens retning(a<\/span><span class=\"T175\">2<\/span><span class=\"T176\">).<\/span><span class=\"T177\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T178\">Disse koordinater skrives som&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T179\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T180\">.<\/span><span class=\"T181\">&#x200b;&#x200b;&nbsp;Alts\u00e5&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><mml:mo>=<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T182\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mi>x<\/mml:mi><mml:mo>-<\/mml:mo><mml:mi>v<\/mml:mi><mml:mi mathvariant=\"normal\">\u00e6<\/mml:mi><mml:mi>r<\/mml:mi><mml:mi>d<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:mrow><mml:mrow><mml:mi>y<\/mml:mi><mml:mo>-<\/mml:mo><mml:mi>v<\/mml:mi><mml:mi mathvariant=\"normal\">\u00e6<\/mml:mi><mml:mi>r<\/mml:mi><mml:mi>d<\/mml:mi><mml:mi>i<\/mml:mi><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T183\">.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T184\">&#x200b;&#x200b;&nbsp;Og dette betyder at<\/span><span class=\"T185\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T186\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;= a<\/span><span class=\"T187\">1<\/span><span class=\"T188\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">I<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T189\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;+ a<\/span><span class=\"T190\">2<\/span><span class=\"T191\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">J<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T192\">.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P193\"><span class=\"T194\">Et eksempel kunne v\u00e6re at man havde en vektor<\/span><span class=\"T195\">, fx&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T196\">,<\/span><span class=\"T197\">&#x200b;&#x200b;&nbsp;der gik 4 i x\u2019s retning og 3 i y\u2019s retning. Dette ville s\u00e5 kunne skrives som&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T198\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;= 4 *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">I<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T199\">&#x200b;&#x200b;&nbsp;+ 3 *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">J<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T200\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T201\">.&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P202\"><span>Har man s\u00e5 to vektorer, s\u00e5 kan disses koordinater jo naturligvis l\u00e6gges sammen<\/span>&#x200b;&#x200b;&nbsp;<span>og<\/span>&#x200b;&#x200b;&nbsp;<span>tr\u00e6kkes fra hinanden. Ved dette g\u00e6lder f\u00f8lgende regler:<\/span><\/p><ul class=\"LFO9\" style=\"counter-reset: ulLFO9_1 0;\"><li style=\"color: initial; font-family: initial;hyphenate: false;font-family: Times New Roman;font-size: 12pt;hyphenate: false;hyphenate: false;font-family: Times New Roman;font-size: 12pt;hyphenate: false;margin-top: 0.0694in !important;margin-bottom: 0.0694in !important;line-height: 1 !important;text-align: justify !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P203\"><span class=\"T204\">Addition:<\/span><\/p><\/li><\/ul><p class=\"P205\"><span>Jeg g\u00e5r ud fra at jeg har f\u00f8lgende to vektorer med dertilh\u00f8rende koordinater:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P206\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T207\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T208\">&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;og &nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T209\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p class=\"P210\"><span>Disse l\u00e6gges s\u00e5 sammen ved at addere henholdsvis de to x-v\u00e6rdier og de to y-v\u00e6rdier, hvilket ses herunder:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P211\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T212\">&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T213\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:msub><mml:mrow><mml:mo>+<\/mml:mo><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><ul class=\"LFO10\" style=\"counter-reset: ulLFO10_1 1;\"><li style=\"color: initial; font-family: initial;hyphenate: false;font-family: Times New Roman;font-size: 12pt;hyphenate: false;font-family: Calibri;textdecor: single;textdecor: underline;textdecor-style: solid;margin-top: 0.0694in !important;margin-bottom: 0.0694in !important;line-height: 1 !important;text-align: justify !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P214\"><span>Subtraktion:<\/span><\/p><\/li><\/ul><p class=\"P215\"><span>Her tr\u00e6kkes henholdsvis de to x-v\u00e6rdier og de to y-v\u00e6rdier fra hinanden. Alts\u00e5 s\u00e5ledes:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P216\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T217\">&#x200b;&#x200b;&nbsp;-<\/span><span class=\"T218\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T219\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:msub><mml:mrow><mml:mo>-<\/mml:mo><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><ul class=\"LFO10\" style=\"counter-reset: ulLFO10_1 3;\"><li style=\"color: initial; font-family: initial;hyphenate: false;font-family: Times New Roman;font-size: 12pt;hyphenate: false;hyphenate: false;font-family: Times New Roman;font-size: 12pt;hyphenate: false;margin-top: 0.0694in !important;margin-bottom: 0.0694in !important;line-height: 1 !important;text-align: justify !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P220\"><span class=\"T221\">Multiplikation:<\/span><\/p><\/li><\/ul><p class=\"P222\"><span>Jeg kan s\u00e5<\/span>&#x200b;&#x200b;&nbsp;<span>som<\/span>&#x200b;&#x200b;&nbsp;<span>ogs\u00e5 v\u00e6lge at gange en vektors koordinater med et tal.<\/span>&#x200b;&#x200b;&nbsp;<span>Vil jeg g\u00f8re dette skal jeg, som det ses herunder, gange tallet med b\u00e5de x- og y-v\u00e6rdien.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P223\"><span class=\"T224\">t *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T225\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mi mathvariant=\"normal\">t<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">t<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P226\"><span class=\"T227\">Jeg vil s\u00e5 bevise s\u00e6tningen&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T228\">&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T229\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:msub><mml:mrow><mml:mo>+<\/mml:mo><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T230\">&#x200b;&#x200b;&nbsp;fra f\u00f8r.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P231\"><span class=\"T232\">Her starter jeg med at benytte definitionen p\u00e5 vektorers koordinater som hed(<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T233\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;= a<\/span><span class=\"T234\">1<\/span><span class=\"T235\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">I<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T236\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;+ a<\/span><span class=\"T237\">2<\/span><span class=\"T238\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">J<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T239\">), s\u00e5ledes at j<\/span><span class=\"T240\">eg kan omskrive ovenst\u00e5ende s\u00e6tning. Derved f\u00e5r jeg:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P241\"><span class=\"T242\">(<\/span><span class=\"T243\">a<\/span><span class=\"T244\">1<\/span><span class=\"T245\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">I<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T246\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;+ a<\/span><span class=\"T247\">2<\/span><span class=\"T248\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">J<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T249\">) + (b<\/span><span class=\"T250\">1<\/span><span class=\"T251\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">I<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T252\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;+ b<\/span><span class=\"T253\">2<\/span><span class=\"T254\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">J<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T255\">)<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P256\"><span class=\"T257\">Ved s\u00e5 at benytte den associative lov fra tidligere som sagde at&#x200b;&#x200b;&nbsp;<\/span><span class=\"T258\">(<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T259\">&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T260\">) +&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T261\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T262\">&#x200b;&#x200b;&nbsp;+ (<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T263\">&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T264\">)<\/span><span class=\"T265\">, samt den kommutative lov som sagde af&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T266\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T267\">, s\u00e5 kan jeg omskrive ovenst\u00e5ende til:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P268\"><span class=\"T269\">(a<\/span><span class=\"T270\">1<\/span><span class=\"T271\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">I<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T272\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;+ b<\/span><span class=\"T273\">1<\/span><span class=\"T274\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">I<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T275\">) + (a<\/span><span class=\"T276\">2<\/span><span class=\"T277\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">J<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T278\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;+ b<\/span><span class=\"T279\">2<\/span><span class=\"T280\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">J<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T281\">)<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P282\"><span class=\"T283\">S\u00e5 kan jeg s\u00e5 benytte den distributive lov<\/span><span class=\"T284\">(<\/span><span class=\"T285\">t *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T286\">&#x200b;&#x200b;&nbsp;+ t *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T287\">&#x200b;&#x200b;&nbsp;= t * (<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T288\">&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T289\">))<\/span><span class=\"T290\">, for n\u00e5r jeg kender denne,<\/span><span class=\"T291\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T292\">ved jeg at ovenst\u00e5ende er det samme som:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P293\"><span class=\"T294\">(a<\/span><span class=\"T295\">1<\/span><span class=\"T296\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T297\">+ b<\/span><span class=\"T298\">1<\/span><span class=\"T299\">) *<\/span><span class=\"T300\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">I<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T301\">&#x200b;&#x200b;&nbsp;+ (a<\/span><span class=\"T302\">2<\/span><span class=\"T303\">&#x200b;&#x200b;&nbsp;+ b<\/span><span class=\"T304\">2<\/span><span class=\"T305\">)&#x200b;&#x200b;&nbsp;<\/span><span class=\"T306\">*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">J<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><\/p><p class=\"P307\"><span>Dette bliver s\u00e5, ved at benytte definitionen af en vektors koordinater bagl\u00e6ns, til:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P308\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T309\">&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T310\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:msub><mml:mrow><mml:mo>+<\/mml:mo><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p class=\"P311\"><span>Og s\u00e6tningen er bevist.<\/span><\/p><p class=\"P312\">&nbsp;<\/p><h2 class=\"P313\"><span id=\"_Toc433991960\"\/><span>Stedvektorer<\/span><\/h2><p class=\"P314\"><span>En vektor indtegnet i et koordinatsystem, som har sit begyndelsespunkt i punktet (0,0), kaldes for en stedvektor. Alts\u00e5 kan man sige at koordinaterne til en stedvektor svarer til koordinaterne til endepunktet p\u00e5 en s\u00e5dan vektor.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P315\"><span class=\"T316\">En stedvektor skrives som&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">O<\/mml:mi><mml:mi mathvariant=\"normal\">P<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T317\">, hvor O st\u00e5r for origo, som er et navn for begyndelsesp<\/span><span class=\"T318\">un<\/span><span class=\"T319\">ktet i et koordinatsystem, mens&#x200b;&#x200b;&nbsp;<\/span><span class=\"T320\">P s\u00e5 st\u00e5r for punktet P som er endepunktet. En stedvektors koordinater skrives s\u00e5 som&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">O<\/mml:mi><mml:mi mathvariant=\"normal\">P<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T321\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">p<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">p<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T322\">.<\/span><span class=\"T323\">&#x200b;&#x200b;&nbsp;Dvs. at jeg lader en vektor der hedder&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T324\">, s\u00e5 har jeg punktet A, og s\u00e5 kommer stedvektoren til at hedde&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">O<\/mml:mi><mml:mi mathvariant=\"normal\">A<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T325\">, og koordinaterne til denne bliver s\u00e5<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T326\">. Alts\u00e5 her et punkt og punktets stedvektor samme koordinater. &nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P327\"><span class=\"T328\">Skal jeg s\u00e5 beregne koordinaterne til en&#x200b;&#x200b;&nbsp;<\/span><span class=\"T329\">hvilken som helst vektor, fx vektor&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T330\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T331\">, benyttes formlen&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T332\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T333\">.&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P334\"><span class=\"T335\">Et eksempel p\u00e5 brugen af denne kunne v\u00e6re at jeg havde&#x200b;&#x200b;&nbsp;<\/span><span class=\"T336\">de to punkter<\/span><span class=\"T337\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T338\">A = (7,3) og B = (2,4)<\/span><span class=\"T339\">. Jeg finder s\u00e5 denne vektors koordinater s\u00e5ledes:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P340\"><span class=\"T341\">\u00a0<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T342\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>2<\/mml:mn><mml:mo>-<\/mml:mo><mml:mn>7<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T343\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T344\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P345\">&nbsp;<\/p><h2 class=\"P346\"><span id=\"_Toc433991961\"\/><span>L\u00e6ngden af vektorer<\/span><\/h2><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P347\"><span class=\"T348\">Som tidligere n\u00e6vnt betegnes l\u00e6ngden af en vektor som&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T349\">.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T350\">Vil jeg s\u00e5 finde&#x200b;&#x200b;&nbsp;<\/span><span class=\"T351\">denne l\u00e6ngde, s\u00e5<\/span><span class=\"T352\">&#x200b;&#x200b;&nbsp;er der to muligheder.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P353\"><span class=\"T354\">Den f\u00f8rste mulighed er hvis vektorens koordinater kendes. Er dette tilf\u00e6ldet benyttes formlen&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T355\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:math><\/span><\/span><span class=\"T356\">, som er udle<\/span><span class=\"T357\">dt af Pythagoras gamle l\u00e6res\u00e6tning a<\/span><span class=\"T358\">2<\/span><span class=\"T359\">&#x200b;&#x200b;&nbsp;+b<\/span><span class=\"T360\">2<\/span><span class=\"T361\">&#x200b;&#x200b;&nbsp;= c<\/span><span class=\"T362\">2<\/span><span class=\"T363\">.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P364\"><span class=\"T365\">Et eksempel p\u00e5 brugen af denne kunne v\u00e6re at jeg har en vektor(<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T366\">), med koordinaterne<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mtable><mml:mtr><mml:mtd><mml:mn>3<\/mml:mn><\/mml:mtd><\/mml:mtr><mml:mtr><mml:mtd><mml:mn>4<\/mml:mn><\/mml:mtd><\/mml:mtr><\/mml:mtable><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T367\">. Jeg kan s\u00e5 beregne l\u00e6ngden af denne ved at inds\u00e6tte tallene i f\u00f8rn\u00e6vnte formel:<\/span><span class=\"T368\">&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P369\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T370\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msqrt><mml:msup><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:math><\/span><\/span><span class=\"T371\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msqrt><mml:mn>25<\/mml:mn><\/mml:msqrt><\/mml:math><\/span><\/span><span class=\"T372\">&#x200b;&#x200b;&nbsp;= 5<\/span><\/p><p class=\"P373\"><span class=\"T374\">Den anden mulighed er at begyndelsespunkt og endepunkt til vektoren kendes.<\/span><span class=\"T375\">&#x200b;&#x200b;&nbsp;Disse pun<\/span><span class=\"T376\">kter kunne man kalde for A og B, og de ville s\u00e5 have koordinaterne&#x200b;&#x200b;&nbsp;<\/span><span class=\"T377\">A = (a<\/span><span class=\"T378\">1<\/span><span class=\"T379\">, a<\/span><span class=\"T380\">2<\/span><span class=\"T381\">) &nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;B = (b<\/span><span class=\"T382\">2<\/span><span class=\"T383\">, b<\/span><span class=\"T384\">2<\/span><span class=\"T385\">)<\/span><span class=\"T386\">.<\/span><span class=\"T387\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T388\">I<\/span><span class=\"T389\">&#x200b;&#x200b;&nbsp;dette tilf\u00e6lde kan jeg s\u00e5 benytte afstandsformlen<\/span><span class=\"T390\">&#x200b;&#x200b;&nbsp;der<\/span><span class=\"T391\">&#x200b;&#x200b;&nbsp;lyder<\/span><span class=\"T392\">&#x200b;&#x200b;&nbsp;s\u00e5ledes.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P393\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T394\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:math><\/span><\/span><\/p><p class=\"P395\"><span>Det man s\u00e5 reelt g\u00f8r med denne formel er, at man finder<\/span>&#x200b;&#x200b;&nbsp;<span>afstanden mellem de to punkter A og B.<\/span>&#x200b;&#x200b;&nbsp;<\/p><p class=\"P396\"><span>Et eksempel p\u00e5 brugen af den kunne v\u00e6re at jeg har to punkter med dertilh\u00f8rende koordinater:<\/span><\/p><p class=\"P397\"><span>A = <\/span>&#x200b;&#x200b;&nbsp;<span>(2,4) og B =<\/span>&#x200b;&#x200b;&nbsp;<span>(7,1)<\/span><\/p><p class=\"P398\"><span>Jeg finder<\/span>&#x200b;&#x200b;&nbsp;<span>nu<\/span>&#x200b;&#x200b;&nbsp;<span>l\u00e6ngden af vektor AB<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P399\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T400\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>7<\/mml:mn><mml:mo>-<\/mml:mo><mml:mn>2<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>-<\/mml:mo><mml:mn>4<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:math><\/span><\/span><span class=\"T401\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msqrt><mml:mn>34<\/mml:mn><\/mml:msqrt><\/mml:math><\/span><\/span><span class=\"T402\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T403\">=<\/span><span class=\"T404\">&#x200b;&#x200b;&nbsp;5,83<\/span><\/p><p class=\"P405\">&nbsp;<\/p><h2 class=\"P406\"><span id=\"_Toc433991962\"\/><span>Skalarprodukt(prikprodukt)<\/span><\/h2><p class=\"P407\"><span>Skalarproduktet ogs\u00e5 kaldet prikproduktet er et begreb der bruges i forbindelse med at finde vinklen mellem to vektorer.<\/span>&#x200b;&#x200b;&nbsp;<span>For n\u00e5r skalarproduktet samt l\u00e6ngden af vektorerne<\/span>&#x200b;&#x200b;&nbsp;<span>kendes,<\/span>&#x200b;&#x200b;&nbsp;<span>kan vinklen mellem dem beregnes.<\/span>&#x200b;&#x200b;&nbsp;<\/p><p class=\"P408\"><span>Grunden til at skalarproduktet ogs\u00e5 kaldes for prikproduktet er, at man i princippet ikke kan gange to vektorer med hinanden, og man er derfor n\u00f8dt til <\/span>&#x200b;&#x200b;&nbsp;<span>i stedet at prikke dem med hinanden.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P409\"><span class=\"T410\">Definitionen p\u00e5 skalarproduktet er som f\u00f8lger<\/span><span class=\"T411\">:&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T412\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T413\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T414\">&#x200b;&#x200b;&nbsp;= a<\/span><span class=\"T415\">1<\/span><span class=\"T416\">&#x200b;&#x200b;&nbsp;* b<\/span><span class=\"T417\">1<\/span><span class=\"T418\">&#x200b;&#x200b;&nbsp;+ a<\/span><span class=\"T419\">2<\/span><span class=\"T420\">&#x200b;&#x200b;&nbsp;* b<\/span><span class=\"T421\">2<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P422\"><span class=\"T423\">Et eksempel<\/span><span class=\"T424\">&#x200b;&#x200b;&nbsp;kunne v\u00e6re at jeg har de to vektorer&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T425\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T426\">&#x200b;&#x200b;&nbsp;og &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T427\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T428\">. Disse tal s\u00e6tter jeg s\u00e5 ind i forskriften<\/span><span class=\"T429\">&#x200b;&#x200b;&nbsp;og f\u00e5r f\u00f8lgende:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P430\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T431\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T432\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T433\">&#x200b;&#x200b;&nbsp;= 2 * 4 + 3 * 1 = 11<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P434\"><span class=\"T435\">En ting der i forbindelse med skalarproduktet er vigtigt at vide er, at hvis vinklen mellem de to vektorer er lig 0, s\u00e5 st\u00e5r disse vinkelret p\u00e5 hinanden.<\/span><span class=\"T436\">&#x200b;&#x200b;&nbsp;Dette skrives s\u00e5ledes at hvis&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T437\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u22a5<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T438\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T439\">&#x200b;&#x200b;&nbsp;s\u00e5 er&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T440\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T441\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T442\">&#x200b;&#x200b;&nbsp;= 0.&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P443\"><span>Man kan ogs\u00e5 finde vinklen mellem to vektorer ved at benyttes cosinus til vinkel, i s\u00e5 fald benyttes en af nedenst\u00e5ende to metoder:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P444\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T445\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T446\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T447\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;= &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T448\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T449\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T450\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T451\">* cos(v) &lt;=&gt; cos(v) =&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P452\"><span class=\"T453\">Jeg vil s\u00e5 komme med et eksempel p\u00e5 brugen af den sidste.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T454\">Her arbejder jeg ud fra, at jeg&#x200b;&#x200b;&nbsp;<\/span><span class=\"T455\">har&#x200b;&#x200b;&nbsp;<\/span><span class=\"T456\">f\u00f8lgende to vektorer:&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T457\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T458\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T459\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>7<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P460\"><span class=\"T461\">Disse s\u00e6tter jeg s\u00e5 ind i&#x200b;&#x200b;&nbsp;<\/span><span class=\"T462\">cos(v) =&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T463\">:<\/span><span class=\"T464\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P465\"><span class=\"T466\">cos(v) =&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mn>2<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mn>7<\/mml:mn><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>+<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mn>3<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mn>7<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T467\">&#x200b;&#x200b;&nbsp;=<\/span><\/p><p class=\"P468\"><span>Jeg reducerer og der udregnes:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P469\"><span class=\"T470\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mn>17<\/mml:mn><\/mml:mrow><mml:mrow><mml:msqrt><mml:mn>13<\/mml:mn><\/mml:msqrt><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:msqrt><mml:mn>50<\/mml:mn><\/mml:msqrt><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T471\">&#x200b;&#x200b;&nbsp;= 0,6668<\/span><\/p><p class=\"P472\"><span>Jeg benytter s\u00e5 cosinus, s\u00e5ledes at jeg f\u00e5r resultatet:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P473\"><span class=\"T474\">v = cos<\/span><span class=\"T475\">-1<\/span><span class=\"T476\">(0,6668) =&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u00b1<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T477\">&#x200b;&#x200b;&nbsp;48,18<\/span><\/p><p class=\"P478\"><span>N\u00e5r man s\u00e5 har beregnet en vinkel, kan man alene ud fra tallet, bed\u00f8mme om vinklen er ret eller stump. Her g\u00e6lder det at:<\/span><\/p><ul xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"LFO13\" style=\"counter-reset: ulLFO13_1 0;\"><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;hyphenate: false;hyphenate: false;mleft: 0.5in !important;text-align: justify !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P479\"><span class=\"T480\">Hvis&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi>\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi>\u00a0<\/mml:mi><mml:mo>&gt;<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:math><\/span><\/span><span class=\"T481\">&#x200b;&#x200b;&nbsp;s\u00e5 er v en spids vinkel: 0\u00b0 \u2264 v &lt; 90\u00b0<\/span><\/p><\/li><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;hyphenate: false;hyphenate: false;mleft: 0.5in !important;text-align: justify !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P482\"><span class=\"T483\">Hvis&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi>\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi>\u00a0<\/mml:mi><mml:mo>&lt;<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:math><\/span><\/span><span class=\"T484\">&#x200b;&#x200b;&nbsp;s\u00e5 er v en stump vinkel: 90\u00b0 &lt; v \u2264 180\u00b0<\/span><\/p><\/li><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;hyphenate: false;hyphenate: false;mleft: 0.5in !important;text-align: justify !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P485\"><span class=\"T486\">Hvis&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi>\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi>\u00a0<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>0<\/mml:mn><\/mml:math><\/span><\/span><span class=\"T487\">&#x200b;&#x200b;&nbsp;s\u00e5 er<\/span><span class=\"T488\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T489\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u22a5<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T490\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T491\">: v = 90\u00b0<\/span><\/p><\/li><\/ul><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P492\"><span class=\"T493\">Nu vil jeg s\u00e5 bevise s\u00e6tningen&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T494\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T495\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T496\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;= &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T497\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T498\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T499\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T500\">* cos(v)<\/span><span class=\"T501\">. For at g\u00f8re dette, foruds\u00e6tter jeg at jeg har<\/span><span class=\"T502\">&#x200b;&#x200b;&nbsp;de to vektorer&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T503\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T504\">. For s\u00e5 at finde vinklen mellem disse, foruds\u00e6tter jeg&#x200b;&#x200b;&nbsp;<\/span><span class=\"T505\">ligeledes&#x200b;&#x200b;&nbsp;<\/span><span class=\"T506\">at de har de begge har deres begyndelsespunkt i (0,0). Jeg kan s\u00e5 tilf\u00f8je&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T507\">, og jeg har nu en vilk\u00e5rlig trekant.<\/span><\/p><p class=\"P508\"><span>Hvis jeg tegnede denne trekant, ville jeg kunne se f\u00f8lgende:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P509\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T510\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi mathvariant=\"normal\">C<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T511\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msqrt><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><\/mml:msqrt><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P512\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T513\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi mathvariant=\"normal\">C<\/mml:mi><mml:mi mathvariant=\"normal\">A<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T514\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msqrt><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><\/mml:msqrt><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P515\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T516\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T517\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:math><\/span><\/span><\/p><p class=\"P518\"><span>Som det ses har jeg tre vektorer, som jo s\u00e5 svarer til siderne a, b og c i en trekant. N\u00e5r jeg ved dette kan jeg benytte cosinusrelationen for en vilk\u00e5rlig trekant som siger f\u00f8lgende:<\/span><\/p><p class=\"P519\"><span class=\"T520\">c<\/span><span class=\"T521\">2<\/span><span class=\"T522\">&#x200b;&#x200b;&nbsp;= a<\/span><span class=\"T523\">2<\/span><span class=\"T524\">&#x200b;&#x200b;&nbsp;+ b<\/span><span class=\"T525\">2<\/span><span class=\"T526\">&#x200b;&#x200b;&nbsp;- 2ab*cos(C)<\/span><\/p><p class=\"P527\"><span>I denne kan jeg kan s\u00e5 inds\u00e6tte udtrykkene for l\u00e6ngerne af vektorerne.<\/span>&#x200b;&#x200b;&nbsp;<\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P528\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T529\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>-<\/mml:mo><mml:mn>2<\/mml:mn><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi mathvariant=\"normal\">c<\/mml:mi><mml:mi mathvariant=\"normal\">o<\/mml:mi><mml:mi mathvariant=\"normal\">s<\/mml:mi><mml:mo>\u2061<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi mathvariant=\"normal\">v<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T530\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P531\"><span>Jeg kan s\u00e5 udregen de forskellige led:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P532\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi>\u00a0<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><mml:mo>+<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi>\u00a0<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfenced><mml:mi>\u00a0<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T533\">&#x200b;&#x200b;&nbsp;= &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>-<\/mml:mo><mml:mn>2<\/mml:mn><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi mathvariant=\"normal\">c<\/mml:mi><mml:mi mathvariant=\"normal\">o<\/mml:mi><mml:mi mathvariant=\"normal\">s<\/mml:mi><mml:mo>\u2061<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi mathvariant=\"normal\">v<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><\/p><p class=\"P534\"><span>Jeg ganger de to parenteser med hinanden:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P535\"><span class=\"T536\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>-<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>\u00a0<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T537\">+ &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>-<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math><\/span><\/span><span class=\"T538\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;= &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>-<\/mml:mo><mml:mn>2<\/mml:mn><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi mathvariant=\"normal\">c<\/mml:mi><mml:mi mathvariant=\"normal\">o<\/mml:mi><mml:mi mathvariant=\"normal\">s<\/mml:mi><mml:mo>\u2061<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi mathvariant=\"normal\">v<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><\/p><p class=\"P539\"><span>Jeg reducerer:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P540\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>-<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mn>2<\/mml:mn><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mi>\u00a0<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T541\">&#x200b;&#x200b;&nbsp;+ &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>-<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msub><mml:mrow><mml:mn>2<\/mml:mn><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><\/mml:math><\/span><\/span><span class=\"T542\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;= &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>-<\/mml:mo><mml:mn>2<\/mml:mn><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi mathvariant=\"normal\">c<\/mml:mi><mml:mi mathvariant=\"normal\">o<\/mml:mi><mml:mi mathvariant=\"normal\">s<\/mml:mi><mml:mo>\u2061<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi mathvariant=\"normal\">v<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P543\"><span class=\"T544\">Jeg kan nu tillade mig at fjerne&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><mml:mo>+<\/mml:mo><mml:msubsup><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msubsup><\/mml:math><\/span><\/span><span class=\"T545\">&#x200b;&#x200b;&nbsp;p\u00e5 begge sider:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P546\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>-<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mn>2<\/mml:mn><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mn>2<\/mml:mn><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math><\/span><\/span><span class=\"T547\">&#x200b;&#x200b;&nbsp;= &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>-<\/mml:mo><mml:mn>2<\/mml:mn><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi mathvariant=\"normal\">c<\/mml:mi><mml:mi mathvariant=\"normal\">o<\/mml:mi><mml:mi mathvariant=\"normal\">s<\/mml:mi><mml:mo>\u2061<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi mathvariant=\"normal\">v<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><\/p><p class=\"P548\"><span>Dette kan jeg s\u00e5 yderligere forkorte ved at dividere med -2 p\u00e5 begge sider:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P549\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>+<\/mml:mo><mml:msub><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:math><\/span><\/span><span class=\"T550\">&#x200b;&#x200b;&nbsp;= &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi mathvariant=\"normal\">c<\/mml:mi><mml:mi mathvariant=\"normal\">o<\/mml:mi><mml:mi mathvariant=\"normal\">s<\/mml:mi><mml:mo>\u2061<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi mathvariant=\"normal\">v<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P551\"><span class=\"T552\">Grundet definitionen af skalarproduktet<\/span><span class=\"T553\">(<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T554\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T555\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T556\">&#x200b;&#x200b;&nbsp;= a<\/span><span class=\"T557\">1<\/span><span class=\"T558\">&#x200b;&#x200b;&nbsp;* b<\/span><span class=\"T559\">1<\/span><span class=\"T560\">&#x200b;&#x200b;&nbsp;+ a<\/span><span class=\"T561\">2<\/span><span class=\"T562\">&#x200b;&#x200b;&nbsp;* b<\/span><span class=\"T563\">2<\/span><span class=\"T564\">)<\/span><span class=\"T565\">&#x200b;&#x200b;&nbsp;bliver dette s\u00e5 til:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P566\"><span class=\"T567\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mo>\u2219<\/mml:mo><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T568\">&#x200b;&#x200b;&nbsp;= &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi mathvariant=\"normal\">c<\/mml:mi><mml:mi mathvariant=\"normal\">o<\/mml:mi><mml:mi mathvariant=\"normal\">s<\/mml:mi><mml:mo>\u2061<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi mathvariant=\"normal\">v<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><\/p><p class=\"P569\"><span>Og s\u00e6tningen er nu bevist.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P570\"><span class=\"T571\">Jeg vil dog ogs\u00e5 gerne bevise s\u00e6tningen&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T572\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u22a5<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T573\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T574\">&#x200b;&#x200b;&nbsp;s\u00e5 er&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T575\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T576\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T577\">&#x200b;&#x200b;&nbsp;= 0, og dette kan jeg g\u00f8re ved at benytte ovenst\u00e5ende s\u00e6tning. For n\u00e5r jeg ved at:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P578\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mo>\u2219<\/mml:mo><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T579\">&#x200b;&#x200b;&nbsp;= &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi mathvariant=\"normal\">c<\/mml:mi><mml:mi mathvariant=\"normal\">o<\/mml:mi><mml:mi mathvariant=\"normal\">s<\/mml:mi><mml:mo>\u2061<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi mathvariant=\"normal\">v<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><\/p><p class=\"P580\"><span>S\u00e5 m\u00e5 dette n\u00f8dvendigvis betyde at hvis:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P581\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T582\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T583\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T584\">&#x200b;&#x200b;&nbsp;= 0&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P585\"><span>S\u00e5 bliver<\/span>&#x200b;&#x200b;&nbsp;<\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P586\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi>*<\/mml:mi><mml:mi mathvariant=\"normal\">c<\/mml:mi><mml:mi mathvariant=\"normal\">o<\/mml:mi><mml:mi mathvariant=\"normal\">s<\/mml:mi><mml:mo>\u2061<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi mathvariant=\"normal\">v<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T587\">&#x200b;&#x200b;&nbsp;= 0<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P588\"><span class=\"T589\">Og da b\u00e5de&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi>\u00a0<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T590\">og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T591\">&#x200b;&#x200b;&nbsp;er egentlige vektorer, og dermed er defineret som vektorer med en l\u00e6ngde, kan disse ikke v\u00e6re nul. Dette betyder s\u00e5 at det m\u00e5 cosinus til v der er lig nul:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P592\"><span class=\"T593\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mi mathvariant=\"normal\">c<\/mml:mi><mml:mi mathvariant=\"normal\">o<\/mml:mi><mml:mi mathvariant=\"normal\">s<\/mml:mi><mml:mo>\u2061<\/mml:mo><mml:mo>(<\/mml:mo><mml:mi mathvariant=\"normal\">v<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T594\">&#x200b;&#x200b;&nbsp;= 0<\/span><\/p><p class=\"P595\"><span>Dette betyder s\u00e5 at vinklen m\u00e5 v\u00e6re:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P596\"><span class=\"T597\">V =&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u00b1<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:mn>90<\/mml:mn><mml:mo>\u00b0<\/mml:mo><\/mml:math><\/span><\/span><\/p><p class=\"P598\"><span>Og alts\u00e5 kan det konkluderes at:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P599\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"center\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T600\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"center\"><mml:mo>\u22a5<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T601\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\" display=\"block\" indentalign=\"center\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><\/p><h2 class=\"P602\"><span id=\"_Toc433991963\"\/><span>Regneregler for skalarprodukt<\/span><\/h2><p class=\"P603\"><span>Der findes en r\u00e6kke regneregler for skalarproduktet,<\/span>&#x200b;&#x200b;&nbsp;<span>som jeg ogs\u00e5 lige vil tage med. Disse er kort beskrevet herunder:<\/span><\/p><ul xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"LFO1\" style=\"counter-reset: ulLFO1_1 0;\"><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;text-align: justify !important;vertical-align: middle !important;margin-bottom: 0in !important;line-height: 1 !important;mleft: 0.375in !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P604\"><span class=\"T605\">Kommutative lov:<\/span><span class=\"T606\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T607\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T608\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T609\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T610\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T611\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><\/p><\/li><\/ul><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P612\"><span class=\"T613\">Her ses at akkurat som at det&#x200b;&#x200b;&nbsp;<\/span><span class=\"T614\">ingen betydning om man f\u00f8rst prikker&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T615\">&#x200b;&#x200b;&nbsp;med &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T616\">, eller om man g\u00f8r det omvendt.<\/span><span class=\"T617\">&#x200b;&#x200b;&nbsp;Dette sker da definitionen p\u00e5 skalarproduktet var&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T618\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T619\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T620\">&#x200b;&#x200b;&nbsp;= a<\/span><span class=\"T621\">1<\/span><span class=\"T622\">&#x200b;&#x200b;&nbsp;* b<\/span><span class=\"T623\">1<\/span><span class=\"T624\">&#x200b;&#x200b;&nbsp;+ a<\/span><span class=\"T625\">2<\/span><span class=\"T626\">&#x200b;&#x200b;&nbsp;* b<\/span><span class=\"T627\">2<\/span><span class=\"T628\">. Og da koordinaterne til vektorerne jo er almindelige tal(fx&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T629\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T630\">&#x200b;&#x200b;&nbsp;og<\/span><span class=\"T631\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T632\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T633\">), s\u00e5 er r\u00e6kkef\u00f8lgen ligegyldig.&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P634\">&nbsp;<\/p><ul class=\"LFO2\" style=\"counter-reset: ulLFO2_1 1;\"><li style=\"color: initial; font-family: initial;hyphenate: false;font-family: Calibri;font-weight: bold;font-size: 12pt;text-align: justify !important;vertical-align: middle !important;margin-bottom: 0in !important;line-height: 1 !important;mleft: 0.375in !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P635\"><span>Distributive lov:<\/span>&#x200b;&#x200b;&nbsp;<\/p><\/li><\/ul><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P636\"><span class=\"T637\">Hvis jeg g\u00e5r ud fra at jeg har tre tilf\u00e6ldige vektorer,&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T638\">,<\/span><span class=\"T639\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T640\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T641\">, s\u00e5 g\u00e6lder det at:<\/span><\/p><ul xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"LFO3\" style=\"counter-reset: ulLFO3_1 1;\"><li style=\"color: initial; font-family: initial;\" class=\"removeBullet\"><ul style=\"counter-reset: ulLFO3_2 0;\"><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;text-align: justify !important;vertical-align: middle !important;margin-bottom: 0in !important;line-height: 1 !important;mleft: 0.75in !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P642\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T643\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T644\">&#x200b;&#x200b;&nbsp;(<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T645\">&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>c<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T646\">) =&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T647\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T648\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T649\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T650\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T651\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">c<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><\/p><\/li><\/ul><\/li><\/ul><ul xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"LFO3\" style=\"counter-reset: ulLFO3_1 1;\"><li style=\"color: initial; font-family: initial;\" class=\"removeBullet\"><ul style=\"counter-reset: ulLFO3_2 1;\"><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;text-align: justify !important;vertical-align: middle !important;margin-bottom: 0in !important;line-height: 1 !important;mleft: 0.75in !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P652\"><span class=\"T653\">t * (<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T654\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T655\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T656\">) = (t*<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T657\">)&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T658\">&#x200b;&#x200b;&nbsp;= (t*<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T659\">)&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T660\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><\/p><\/li><\/ul><\/li><\/ul><p class=\"P661\"><span>Alts\u00e5 det er lige meget hvor parentesen placeres.<\/span>&#x200b;&#x200b;&nbsp;<\/p><p class=\"P662\">&nbsp;<\/p><ul xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"LFO4\" style=\"counter-reset: ulLFO4_1 2;\"><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;text-align: justify !important;vertical-align: middle !important;margin-bottom: 0in !important;line-height: 1 !important;mleft: 0.375in !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P663\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T664\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T665\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T666\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msup><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><\/p><\/li><\/ul><p class=\"P667\"><span>Alts\u00e5 vektor a prik vektor a er lig l\u00e6ngden af vektor a<\/span><\/p><p class=\"P668\"><span>\u00a0<\/span><\/p><ul class=\"LFO5\" style=\"counter-reset: ulLFO5_1 3;\"><li style=\"color: initial; font-family: initial;hyphenate: false;font-family: Calibri;font-weight: bold;font-size: 12pt;text-align: justify !important;vertical-align: middle !important;margin-bottom: 0in !important;line-height: 1 !important;mleft: 0.375in !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P669\"><span>Vinkelrette vektorer:<\/span><\/p><\/li><\/ul><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P670\"><span class=\"T671\">Hvis&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T672\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u22a5<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T673\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T674\">&#x200b;&#x200b;&nbsp;s\u00e5 er&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T675\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T676\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T677\">&#x200b;&#x200b;&nbsp;= 0<\/span><span class=\"T678\">.&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P679\"><span class=\"T680\">Dette betyder at hvis&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T681\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T682\">er vinkelret p\u00e5<\/span><span class=\"T683\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T684\">, s\u00e5 er skalarproduktet, alts\u00e5 de to vektorer prikket med hinanden, lig nul.<\/span><\/p><p class=\"P685\"><span>\u00a0<\/span><\/p><ul class=\"LFO6\" style=\"counter-reset: ulLFO6_1 4;\"><li style=\"color: initial; font-family: initial;hyphenate: false;font-family: Calibri;font-weight: bold;font-size: 12pt;text-align: justify !important;vertical-align: middle !important;margin-bottom: 0in !important;line-height: 1 !important;mleft: 0.375in !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P686\"><span>Parallelle vektorer:<\/span><\/p><\/li><\/ul><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P687\"><span class=\"T688\">Hvis at&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T689\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T690\">&#x200b;&#x200b;&nbsp;er parallelle vektorer(d<\/span><span class=\"T691\">vs.&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T692\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2225<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T693\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T694\">), s\u00e5 kan<\/span><span class=\"T695\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T696\">&#x200b;&#x200b;&nbsp;skrives som:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P697\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T698\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T699\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><\/p><p class=\"P700\">&nbsp;<\/p><h2 class=\"P701\"><span id=\"_Toc433991964\"\/><span>Tv\u00e6rvektor<\/span><\/h2><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P702\"><span class=\"T703\">Hvis vi siger at vi har en vektor der hedder&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T704\">, s\u00e5 finder man tv\u00e6rvektoren til denne vektor ved at dreje vektoren 90\u00b0 i positiv oml\u00f8bsretning. Positiv oml\u00f8bsretning vil s\u00e5&#x200b;&#x200b;&nbsp;<\/span><span class=\"T705\">sige at man drejer den mod uret, s\u00e5ledes at vinklen mellem den oprindelige vektor og den drejede vektor er positiv.&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P706\"><span class=\"T707\">En tv\u00e6rvektor skrives som&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:math><\/span><\/span><span class=\"T708\">. Alts\u00e5&#x200b;&#x200b;&nbsp;<\/span><span class=\"T709\">er tv\u00e6rvektor<\/span><span class=\"T710\">&#x200b;&#x200b;&nbsp;til&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T711\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:math><\/span><\/span><span class=\"T712\">.&#x200b;&#x200b;&nbsp;<\/span><span class=\"T713\">Dette udtales vektor a-hat.&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P714\"><span>Og ud fra dette kan man s\u00e5 konstatere nogle ting. Nemlig<\/span>&#x200b;&#x200b;&nbsp;<span>at:<\/span>&#x200b;&#x200b;&nbsp;<\/p><ul xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"LFO7\" style=\"counter-reset: ulLFO7_1 0;\"><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;text-align: justify !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P715\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T716\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T717\">. Hvilket vil sige at<\/span><span class=\"T718\">&#x200b;&#x200b;&nbsp;l\u00e6ngden af tv\u00e6rkvektor a er lig l\u00e6ngden af vektor a<\/span><span class=\"T719\">.<\/span><\/p><\/li><\/ul><ul xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"LFO8\" style=\"counter-reset: ulLFO8_1 1;\"><li style=\"color: initial; font-family: initial;hyphenate: false;hyphenate: false;text-align: justify !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P720\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T721\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T722\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:math><\/span><\/span><span class=\"T723\">&#x200b;&#x200b;&nbsp;= 0. A<\/span><span class=\"T724\">lt<\/span><span class=\"T725\">s\u00e5&#x200b;&#x200b;&nbsp;<\/span><span class=\"T726\">at&#x200b;&#x200b;&nbsp;<\/span><span class=\"T727\">vektor a prik tv\u00e6rvektor a er lig nul<\/span><span class=\"T728\">.<\/span><\/p><\/li><\/ul><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P729\"><span class=\"T730\">Det man s\u00e5 i praksis g\u00f8r, for at g\u00e5 fra&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T731\">&#x200b;&#x200b;&nbsp;til&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:math><\/span><\/span><span class=\"T732\">&#x200b;&#x200b;&nbsp;er, at man bytter om p\u00e5 x- og y-koordinaten, samt s\u00e6tter et minus foran x-koordinaten. Dermed kan man konkludere at reglen hedder sig, at&#x200b;&#x200b;&nbsp;<\/span><span class=\"T733\">h<\/span><span class=\"T734\">vis&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T735\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T736\">&#x200b;&#x200b;&nbsp;s\u00e5 er&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:math><\/span><\/span><span class=\"T737\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mo>-<\/mml:mo><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T738\">.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P739\"><span class=\"T740\">Et eksempel p\u00e5 dette kunne v\u00e6re, at man har en&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T741\">&#x200b;&#x200b;&nbsp;med koordinaterne&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T742\">. I s\u00e5 fald vil&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:math><\/span><\/span><span class=\"T743\">&#x200b;&#x200b;&nbsp;s\u00e5 blive&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>4<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>8<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T744\">.<\/span><\/p><p class=\"P745\"><span>Der er s\u00e5 ogs\u00e5 nogle egenskaber som man g\u00f8r kende i forhold til en vektor og et tal.<\/span><\/p><ul xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"LFO11\" style=\"counter-reset: ulLFO11_1 0;\"><li style=\"color: initial; font-family: initial;hyphenate: false;font-family: Times New Roman;font-size: 12pt;hyphenate: false;hyphenate: false;font-family: Times New Roman;font-size: 12pt;hyphenate: false;margin-top: 0.0694in !important;margin-bottom: 0.0694in !important;line-height: 1 !important;text-align: justify !important;margin-top: 0in !important;margin-bottom: 0.1388in !important;line-height: 1.15 !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P746\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mover accent=\"true\"><mml:mrow><mml:mi>t<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:math><\/span><\/span><span class=\"T747\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mi>t<\/mml:mi><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:math><\/span><\/span><\/p><\/li><li style=\"color: initial; font-family: initial;hyphenate: false;font-family: Times New Roman;font-size: 12pt;hyphenate: false;hyphenate: false;font-family: Times New Roman;font-size: 12pt;hyphenate: false;margin-top: 0.0694in !important;margin-bottom: 0.0694in !important;line-height: 1 !important;text-align: justify !important;margin-top: 0in !important;margin-bottom: 0.1388in !important;line-height: 1.15 !important;\"><p style=\"display: table-cell;margin-left:0 !important;text-indent: 0 !important;\" class=\"P748\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mo>+<\/mml:mo><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><mml:mi>\u00a0<\/mml:mi><\/mml:math><\/span><\/span><span class=\"T749\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:math><\/span><\/span><span class=\"T750\">&#x200b;&#x200b;&nbsp;+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><\/mml:math><\/span><\/span><\/p><\/li><\/ul><p class=\"P751\">&nbsp;<\/p><h2 class=\"P752\"><span id=\"_Toc433991965\"\/><span>Vektorprojektioner<\/span><\/h2><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P753\"><span class=\"T754\">Dette handler om at man projicerer en vektor ned p\u00e5 en anden vektor. Dvs. at hvis man har en&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T755\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T756\">og en&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T757\">, s\u00e5 &#x200b;&#x200b;&nbsp;er v<\/span><span class=\"T758\">ektorprojektion<\/span><span class=\"T759\">&#x200b;&#x200b;&nbsp;d<\/span><span class=\"T760\">en vektor der fremkommer n\u00e5r man&#x200b;&#x200b;&nbsp;<\/span><span class=\"T761\">tager endepunktet ved&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T762\">&#x200b;&#x200b;&nbsp;og nedf\u00e6lder den vinkelret p\u00e5<\/span><span class=\"T763\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T764\">.<\/span><span class=\"T765\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T766\">Sk\u00e6ringspunktet man s\u00e5 rammer kaldes for P. Og den vektor man f\u00e5r kaldes for&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">O<\/mml:mi><mml:mi mathvariant=\"normal\">P<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T767\">, da den begynder i punktet O(<\/span><span class=\"T768\">punktet hvor b\u00e5de&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T769\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T770\">&#x200b;&#x200b;&nbsp;begynder), og slutter i punktet P.<\/span><span class=\"T771\">&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P772\"><span class=\"T773\">Dog beh\u00f8ver man ikke tegne vektorerne for at finde koordinaterne til projektionen. Man ka<\/span><span class=\"T774\">n i stedet blot benytte formlen<\/span><span class=\"T775\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T776\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T777\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T778\">.<\/span><\/p><p class=\"P779\"><span>Herunder vil jeg komme med et eksempel p\u00e5 brugen af denne.<\/span><\/p><p class=\"P780\"><span>Jeg har de to vektorer:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P781\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T782\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T783\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T784\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T785\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P786\"><span>Jeg s\u00e6tter tallene ind i forskriften:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P787\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi mathvariant=\"normal\">b<\/mml:mi><\/mml:mrow><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T788\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mn>4<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:mfenced><mml:mo>+<\/mml:mo><mml:mn>1<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:msqrt><mml:msup><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T789\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><\/mml:math><\/span><\/span><\/p><p class=\"P790\"><span>Jeg reduserer og udregner:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P791\"><span class=\"T792\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>17<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T793\">&#x200b;&#x200b;&nbsp;*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T794\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mfrac><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>17<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>17<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P795\"><span class=\"T796\">\u00a0<\/span><span class=\"T797\">Og dermed er&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mfrac><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>17<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><mml:mrow><mml:mfrac><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>17<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T798\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T799\">s\u00e5 projektionen af vektor b p\u00e5 vektor a<\/span><span class=\"T800\">.<\/span><\/p><p class=\"P801\">&nbsp;<\/p><h2 class=\"P802\"><span id=\"_Toc433991966\"\/><span>Areal af parallelogram<\/span><\/h2><p class=\"P803\"><span>Ved hj\u00e6lp af tv\u00e6rvektoren og vektorprojektion kan man bestemme arealet af parallelogrammet udsp\u00e6ndt af to vektorer.<\/span>&#x200b;&#x200b;&nbsp;<\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P804\"><span class=\"T805\">Hvis man antager at man har&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T806\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T807\">, s\u00e5 er arealet af parallelogrammet bestemt ved&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mover accent=\"true\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mo>^<\/mml:mo><\/mml:mover><mml:mi>\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi>\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi>\u00a0<\/mml:mi><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T808\">.&#x200b;&#x200b;&nbsp;<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P809\"><span class=\"T810\">Et eksempel p\u00e5 brugen af dette kunne v\u00e6re, at jeg har de to vektorer&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T811\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T812\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T813\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>9<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T814\">. Disse tal&#x200b;&#x200b;&nbsp;<\/span><span class=\"T815\">kan jeg s\u00e5 s\u00e6tte ind i den f\u00f8rn\u00e6vnte s\u00e6tning s\u00e5ledes:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P816\"><span class=\"T817\">\u00a0A(Arealet) =&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>9<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T818\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T819\">=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:mn>4<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mn>9<\/mml:mn><mml:mo>+<\/mml:mo><mml:mn>3<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T820\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T821\">= 48<\/span><\/p><p class=\"P822\"><span>Alts\u00e5 bliver arealet 48.<\/span><\/p><p class=\"P823\">&nbsp;<\/p><h1 class=\"P824\"><span id=\"_Toc433991967\"\/><span>Opgaver<\/span><\/h1><h2 class=\"P825\"><span id=\"_Toc433991968\"\/><span>Opgave 1<\/span><\/h2><p class=\"P826\"><span>I denne opgave skal jeg komme med nogle beviser, og dette er noget som jeg har gjort<\/span>&#x200b;&#x200b;&nbsp;<span>l\u00f8bende gennem opgaven.<\/span><\/p><p class=\"P827\">&nbsp;<\/p><p class=\"P828\">&nbsp;<\/p><h2 class=\"P829\"><span id=\"_Toc433991969\"\/><span>Opgave 2<\/span><\/h2><p class=\"P830\"><span class=\"T831\">En firkant ABCD er bestemt af punkterne &nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span class=\"T832\">A<\/span><span class=\"T833\">(<\/span><span class=\"T834\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T835\">\u2013<\/span><span class=\"T836\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T837\">3,\u2013<\/span><span class=\"T838\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T839\">9), &nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span class=\"T840\">B<\/span><span class=\"T841\">(<\/span><span class=\"T842\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T843\">5, 4), &nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span class=\"T844\">C<\/span><span class=\"T845\">(<\/span><span class=\"T846\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T847\">\u2013<\/span><span class=\"T848\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T849\">3,<\/span><span class=\"T850\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T851\">9) &#x200b;&#x200b;&nbsp;og &#x200b;&#x200b;&nbsp;<\/span><span class=\"T852\">D<\/span><span class=\"T853\">(<\/span><span class=\"T854\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T855\">\u2013<\/span><span class=\"T856\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T857\">6, 4).<\/span><\/p><p class=\"P858\"><span class=\"T859\">Beregn vinklerne i firkant ABCD ved hj\u00e6lp af s\u00e6tningen&#x200b;&#x200b;&nbsp;<\/span><span class=\"T860\"><span><img src=\"data:image\/ObjectReplacements\/Object 346;base64,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\" style=\"z-index: 100;width: 1.27083in;height: 0.33333in;width: 1.27083in;display: inline;\" title=\"\" alt=\"\"\/><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P861\"><span class=\"T862\">Jeg ved at:&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T863\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T864\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T865\">&nbsp;&#x200b;&#x200b;&#x200b;&#x200b;&nbsp;= &#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T866\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mo>\u2219<\/mml:mo><\/mml:math><\/span><\/span><span class=\"T867\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T868\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T869\">* cos(v) &lt;=&gt; cos(v) =&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P870\"><span class=\"T871\">Alts\u00e5 kan s\u00e6tningen cos(v) =&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T872\">&#x200b;&#x200b;&nbsp;benyttes.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P873\"><span class=\"T874\">Ud fra de fire ovenst\u00e5ende punkter kan jeg danne en firkant med de fire vektorer&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T875\">,&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">B<\/mml:mi><mml:mi mathvariant=\"normal\">C<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T876\">,&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">C<\/mml:mi><mml:mi mathvariant=\"normal\">D<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T877\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">D<\/mml:mi><mml:mi mathvariant=\"normal\">A<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T878\">. Det<\/span><span class=\"T879\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T880\">jeg&#x200b;&#x200b;&nbsp;<\/span><span class=\"T881\">s\u00e5&#x200b;&#x200b;&nbsp;<\/span><span class=\"T882\">skal g\u00f8re er at&#x200b;&#x200b;&nbsp;<\/span><span class=\"T883\">finde vinklerne<\/span><span class=\"T884\">&#x200b;&#x200b;&nbsp;til disse. F<\/span><span class=\"T885\">or at g\u00f8re dette benytter jeg formlen&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:msub><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msup><mml:mrow><mml:mi>x<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:mrow><mml:mrow><mml:msub><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>-<\/mml:mo><mml:msub><mml:mrow><mml:mi>y<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T886\">.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P887\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T888\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>5<\/mml:mn><mml:mo>-<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><mml:mo>-<\/mml:mo><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>9<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T889\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P890\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">B<\/mml:mi><mml:mi mathvariant=\"normal\">C<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T891\">&#x200b;&#x200b;&nbsp;=<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:mfenced><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>9<\/mml:mn><mml:mo>-<\/mml:mo><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T892\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P893\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">C<\/mml:mi><mml:mi mathvariant=\"normal\">D<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T894\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>6<\/mml:mn><\/mml:mrow><\/mml:mfenced><mml:mo>-<\/mml:mo><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>4<\/mml:mn><mml:mo>-<\/mml:mo><mml:mn>9<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T895\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P896\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">D<\/mml:mi><mml:mi mathvariant=\"normal\">A<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T897\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><\/mml:mfenced><mml:mo>-<\/mml:mo><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>6<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>9<\/mml:mn><\/mml:mrow><\/mml:mfenced><mml:mo>-<\/mml:mo><mml:mn>4<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T898\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p class=\"P899\">&nbsp;<\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P900\"><span class=\"T901\">Nu finder jeg s\u00e5 vinklen mellem&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T902\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">D<\/mml:mi><mml:mi mathvariant=\"normal\">A<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T903\">.<\/span><span class=\"T904\">&#x200b;&#x200b;&nbsp;<\/span><span class=\"T905\">F<\/span><span class=\"T906\">or at g\u00f8re vender jeg&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">D<\/mml:mi><mml:mi mathvariant=\"normal\">A<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T907\">&#x200b;&#x200b;&nbsp;om ved at s\u00e6tte minus foran denne, alts\u00e5 jeg finder -<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">D<\/mml:mi><mml:mi mathvariant=\"normal\">A<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T908\">.<\/span><span class=\"T909\">&#x200b;&#x200b;&nbsp;Dermed f\u00e5r jeg:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P910\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T911\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P912\"><span class=\"T913\">Til at finde l\u00e6ngden af en vektor benytter jeg formlen&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T914\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:msub><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P915\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>\u2219<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>8<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>13<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>13<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T916\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mn>8<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><mml:mo>)<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>13<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mn>13<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>8<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>13<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>13<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T917\">&#x200b;&#x200b;&nbsp;= 0,7120&#x200b;&#x200b;&nbsp;<\/span><\/p><p class=\"P918\"><span class=\"T919\">Cos<\/span><span class=\"T920\">-1<\/span><span class=\"T921\">(0,7120) =&#x200b;&#x200b;&nbsp;<\/span><span class=\"T922\">44,6\u00b0<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P923\"><span class=\"T924\">V<\/span><span class=\"T925\">inklen mellem&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T926\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">B<\/mml:mi><mml:mi mathvariant=\"normal\">C<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T927\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T928\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P929\"><span class=\"T930\">Jeg vender&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">A<\/mml:mi><mml:mi mathvariant=\"normal\">B<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T931\">&#x200b;&#x200b;&nbsp;om s\u00e5ledes at den bliver&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P932\"><span class=\"T933\">Alts\u00e5&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T934\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P935\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>+<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mn>5<\/mml:mn><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>13<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>5<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T936\">&#x200b;&#x200b;&nbsp;= -0,0069<\/span><\/p><p class=\"P937\"><span class=\"T938\">Cos<\/span><span class=\"T939\">-1<\/span><span class=\"T940\">(-0,0069) =&#x200b;&#x200b;&nbsp;<\/span><span class=\"T941\">90,4\u00b0<\/span><\/p><p class=\"P942\">&nbsp;<\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P943\"><span class=\"T944\">Vinklen mellem&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">B<\/mml:mi><mml:mi mathvariant=\"normal\">C<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T945\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">C<\/mml:mi><mml:mi mathvariant=\"normal\">D<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T946\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T947\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P948\"><span class=\"T949\">Jeg vender&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">B<\/mml:mi><mml:mi mathvariant=\"normal\">C<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T950\">&#x200b;&#x200b;&nbsp;om s\u00e5ledes at den bliver&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P951\"><span class=\"T952\">Alts\u00e5&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>8<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T953\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P954\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mn>8<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><mml:mo>)<\/mml:mo><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>+<\/mml:mo><mml:mfenced separators=\"|\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfenced><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>8<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T955\">&#x200b;&#x200b;&nbsp;= 0,0182<\/span><\/p><p class=\"P956\"><span class=\"T957\">Cos<\/span><span class=\"T958\">-1<\/span><span class=\"T959\">(0,0182) =&#x200b;&#x200b;&nbsp;<\/span><span class=\"T960\">89\u00b0<\/span><\/p><p class=\"P961\">&nbsp;<\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P962\"><span class=\"T963\">Vinklen mellem&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">C<\/mml:mi><mml:mi mathvariant=\"normal\">D<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T964\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">D<\/mml:mi><mml:mi mathvariant=\"normal\">A<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T965\">&#x200b;&#x200b;&nbsp;=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T966\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P967\"><span class=\"T968\">Jeg vender&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"normal\">C<\/mml:mi><mml:mi mathvariant=\"normal\">D<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T969\">&#x200b;&#x200b;&nbsp;om s\u00e5ledes at den bliver&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P970\"><span class=\"T971\">Alts\u00e5&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mn>5<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T972\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced separators=\"|\"><mml:mrow><mml:mfrac linethickness=\"0pt\"><mml:mrow><mml:mn>3<\/mml:mn><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>13<\/mml:mn><\/mml:mrow><\/mml:mfrac><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"P973\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfrac><mml:mrow><mml:mn>3<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mn>3<\/mml:mn><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mo>+<\/mml:mo><mml:mn>5<\/mml:mn><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>13<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>3<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>5<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:mi mathvariant=\"normal\">*<\/mml:mi><mml:mi mathvariant=\"normal\">\u00a0<\/mml:mi><mml:msqrt><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mn>3<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>+<\/mml:mo><mml:msup><mml:mrow><mml:mo>(<\/mml:mo><mml:mo>-<\/mml:mo><mml:mn>13<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:msqrt><\/mml:mrow><\/mml:mfrac><\/mml:math><\/span><\/span><span class=\"T974\">&#x200b;&#x200b;&nbsp;= -0,7198<\/span><\/p><p class=\"P975\"><span class=\"T976\">Cos<\/span><span class=\"T977\">-1<\/span><span class=\"T978\">(-0,7198) =&#x200b;&#x200b;&nbsp;<\/span><span class=\"T979\">136,04\u00b0<\/span><\/p><p class=\"P980\">&nbsp;<\/p><p class=\"P981\"><span>Jeg kan s\u00e5<\/span>&#x200b;&#x200b;&nbsp;<span>evt.<\/span>&#x200b;&#x200b;&nbsp;<span>benytte fx GeoGebra til at se om jeg har regnet rigtigt.<\/span><\/p><p class=\"P982\"><span class=\"T983\"><span><img 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\" style=\"z-index: 100;width: 2.03896in;height: 2.29896in;width: 2.03896in;display: inline;\" title=\"\" alt=\"\"\/><\/span><\/span><\/p><h2 class=\"P984\"><span id=\"_Toc433991970\"\/><span>Opgave 3<\/span><\/h2><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T985\">For vektorerne&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T986\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T987\">&#x200b;&#x200b;&nbsp;g\u00e6lder det at<\/span><span class=\"T988\">&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T989\">&#x200b;&#x200b;&nbsp;= 10,&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T990\">&#x200b;&#x200b;&nbsp;= 5 og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a<\/mml:mi><\/mml:mrow><\/mml:munder><mml:mo>-<\/mml:mo><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T991\">&#x200b;&#x200b;&nbsp;= 13<\/span><span class=\"T992\">.<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span class=\"T993\">Beregn vinklen mellem&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T994\">&#x200b;&#x200b;&nbsp;og&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi mathvariant=\"bold-italic\">b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:math><\/span><\/span><span class=\"T995\">.<\/span><\/p><p class=\"P996\"><span>Her kan jeg benytte Pythagoras:<\/span><\/p><p class=\"Normal\"><span class=\"T997\">c<\/span><span class=\"T998\">2<\/span><span class=\"T999\">&#x200b;&#x200b;&nbsp;= a<\/span><span class=\"T1000\">2<\/span><span class=\"T1001\">&#x200b;&#x200b;&nbsp;+ b<\/span><span class=\"T1002\">2<\/span><span class=\"T1003\">&#x200b;&#x200b;&nbsp;- 2*a*b*cos(v)<\/span><\/p><p class=\"P1004\"><span>Derved bliver det:<\/span><\/p><p xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"Normal\"><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msup><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><mml:mo>-<\/mml:mo><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T1005\">=&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msup><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T1006\">+&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:msup><mml:mrow><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:mrow><mml:mrow><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math><\/span><\/span><span class=\"T1007\">- 2 *&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>a<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T1008\">*&#x200b;&#x200b;&nbsp;<\/span><span><span xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" class=\"doxMathFormula\"><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" xmlns:m=\"http:\/\/schemas.openxmlformats.org\/officeDocument\/2006\/math\"><mml:mfenced open=\"|\" close=\"|\" separators=\"|\"><mml:mrow><mml:munder accentunder=\"false\"><mml:mo>\u2192<\/mml:mo><mml:mrow><mml:mi>b<\/mml:mi><\/mml:mrow><\/mml:munder><\/mml:mrow><\/mml:mfenced><\/mml:math><\/span><\/span><span class=\"T1009\">*cos(v)<\/span><\/p><p class=\"P1010\"><span>Jeg s\u00e6tter s\u00e5 mine tal ind i denne s\u00e6tning og f\u00e5r s\u00e5ledes:<\/span><\/p><p class=\"Normal\"><span class=\"T1011\">13<\/span><span class=\"T1012\">2<\/span><span class=\"T1013\">&#x200b;&#x200b;&nbsp;= 10<\/span><span class=\"T1014\">2<\/span><span class=\"T1015\">&#x200b;&#x200b;&nbsp;+ 5<\/span><span class=\"T1016\">2<\/span><span class=\"T1017\">&#x200b;&#x200b;&nbsp;- 2 * 10 * 5 * cos(v)<\/span><\/p><p class=\"P1018\"><span>Der udregnes:<\/span><\/p><p class=\"P1019\"><span>169 = 100 + 25 - 100 * cos(v)<\/span><\/p><p class=\"P1020\"><span>Der udregnes:<\/span><\/p><p class=\"P1021\"><span>169 = 125 - 100 * cos(v)<\/span><\/p><p class=\"P1022\"><span>Der tr\u00e6kkes 125 fra p\u00e5 begge sider<\/span><\/p><p class=\"P1023\"><span>44 = -100 * cos(v)<\/span><\/p><p class=\"P1024\"><span>Der divideres med -100 p\u00e5 begge sider<\/span><\/p><p class=\"P1025\"><span>-0,44 = cos(v)<\/span><\/p><p class=\"P1026\"><span>cos(v) = -0,44<\/span><\/p><p class=\"P1027\"><span>Jeg benytter s\u00e5 cosinus og f\u00e5r derved vinklen<\/span><\/p><p class=\"Normal\"><span class=\"T1028\">c<\/span><span class=\"T1029\">os<\/span><span class=\"T1030\">-1<\/span><span class=\"T1031\">(-0,44) =&#x200b;&#x200b;&nbsp;<\/span><span class=\"T1032\">116,1\u00b0<\/span><\/p><\/div><\/section><script>(function( $ ) {\n    \/\/footnotes\n    $('.defaultNote').children(\":first\").css('cursor', 'pointer');\n    $('.defaultNote').children(\":first\").click(function(){\n        var gotoID = '#' + $(this).parent().attr('id');\n        $('html, body').animate(\n            {\n              scrollTop: $('a[href=\"' + gotoID + '\"]').offset().top - 60,\n            },\n            250,\n            'linear'\n          )\n    });\n}(jQuery));\n\n\n<\/script><script>\nvar numFormat = \",.\";\n(function( $ ) {\n    \"use strict\";\n    \/\/do some cleaning on load\n    var sortCells = $('td[data-sorting], th[data-sorting]');\n    sortCells.each(function(){\n        var first = $(this).text().trim().charAt(0);\n        if (first 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