{"id":3178,"date":"2024-04-18T00:08:48","date_gmt":"2024-04-17T22:08:48","guid":{"rendered":"https:\/\/mxth.dk\/?page_id=3178"},"modified":"2024-05-02T22:50:29","modified_gmt":"2024-05-02T20:50:29","slug":"opgaver-til-vektorer-i-rummet","status":"publish","type":"page","link":"https:\/\/mxth.dk\/?page_id=3178","title":{"rendered":"Opgaver til vektorer i rummet"},"content":{"rendered":"\n<p class=\" eplus-wrapper\">Linjens parameterfremstilling &#8211; punkt p\u00e5 linjen<\/p>\n\n\n\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-id3178_5bee60-df kt-accordion-has-4-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"false\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane3178_335248-9a\" id=\"VRLPPL001\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VRLPPL001<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">En linje har parameterfremstillingen \\(\\begin{pmatrix}x\\\\ y\\\\ z\\end{pmatrix}=\\begin{pmatrix}-3\\\\ 2\\\\ -1\\end{pmatrix}+t\\cdot\\begin{pmatrix}1\\\\ 1\\\\ 1\\end{pmatrix}\\). Ligger punktet P(-2,3,1) p\u00e5 linjen?<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-2 kt-pane3178_d64cdd-ba\" id=\"VRLPPL002\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VRLPPL002<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Lad os sige, vi har punktet P(3, -2, 1) og linjen gennem punktet A(2, 1, 0) og med retningsvektoren \\(\\vec{v}=\\begin{pmatrix}1\\\\ -1\\\\ 2\\end{pmatrix}\\). Ligger P p\u00e5 den n\u00e6vnte linje?<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-3 kt-pane3178_0667fa-33\" id=\"VRLPPL003\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VRLPPL003<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Givet punkterne P(2, 3, 4) og linjen gennem punktet A(1, -1, 2) og B(5, 2, -3). Ligger punktet P p\u00e5 linjen gennem A og B?<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-4 kt-pane3178_ed6f13-f2\" id=\"VRLPPL004\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VRLPPL004<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Du er givet punktet P(4, 2, 3) og to linjer. Den f\u00f8rste linje g\u00e5r gennem punkterne A(1, -1, 2) og B(3, 1, -1), og den anden linje g\u00e5r gennem punkterne C(2, 0, 4) og D(5, 3, 1). Ligger punktet P p\u00e5 en af disse linjer, begge linjer, eller ingen af dem?<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p class=\" eplus-wrapper\">Planets parameterfremstilling<\/p>\n\n\n\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-id3178_27b232-e3 kt-accordion-has-2-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"false\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane3178_95208e-40\" id=\"VRPP001\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VRPP001**<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Du er givet to linjer i rummer.<\/p>\n\n\n\n<p class=\" eplus-wrapper\">\\(\\ell_1:\\begin{pmatrix}x\\\\ y\\\\ z\\end{pmatrix}=\\begin{pmatrix}2,5\\\\ 3,5\\\\ 0\\end{pmatrix}+t\\cdot\\begin{pmatrix}1\\\\ 1\\\\ 2\\end{pmatrix}\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">\\(\\ell_2:\\begin{pmatrix}x\\\\ y\\\\ z\\end{pmatrix}=\\begin{pmatrix}1\\\\ 2\\\\ -3\\end{pmatrix}+t\\cdot\\begin{pmatrix}2\\\\ 1\\\\ 1\\end{pmatrix}\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Kan du ud fra oplysningerne om de to linjer opstille parameterfremstillingen for et plan? Hvis ja, hvad er s\u00e5 planets parameterfremstilling. Hvis nej, hvorfor kan det ikke lade sig g\u00f8re?<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p class=\" eplus-wrapper\">Planets parameterfremstilling ud fra tre punkter<\/p>\n\n\n\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-id3178_ba4265-b5 kt-accordion-has-3-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"false\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane3178_1815bf-88\" id=\"VRPPUTP001\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VRPPUTP001<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Find parameterfremstillingen for planet gennem punkterne (1, 2, 3), (4, 5, 6) og (-1, 0, 2).<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-2 kt-pane3178_35c786-c0\" id=\"VRPPUTP002\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VRPPUTP002<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Find parameterfremstillingen for planet, der g\u00e5r gennem punkterne (1, 1, 1), (2, -1, 3) og (-1, 2, -1).<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-3 kt-pane3178_8ba2cc-42\" id=\"VRPPUTP003\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VRPPUTP003*<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">Bestem parameterfremstillingen for planet, det g\u00e5r gennem punktet (2, -1, 3) og indeholder linjen \\(\\begin{pmatrix}x\\\\y\\\\z\\end{pmatrix}=\\begin{pmatrix}3+t\\\\-1-2t\\\\2t\\end{pmatrix}\\)<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n\n\n\n<p class=\" eplus-wrapper\">Sk\u00e6ringspunkt mellem en linje og et plan<\/p>\n\n\n\n<div class=\"wp-block-kadence-accordion alignnone\"><div class=\"kt-accordion-wrap kt-accordion-id3178_964a55-eb kt-accordion-has-3-panes kt-active-pane-0 kt-accordion-block kt-pane-header-alignment-left kt-accodion-icon-style-basic kt-accodion-icon-side-right\" style=\"max-width:none\"><div class=\"kt-accordion-inner-wrap\" data-allow-multiple-open=\"false\" data-start-open=\"none\">\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-1 kt-pane3178_5ed278-79\" id=\"VRSMLP001\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VRSMLP001<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">En linje \\(\\ell\\) har parameterfremstillingen<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(\\begin{pmatrix}x\\\\y\\\\z\\end{pmatrix}=\\begin{pmatrix}2\\\\-1\\\\4\\end{pmatrix}+t\\cdot\\begin{pmatrix}3\\\\2\\\\-1\\end{pmatrix}\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">og et plan \\(\\alpha\\) er givet ved ligningen<\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(2x-3y+4z=12\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Find sk\u00e6ringspunktet mellem linjen \\(\\ell\\) og planet \\(\\alpha\\).<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-2 kt-pane3178_d35555-cc\" id=\"VRSMLP002\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VRSMLP002<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">En linje \\(\\ell\\) er givet ved to punkter A(1,-2,3) og B(3,1,5). Et plan \\(\\alpha\\) g\u00e5r gennem punktet C(2,0,4) og er vinkelret p\u00e5 linjen \\(\\ell\\). Find sk\u00e6ringspunktet mellem linjen \\(\\ell\\) og planet \\(\\alpha\\).<\/p>\n<\/div><\/div><\/div>\n\n\n\n<div class=\"wp-block-kadence-pane kt-accordion-pane kt-accordion-pane-3 kt-pane3178_c90870-3d\" id=\"VRSMLP003\"><div class=\"kt-accordion-header-wrap\"><button class=\"kt-blocks-accordion-header kt-acccordion-button-label-show\"><span class=\"kt-blocks-accordion-title-wrap\"><span class=\"kt-blocks-accordion-title\">VRSMLP003<\/span><\/span><span class=\"kt-blocks-accordion-icon-trigger\"><\/span><\/button><\/div><div class=\"kt-accordion-panel kt-accordion-panel-hidden\"><div class=\"kt-accordion-panel-inner\">\n<p class=\" eplus-wrapper\">En linje \\(\\ell\\) har parameterfremstillingen <\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(\\vec{r}=\\begin{pmatrix}1\\\\2\\\\-3\\end{pmatrix}+t\\cdot\\begin{pmatrix}2\\\\-1\\\\4\\end{pmatrix}\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">og et plan \\(\\alpha\\) er givet ved ligningen <\/p>\n\n\n\n<p class=\" has-text-align-center eplus-wrapper\">\\(2x+3y-z=5\\)<\/p>\n\n\n\n<p class=\" eplus-wrapper\">Unders\u00f8g om linjen \\(\\ell\\) sk\u00e6rer planen \\(\\alpha\\) og hvis ja, find da sk\u00e6ringspunktet.<\/p>\n<\/div><\/div><\/div>\n<\/div><\/div><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Linjens parameterfremstilling &#8211; punkt p\u00e5 linjen Planets parameterfremstilling Planets parameterfremstilling ud fra tre punkter Sk\u00e6ringspunkt mellem en linje og et plan<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"ub_ctt_via":"","editor_plus_copied_stylings":"{}","footnotes":""},"categories":[],"tags":[],"class_list":["post-3178","page","type-page","status-publish","hentry"],"featured_image_src":null,"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/pages\/3178","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/types\/page"}],"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=3178"}],"version-history":[{"count":6,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/pages\/3178\/revisions"}],"predecessor-version":[{"id":3227,"href":"https:\/\/mxth.dk\/index.php?rest_route=\/wp\/v2\/pages\/3178\/revisions\/3227"}],"wp:attachment":[{"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3178"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3178"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mxth.dk\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3178"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}