{"id":751,"date":"2021-05-04T09:54:50","date_gmt":"2021-05-04T07:54:50","guid":{"rendered":"https:\/\/mxth.dk\/?p=751"},"modified":"2022-09-13T10:53:09","modified_gmt":"2022-09-13T08:53:09","slug":"vektortyper","status":"publish","type":"post","link":"https:\/\/mxth.dk\/?p=751","title":{"rendered":"Vektortyper"},"content":{"rendered":"\n<h5 class=\"kt-adv-heading_429d90-05 wp-block-kadence-advancedheading\" data-kb-block=\"kb-adv-heading_429d90-05\">Parallelle vektorer<\/h5>\n\n\n\n<p class=\" eplus-wrapper\">Hvis to vektorer er parallelle vil de pleje i samme retning, men de kan godt have forskellig l\u00e6ngde. Der vil g\u00e6lde, at<\/p>\n\n\n\n<p class=\"has-text-align-center eplus-wrapper\">$\\small \\vec{a}=k\\cdot\\vec{b}$<\/p>\n\n\n\n<style>\n  table, th, td {\n    border: 1px solid black;\n    border-collapse: collapse;\n  }\n<\/style>\n<table width=\"100%\" border=\"1\">\n  <thead>\n    <tr>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#3CB371\">Gr\u00f8n<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#FFA500\">Gul<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#DC143C\">R\u00f8d<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTPV001\" taget=\"_blank\" rel=\"noopener\">VTPV001<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTPV003\" taget=\"_blank\" rel=\"noopener\">VTPV003<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTPV002 \" taget=\"_blank\" rel=\"noopener\">VTPV002<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n\n\n\n<h5 class=\"kt-adv-heading_70b16f-d6 wp-block-kadence-advancedheading\" data-kb-block=\"kb-adv-heading_70b16f-d6\">Enhedsvektorer<\/h5>\n\n\n\n<p class=\" eplus-wrapper\">En enhedsvektor er en vektor som har l\u00e6ngden 1. Vi kan lave alle vektorer om til enhedsvektorer.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">N\u00e5r en vektorer skal laves om til en enhedsvektor vil der g\u00e6lde, at<\/p>\n\n\n\n<p class=\"has-text-align-center eplus-wrapper\">$\\small\\vec{e_a}=\\frac{\\vec{a}}{|\\vec{a}|}=\\frac{1}{|\\vec{a}|}\\cdot\\vec{a}$<\/p>\n\n\n\n<style>\n  table, th, td {\n    border: 1px solid black;\n    border-collapse: collapse;\n  }\n<\/style>\n<table width=\"100%\" border=\"1\">\n  <thead>\n    <tr>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#3CB371\">Gr\u00f8n<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#FFA500\">Gul<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#DC143C\">R\u00f8d<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTEV002 \" taget=\"_blank\" rel=\"noopener\">VTEV002<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTEV001\" taget=\"_blank\" rel=\"noopener\">VTEV001<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTEV004\" taget=\"_blank\" rel=\"noopener\">VTEV004<\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTEV003\" taget=\"_blank\" rel=\"noopener\">VTEV003<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matbhtx.systime.dk\/?id=224#c1313\" taget=\"_blank\" rel=\"noopener\">matBhtx 5.10<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTEV005\" taget=\"_blank\" rel=\"noopener\">VTEV005<\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n\n\n\n<h5 class=\"kt-adv-heading_938dc3-c1 wp-block-kadence-advancedheading\" data-kb-block=\"kb-adv-heading_938dc3-c1\">Basisvektorer<\/h5>\n\n\n\n<p class=\" eplus-wrapper\">Basisvektorer er typer af enhedsvektorer. Vi arbejder med tre forskellige, en som pejer i hver af koordinatakserne. Basisvektoren som g\u00e5r langs f\u00f8rste-aksen ben\u00e6vnes <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cvec%7Bi%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;vec{i}\" class=\"latex\" \/> og den som g\u00e5r langs anden-aksen ben\u00e6vnes <img decoding=\"async\" src=\"https:\/\/s0.wp.com\/latex.php?latex=%5Cvec%7Bj%7D&#038;bg=ffffff&#038;fg=000&#038;s=0&#038;c=20201002\" alt=\"&#92;vec{j}\" class=\"latex\" \/>. Basisvektorerne har f\u00f8lgende koordinater<\/p>\n\n\n\n<p class=\"has-text-align-center eplus-wrapper\">$\\small\\vec{i}=\\begin{pmatrix}1\\\\0\\end{pmatrix}$ og $\\small\\vec{j}=\\begin{pmatrix}0\\\\1\\end{pmatrix}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Basisvektorerne har begge en l\u00e6ngde p\u00e5 en og er derved en enhedsvektor og er vinkelrette p\u00e5 hinanden.<\/p>\n\n\n\n<style>\n  table, th, td {\n    border: 1px solid black;\n    border-collapse: collapse;\n  }\n<\/style>\n<table width=\"100%\" border=\"1\">\n  <thead>\n    <tr>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#3CB371\">Gr\u00f8n<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#FFA500\">Gul<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#DC143C\">R\u00f8d<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTBV002 \" taget=\"_blank\" rel=\"noopener\">VTBV002<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTBV001\" taget=\"_blank\" rel=\"noopener\">VTBV001<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n\n\n\n<h5 class=\"kt-adv-heading_3caba2-09 wp-block-kadence-advancedheading\" data-kb-block=\"kb-adv-heading_3caba2-09\">Tv\u00e6rvektorer<\/h5>\n\n\n\n<p class=\" eplus-wrapper\">Hvis vi drejer en vektor 90 grader mod uret, s\u00e5ledes at den st\u00e5r vinkelret p\u00e5 den oprindelige vektor, f\u00e5r vi en tv\u00e6rvektor (fordi den st\u00e5r p\u00e5 tv\u00e6rs i forhold til den oprindelige vektor), som ben\u00e6vnes $\\small\\hat{\\vec{a}}$.  F\u00f8lgende vil g\u00e6lde for de to vektorer<\/p>\n\n\n\n<p class=\"has-text-align-center eplus-wrapper\">hvis $\\small\\vec{a}=\\begin{pmatrix}x\\\\y\\end{pmatrix}$ s\u00e5 er $\\small\\widehat{\\vec{a}}=\\begin{pmatrix}-y\\\\x\\end{pmatrix}$<\/p>\n\n\n\n<style>\n  table, th, td {\n    border: 1px solid black;\n    border-collapse: collapse;\n  }\n<\/style>\n<table width=\"100%\" border=\"1\">\n  <thead>\n    <tr>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#3CB371\">Gr\u00f8n<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#FFA500\">Gul<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#DC143C\">R\u00f8d<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTTV001\" taget=\"_blank\" rel=\"noopener\">VTTV001<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTTV002\" taget=\"_blank\" rel=\"noopener\">VTTV002<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTTV005\" taget=\"_blank\" rel=\"noopener\">VTTV005<\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTTV003\" taget=\"_blank\" rel=\"noopener\">VTTV003<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTTV004\" taget=\"_blank\" rel=\"noopener\">VTTV004<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTTV006\" taget=\"_blank\" rel=\"noopener\">VTTV006<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n\n\n\n<h5 class=\"kt-adv-heading_cfc417-aa wp-block-kadence-advancedheading\" data-kb-block=\"kb-adv-heading_cfc417-aa\">Normalvektor<\/h5>\n\n\n\n<p class=\" eplus-wrapper\">En anden vektor som er vinkelret p\u00e5 en anden er normalvektoren. For rette linjer kan vi opstille en vektor, der st\u00e5r vinkelret p\u00e5 linjen.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Da normalvektoren ikke udg\u00e5r fra en bestemt vektor, eller sted p\u00e5 linjen, kan den alts\u00e5 v\u00e6re vilk\u00e5rlig lang og derved har vi at<\/p>\n\n\n\n<p class=\"has-text-align-center eplus-wrapper\">$\\small\\vec{n}=k\\cdot\\widehat{\\vec{r}}=k\\cdot\\begin{pmatrix}-a\\\\1\\end{pmatrix}=\\begin{pmatrix}-k\\cdot a\\\\k\\end{pmatrix}$<\/p>\n\n\n\n<p class=\" eplus-wrapper\">hvor $\\small\\vec{n}$ er normalvektoren, $\\small k$ er en vilk\u00e5rlig konstant og $\\small\\vec{r}$ er retningsvektoren for linjen. <\/p>\n\n\n\n<style>\n  table, th, td {\n    border: 1px solid black;\n    border-collapse: collapse;\n  }\n<\/style>\n<table width=\"100%\" border=\"1\">\n  <thead>\n    <tr>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#3CB371\">Gr\u00f8n<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#FFA500\">Gul<\/th>\n      <th scope=\"col\" width=\"33%\" bgcolor=\"#DC143C\">R\u00f8d<\/th>\n    <\/tr>\n  <\/thead>\n  <tbody>\n    <tr>\n      <td>\n        <p><a href=\"https:\/\/matstxab2opgaver.systime.dk\/?id=220#c2391\" taget=\"_blank\" rel=\"noopener\">matAB2stx 5.02<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/mxth.dk\/?page_id=1769\/#VTNV001\" taget=\"_blank\" rel=\"noopener\">VTNV001<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab2opgaver.systime.dk\/?id=220#c2396\" taget=\"_blank\" rel=\"noopener\">matAB2stx 5.06<\/a><\/p>\n      <\/td>\n    <\/tr>\n    <tr>\n      <td>\n        <p><a href=\"\" taget=\"_blank\" rel=\"noopener\"><\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab2opgaver.systime.dk\/?id=220#c2393\" taget=\"_blank\" rel=\"noopener\">matAB2stx 5.04<\/a><\/p>\n      <\/td>\n      <td>\n        <p><a href=\"https:\/\/matstxab2opgaver.systime.dk\/?id=220#c2395\" taget=\"_blank\" rel=\"noopener\">matAB2stx 5.05<\/a><\/p>\n      <\/td>\n    <\/tr>\n  <\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Vi definerer her forskellige typer af vektorer.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","editor_plus_copied_stylings":"{}","_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[3,9],"tags":[],"class_list":["post-751","post","type-post","status-publish","format-standard","hentry","category-matematik","category-vektorer"],"featured_image_src":null,"author_info":{"display_name":"Henriksen","author_link":"https:\/\/mxth.dk\/?author=1"},"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/751","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=751"}],"version-history":[{"count":18,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/751\/revisions"}],"predecessor-version":[{"id":1881,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/posts\/751\/revisions\/1881"}],"wp:attachment":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=751"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=751"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=751"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}