{"id":303,"date":"2023-07-10T07:15:20","date_gmt":"2023-07-10T07:15:20","guid":{"rendered":"https:\/\/mathority.org\/nl\/coplanaire-of-coplanaire-vectoren\/"},"modified":"2023-07-10T07:15:20","modified_gmt":"2023-07-10T07:15:20","slug":"coplanaire-of-coplanaire-vectoren","status":"publish","type":"post","link":"https:\/\/mathority.org\/nl\/coplanaire-of-coplanaire-vectoren\/","title":{"rendered":"Coplanaire (of coplanaire) vectoren"},"content":{"rendered":"<p>Op deze pagina leert u wat coplanaire vectoren zijn en hoe u kunt bepalen of 2, 3, 4 of meer vectoren coplanair zijn. Bovendien kunt u voorbeelden en oefeningen zien die stap voor stap zijn opgelost met coplanaire vectoren. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-vectores-coplanarios\"><\/span> Wat zijn coplanaire vectoren?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> In de analytische meetkunde is de betekenis van coplanaire (of coplanaire) vectoren als volgt:<\/p>\n<p> <strong>Coplanaire vectoren zijn vectoren die tot hetzelfde vlak behoren.<\/strong><\/p>\n<p> Daarom zijn twee vectoren altijd coplanair omdat een vlak kan worden gevormd met slechts twee vectoren. Aan de andere kant, als er 3, 4 of meer vectoren zijn, is het mogelijk dat een van de vectoren zich niet in hetzelfde vlak bevindt en daarom niet coplanair is. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/vecteurs-coplanaires-ou-coplanaires.webp\" alt=\"voorbeelden van coplanaire of coplanaire vectoren\" class=\"wp-image-3133\" width=\"353\" height=\"171\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> In de grafiek hierboven kun je bijvoorbeeld zien dat de vectoren<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cac24ae79c1e4cbc459f01ed5e4f824e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> En<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> ze liggen in \u00e9\u00e9n vlak met elkaar, omdat ze zich in hetzelfde vlak bevinden. Aan de andere kant zijn deze twee vectoren niet coplanair met de vector<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3b4bbbc56786695092eac40831aee80d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> , omdat er geen vlak kan worden gevormd in de ruimte die de drie vectoren bevat.<\/p>\n<p> Uit deze eigenschap kunnen we afleiden dat als 3 of meer vectoren coplanair zijn, de punten die deze vectoren defini\u00ebren (begin en einde van de vector) ook coplanaire punten zijn. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcuando-los-vectores-son-coplanarios\"><\/span> Wanneer zijn vectoren coplanair?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Zoals we hebben gezien bij de definitie van coplanaire (of coplanaire) vectoren, zijn twee vectoren altijd coplanair, maar hoeven meer dan twee vectoren de coplanariteitsrelatie niet te respecteren.<\/p>\n<p> Er zijn dus verschillende methoden om te bepalen of drie of meer vectoren coplanair zijn:<\/p>\n<ul>\n<li> Als het gemengde product van drie vectoren (of het drievoudige puntproduct) gelijk is aan nul, betekent dit dat de drie vectoren coplanair zijn. Als het u niet helemaal duidelijk is hoe deze bewerking wordt berekend, raad ik u aan om te kijken naar <a href=\"https:\/\/mathority.org\/nl\/voorbeelden-van-gemengde-producten-van-drie-vectoren-of-drievoudige-scalaire-producten\/\">wat het gemengde product is van drie vectoren<\/a> . Hier vindt u de uitleg, evenals voorbeelden en opgeloste oefeningen.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-849b07c1e268c4903e7bd13ef56bcaf3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr] =0\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"92\" style=\"vertical-align: -7px;\"><\/p>\n<\/p>\n<ul>\n<li> Als een reeks vectoren kan worden uitgedrukt als een <a href=\"https:\/\/mathority.org\/nl\/lineaire-combinatie-van-vectoren-voorbeelden-opgeloste-oefeningen\/\">lineaire combinatie van twee vectoren,<\/a> impliceert dit dat ze coplanair zijn, wat betekent dat 3 of meer vectoren coplanair zijn als en slechts als ze lineair afhankelijk zijn. Om aan te tonen dat drie of meer vectoren een lineaire combinatie van twee vectoren zijn, volstaat het dat de rangorde van de matrix gevormd door alle vectoren gelijk is aan 2.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef18656c1a261aa20598fc8f6a587323_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"rg(A) = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Het is belangrijk dat je een goed begrip hebt van het concept van <a href=\"https:\/\/mathority.org\/nl\/onafhankelijke-en-lineair-afhankelijke-vectoren-onafhankelijkheid-lineaire-afhankelijkheid\/\">lineaire afhankelijkheid en onafhankelijkheid<\/a> , dat wil zeggen wanneer twee vectoren lineair afhankelijk of lineair onafhankelijk zijn, en wat dat betekent. Mocht het je niet helemaal duidelijk zijn, dan vind je in de link een zeer gedetailleerde uitleg, waarbij je bovendien voorbeelden en oefeningen stap voor stap opgelost ziet.<\/p>\n<ul>\n<li> Als de vectoren in kwestie <a href=\"https:\/\/mathority.org\/nl\/parallelle-vectoren\/\">parallelle vectoren<\/a> zijn, betekent dit dat ze ook coplanair zijn, dat wil zeggen dat alle parallelle vectoren zich in hetzelfde vlak bevinden. <\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4efa93d26f00c6abc1180201f84d126a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\parallel  \\vv{\\text{v}} \\parallel \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-vectores-coplanarios\"><\/span> Opgeloste problemen van coplanaire vectoren<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Oefening 1<\/h3>\n<p> Bepaal of de volgende drie vectoren coplanair zijn: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-16f2fe8ce9dccfd2f5f2b26461ca54e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (3,1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ce008f944cfd9efa2c48d0083a479c89_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}} = (2,3,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"102\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d8759d1ec233d68fc5f81dfb3b67beb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} = (-1,-5,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"119\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Om te controleren of dit 3 coplanaire vectoren zijn, moeten we het gemengde product tussen de drie vectoren berekenen:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2a1e4b0655c0a3f0165c880f5e64cce0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\bigl[\\vv{\\text{u}},\\vv{\\text{v}},\\vv{\\text{w}}\\bigr]&amp; =\\begin{vmatrix} 3 &amp; 1 &amp; 2 \\\\[1.1ex] 2 &amp; 3 &amp; -1 \\\\[1.1ex] -1 &amp; -5 &amp; 4 \\end{vmatrix} \\\\[2ex] &amp;= 36+1-20+6-15-8 \\\\[2ex] &amp; = \\bm{0} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"166\" width=\"271\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Het gemengde product van de drie vectoren is nul, dus de <strong>3 vectoren zijn coplanair<\/strong> .<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Oefening 2<\/h3>\n<p> Bepaal of de volgende drie vectoren coplanair zijn: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a7c6550cedc0ccb79a9bfdebdd9987cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (4,-2,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-730594e946d69d5c0bd66b4b6d0f443c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}} = (-2,1,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a85ef59d300497c53b26259f19df79f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{w}} = (6,-3,9)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> E\u00e9n manier om te controleren of we met drie coplanaire vectoren te maken hebben, is door het gemengde product tussen de drie vectoren op te lossen. Als we echter goed naar de componenten van de vectoren kijken, kunnen we zien dat ze proportioneel zijn. Daarom zijn de drie vectoren evenwijdig aan elkaar.<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0c5ac41bb15ea29bdc9736f100d1cf74_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\parallel \\vv{\\text{v}} \\parallel \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> En aangezien alle vectoren evenwijdig zijn, <strong>zijn het in feite drie coplanaire vectoren<\/strong> .<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Oefening 3<\/h3>\n<p> Bepaal of de volgende vier vectoren coplanair zijn: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-10d37864162c9c1d2eae8f5b7c7df066_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{a}} = (2,1,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0ab59c17a2058de83ef95ee9b7021751_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{b}} = (1,-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-1822f69d3738974e93084ea4c454d63f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{c}} = (-1,0,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f76c271b5fc0fe45fe9b2591346f083f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{d}} = (3,1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Om te weten of de vier vectoren coplanair zijn, moeten we de rangorde berekenen van de matrix die uit alle vectoren bestaat:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8384924c86edafd568505d5f80e1705d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} 2&amp;1&amp;1 \\\\[1.1ex] 1&amp;-1&amp;2 \\\\[1.1ex] -1&amp;0&amp;-1 \\\\[1.1ex] 3&amp;1&amp;2\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"164\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> In dit geval berekenen we de reikwijdte van genoemde matrix aan de hand van determinanten: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5db59e1c8bbf94b95483870d47cea1b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"rg(A) = \\ ?\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2778435c7f53952adf072419af8b268c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2&amp;1&amp;1 \\\\[1.1ex] 1&amp;-1&amp;2 \\\\[1.1ex] -1&amp;0&amp;-1 \\end{vmatrix}=0 \\quad  \\begin{vmatrix} 2&amp;1&amp;1 \\\\[1.1ex] 1&amp;-1&amp;2 \\\\[1.1ex]3&amp;1&amp;2\\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"280\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-82f278494a221879cc86da92ab4378c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2&amp;1&amp;1 \\\\[1.1ex] -1&amp;0&amp;-1 \\\\[1.1ex] 3&amp;1&amp;2\\end{vmatrix}=0 \\quad \\begin{vmatrix} 1&amp;-1&amp;2 \\\\[1.1ex] -1&amp;0&amp;-1 \\\\[1.1ex] 3&amp;1&amp;2\\end{vmatrix}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"294\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-889142ac348173dd6c838633007f2d06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} 2&amp;1 \\\\[1.1ex] 1&amp;-1\\end{vmatrix}= -3\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"136\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef18656c1a261aa20598fc8f6a587323_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"rg(A) = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> De rangorde van de matrix gevormd door alle vectoren is gelijk aan 2, daarom <strong>zijn de 4 vectoren coplanair<\/strong> .<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Oefening 4<\/h3>\n<p> Bereken parameterwaarde<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> zodat de volgende 4 punten coplanair zijn: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9483cca4fc2a94923b7c72ed89fc2d5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(3,1,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a5b113e265916c03a6de0547cfeb380b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(2,1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-add9ad10fb8badd9de84f8ad1dcfe38d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C(0,-1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8d5054a3ce659090916cda61b74f60bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D(3,2,k)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>zie oplossing<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Om de vier punten coplanair te laten zijn, moeten de daardoor bepaalde vectoren coplanair zijn. We berekenen daarom deze vectoren: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-996a90c58f67665e4a68e9dd4de6c718_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B- A = (2,1,2)-(3,1,4) = (-1,0,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"365\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ae552981d5729c931f5bbb26c133ecc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C- A = (0,-1,3)-(3,1,4) = (-3,-2,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"392\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-91aec8ed49541d8c2d8ca0b3b1f8a20d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AD} = D- A = (3,2,k)-(3,1,4) = (0,1,k-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"371\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Wiens vectormatrix is:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c3d801efcf5b56dd858890720797d6a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} -1&amp;0&amp;-2 \\\\[1.1ex] -3&amp;-2&amp;-1 \\\\[1.1ex] 0&amp;1&amp;k-4\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"181\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Om de resulterende vectoren coplanair te laten zijn, moet de rangorde van de matrix 2 zijn. En daarom moet de determinant van de gehele 3&#215;3-matrix 0 zijn: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-bb7d3b31c10096d100843d781a85b621_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} -1&amp;0&amp;-2 \\\\[1.1ex] -3&amp;-2&amp;-1 \\\\[1.1ex] 0&amp;1&amp;k-4\\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"160\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b265559f1f5505b8c40a89f0d69f0c10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2k-3 =0\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"82\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Eindelijk lossen we het onbekende op <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7cf6d2c84f82625cb8a795ee1394251f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f79b9d3960668149408038b9cb1d1e0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2k =3\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"51\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-e67be6e218d206fe735f54a6125b3d2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{k =}\\mathbf{\\cfrac{3}{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"41\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Op deze pagina leert u wat coplanaire vectoren zijn en hoe u kunt bepalen of 2, 3, 4 of meer vectoren coplanair zijn. Bovendien kunt u voorbeelden en oefeningen zien die stap voor stap zijn opgelost met coplanaire vectoren. Wat zijn coplanaire vectoren? In de analytische meetkunde is de betekenis van coplanaire (of coplanaire) vectoren &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/nl\/coplanaire-of-coplanaire-vectoren\/\"> <span class=\"screen-reader-text\">Coplanaire (of coplanaire) vectoren<\/span> Lees meer &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"","footnotes":""},"categories":[54],"tags":[],"class_list":["post-303","post","type-post","status-publish","format-standard","hentry","category-vectoren"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Coplanaire (of coplanaire) vectoren - Mathority<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathority.org\/nl\/coplanaire-of-coplanaire-vectoren\/\" \/>\n<meta property=\"og:locale\" content=\"nl_NL\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Coplanaire (of coplanaire) vectoren - Mathority\" \/>\n<meta property=\"og:description\" content=\"Op deze pagina leert u wat coplanaire vectoren zijn en hoe u kunt bepalen of 2, 3, 4 of meer vectoren coplanair zijn. 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