{"id":258,"date":"2023-07-10T07:15:20","date_gmt":"2023-07-10T07:15:20","guid":{"rendered":"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/"},"modified":"2023-07-10T07:15:20","modified_gmt":"2023-07-10T07:15:20","slug":"vetores-coplanares-ou-coplanares","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/","title":{"rendered":"Vetores coplanares (ou coplanares)"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea aprender\u00e1 o que s\u00e3o vetores coplanares e como saber se 2, 3, 4 ou mais vetores s\u00e3o coplanares. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e exerc\u00edcios resolvidos passo a passo de vetores coplanares. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-vectores-coplanarios\"><\/span> O que s\u00e3o vetores coplanares?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Na geometria anal\u00edtica, o significado dos vetores coplanares (ou coplanares) \u00e9 o seguinte:<\/p>\n<p> <strong>Vetores coplanares s\u00e3o vetores que pertencem ao mesmo plano.<\/strong><\/p>\n<p> Portanto, dois vetores s\u00e3o sempre coplanares porque um plano pode ser formado com apenas 2 vetores. Por outro lado, quando existem 3, 4 ou mais vetores, \u00e9 poss\u00edvel que um dos vetores n\u00e3o esteja contido no mesmo plano e, portanto, n\u00e3o sejam coplanares. <\/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=\"exemplos de vetores coplanares ou coplanares\" class=\"wp-image-3133\" width=\"353\" height=\"171\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Por exemplo, no gr\u00e1fico acima voc\u00ea pode ver que os vetores<\/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> E<\/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> eles s\u00e3o coplanares entre si, pois est\u00e3o contidos no mesmo plano. Por outro lado, estes dois vetores n\u00e3o s\u00e3o coplanares com o vetor<\/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> , porque nenhum plano pode ser formado no espa\u00e7o que cont\u00e9m os tr\u00eas vetores.<\/p>\n<p> Desta propriedade podemos deduzir que se 3 ou mais vetores s\u00e3o coplanares, os pontos que definem esses vetores (in\u00edcio e fim do vetor) tamb\u00e9m s\u00e3o pontos coplanares. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcuando-los-vectores-son-coplanarios\"><\/span> Quando os vetores s\u00e3o coplanares?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Como vimos na defini\u00e7\u00e3o de vetores coplanares (ou coplanares), dois vetores s\u00e3o sempre coplanares, mas mais de dois vetores n\u00e3o precisam respeitar a rela\u00e7\u00e3o de coplanaridade.<\/p>\n<p> Assim, existem v\u00e1rios m\u00e9todos para determinar se tr\u00eas ou mais vetores s\u00e3o coplanares:<\/p>\n<ul>\n<li> Se o produto misto de tr\u00eas vetores (ou produto escalar triplo) for igual a zero, significa que os tr\u00eas vetores s\u00e3o coplanares. Se voc\u00ea n\u00e3o tem muita clareza sobre como essa opera\u00e7\u00e3o \u00e9 calculada, recomendo que voc\u00ea d\u00ea uma olhada no <a href=\"https:\/\/mathority.org\/pt\/exemplos-de-produtos-mistos-de-tres-vetores-ou-produtos-escalares-triplos\/\">que \u00e9 o produto misto de tr\u00eas vetores<\/a> , aqui voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o, al\u00e9m de exemplos e exerc\u00edcios resolvidos.<\/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> Se um conjunto de vetores pode ser expresso como uma <a href=\"https:\/\/mathority.org\/pt\/combinacao-linear-de-exemplos-de-vetores-exercicios-resolvidos\/\">combina\u00e7\u00e3o linear de dois vetores,<\/a> isso implica que eles s\u00e3o coplanares, o que significa que 3 ou mais vetores s\u00e3o coplanares se e somente se forem linearmente dependentes. Para mostrar que tr\u00eas ou mais vetores s\u00e3o uma combina\u00e7\u00e3o linear de dois vetores, basta que o posto da matriz formada por todos os vetores seja igual a 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> \u00c9 importante que voc\u00ea tenha um bom entendimento do conceito de <a href=\"https:\/\/mathority.org\/pt\/vetores-independentes-e-linearmente-dependentes-independencia-dependencia-linear\/\">depend\u00eancia e independ\u00eancia linear<\/a> , ou seja, quando dois vetores s\u00e3o linearmente dependentes ou linearmente independentes e o que isso significa. Se n\u00e3o estiver totalmente claro, no link voc\u00ea encontrar\u00e1 uma explica\u00e7\u00e3o bem detalhada, onde, al\u00e9m disso, poder\u00e1 ver exemplos e exerc\u00edcios resolvidos passo a passo.<\/p>\n<ul>\n<li> Se os vetores em quest\u00e3o forem <a href=\"https:\/\/mathority.org\/pt\/vetores-paralelos\/\">vetores paralelos<\/a> , isso significa que tamb\u00e9m s\u00e3o coplanares, ou seja, todos os vetores paralelos est\u00e3o contidos no mesmo plano. <\/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> Problemas resolvidos de vetores coplanares<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Determine se os tr\u00eas vetores a seguir s\u00e3o coplanares: <\/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>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para verificar se se trata de 3 vetores coplanares, devemos calcular o produto misto entre os tr\u00eas vetores:<\/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\"> O produto misto dos tr\u00eas vetores \u00e9 zero, ent\u00e3o os <strong>3 vetores s\u00e3o coplanares<\/strong> .<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 2<\/h3>\n<p> Determine se os tr\u00eas vetores a seguir s\u00e3o coplanares: <\/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>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Uma forma de verificar se estamos lidando com 3 vetores coplanares seria resolver o produto misto entre os tr\u00eas vetores. Por\u00e9m, se olharmos atentamente para as componentes dos vetores, podemos ver que elas s\u00e3o proporcionais. Portanto, os tr\u00eas vetores s\u00e3o paralelos entre si.<\/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\"> E como todos os vetores s\u00e3o paralelos, <strong>eles s\u00e3o efetivamente 3 vetores coplanares<\/strong> .<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 3<\/h3>\n<p> Determine se os quatro vetores a seguir s\u00e3o coplanares: <\/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>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para saber se os quatro vetores s\u00e3o coplanares, devemos calcular o posto da matriz composta por todos os vetores:<\/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\"> Neste caso, calculamos o escopo da referida matriz por determinantes: <\/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\"> O posto da matriz formada por todos os vetores equivale a 2, portanto <strong>os 4 vetores s\u00e3o coplanares<\/strong> .<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 4<\/h3>\n<p> Calcular o valor do par\u00e2metro<\/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> de modo que os 4 pontos a seguir sejam coplanares: <\/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>veja solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para que os quatro pontos sejam coplanares, os vetores por eles determinados devem ser coplanares. Portanto, calculamos estes vetores: <\/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\"> Cuja matriz vetorial \u00e9:<\/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\"> Para que os vetores resultantes sejam coplanares, o posto da matriz deve ser 2. E, portanto, o determinante de toda a matriz 3&#215;3 deve ser 0: <\/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\"> Finalmente, resolvemos o desconhecido <\/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>Nesta p\u00e1gina voc\u00ea aprender\u00e1 o que s\u00e3o vetores coplanares e como saber se 2, 3, 4 ou mais vetores s\u00e3o coplanares. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e exerc\u00edcios resolvidos passo a passo de vetores coplanares. O que s\u00e3o vetores coplanares? Na geometria anal\u00edtica, o significado dos vetores coplanares (ou coplanares) \u00e9 o seguinte: Vetores &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/\"> <span class=\"screen-reader-text\">Vetores coplanares (ou coplanares)<\/span> Leia mais &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":[27],"tags":[],"class_list":["post-258","post","type-post","status-publish","format-standard","hentry","category-vetores"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Vetores coplanares (ou coplanares) - 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\/pt\/vetores-coplanares-ou-coplanares\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Vetores coplanares (ou coplanares) - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea aprender\u00e1 o que s\u00e3o vetores coplanares e como saber se 2, 3, 4 ou mais vetores s\u00e3o coplanares. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e exerc\u00edcios resolvidos passo a passo de vetores coplanares. O que s\u00e3o vetores coplanares? Na geometria anal\u00edtica, o significado dos vetores coplanares (ou coplanares) \u00e9 o seguinte: Vetores &hellip; Vetores coplanares (ou coplanares) Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-10T07:15:20+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/vecteurs-coplanaires-ou-coplanaires.webp\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"4 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Vetores coplanares (ou coplanares)\",\"datePublished\":\"2023-07-10T07:15:20+00:00\",\"dateModified\":\"2023-07-10T07:15:20+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/\"},\"wordCount\":745,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Vetores\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/\",\"url\":\"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/\",\"name\":\"Vetores coplanares (ou coplanares) - 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Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e exerc\u00edcios resolvidos passo a passo de vetores coplanares. O que s\u00e3o vetores coplanares? Na geometria anal\u00edtica, o significado dos vetores coplanares (ou coplanares) \u00e9 o seguinte: Vetores &hellip; Vetores coplanares (ou coplanares) Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/","article_published_time":"2023-07-10T07:15:20+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/vecteurs-coplanaires-ou-coplanaires.webp"}],"author":"Equipe Mathoridade","twitter_card":"summary_large_image","twitter_misc":{"Escrito por":"Equipe Mathoridade","Est. tempo de leitura":"4 minutos"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Vetores coplanares (ou coplanares)","datePublished":"2023-07-10T07:15:20+00:00","dateModified":"2023-07-10T07:15:20+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/"},"wordCount":745,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Vetores"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/","url":"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/","name":"Vetores coplanares (ou coplanares) - 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