{"id":259,"date":"2023-07-10T06:51:49","date_gmt":"2023-07-10T06:51:49","guid":{"rendered":"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/"},"modified":"2023-07-10T06:51:49","modified_gmt":"2023-07-10T06:51:49","slug":"exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/","title":{"rendered":"Pontos coplanares (ou coplanares)"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea descobrir\u00e1 o que s\u00e3o pontos coplanares (ou coplanares) e como saber se determinados pontos s\u00e3o coplanares ou n\u00e3o. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e praticar com exerc\u00edcios de pontos coplanares resolvidos. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-puntos-coplanarios\"><\/span> O que s\u00e3o pontos coplanares?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Na geometria anal\u00edtica, o significado dos pontos coplanares (ou coplanares) \u00e9 o seguinte:<\/p>\n<p> <strong>Pontos coplanares s\u00e3o pontos que pertencem ao mesmo plano.<\/strong><\/p>\n<p> Portanto, 2 ou 3 pontos s\u00e3o sempre coplanares porque um plano pode ser formado com apenas 3 pontos. Por outro lado, quando existem 4, 5 ou mais pontos, \u00e9 poss\u00edvel que alguns dos pontos n\u00e3o estejam contidos 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\/points-coplanaires.webp\" alt=\"pontos coplanares\" class=\"wp-image-3165\" width=\"411\" height=\"215\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<p> Por exemplo, na representa\u00e7\u00e3o gr\u00e1fica acima, voc\u00ea pode ver que os pontos A, C, D e F s\u00e3o coplanares entre si, pois est\u00e3o contidos no mesmo plano. Por outro lado, estes 4 pontos n\u00e3o s\u00e3o coplanares com os pontos B, E e G, porque nenhum plano pode ser formado no espa\u00e7o que cont\u00e9m todos os pontos.<\/p>\n<p> Desta propriedade podemos deduzir que os vetores definidos por pontos coplanares tamb\u00e9m s\u00e3o vetores coplanares, ou seja, est\u00e3o contidos no mesmo plano. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcuando-los-puntos-son-coplanarios\"><\/span> Quando os pontos s\u00e3o coplanares?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Como vimos na defini\u00e7\u00e3o de pontos coplanares (ou coplanares), dois ou tr\u00eas pontos s\u00e3o sempre coplanares, mas mais de tr\u00eas pontos n\u00e3o precisam respeitar a rela\u00e7\u00e3o de coplanaridade.<\/p>\n<p> Portanto, existem basicamente 2 m\u00e9todos para determinar se quatro ou mais pontos s\u00e3o coplanares:<\/p>\n<ul>\n<li> Uma forma de saber se os pontos s\u00e3o coplanares \u00e9 pelos vetores que s\u00e3o determinados pelos pontos: <strong>se esses vetores s\u00e3o coplanares<\/strong> , ent\u00e3o os pontos tamb\u00e9m s\u00e3o coplanares.<\/li>\n<\/ul>\n<p> Obviamente, para aplicar este m\u00e9todo \u00e9 necess\u00e1rio saber quando os vetores s\u00e3o coplanares. Mas como tamb\u00e9m existem v\u00e1rias maneiras de determinar se um conjunto de vetores \u00e9 coplanar, recomendamos verificar <a href=\"https:\/\/mathority.org\/pt\/vetores-coplanares-ou-coplanares\/\">como saber se os vetores s\u00e3o coplanares<\/a> . Aqui voc\u00ea encontrar\u00e1 todos os procedimentos existentes para descobrir quando 2, 3, 4 ou mais vetores s\u00e3o coplanares, al\u00e9m de exemplos e exerc\u00edcios resolvidos.<\/p>\n<ul>\n<li> Outra forma de saber se um conjunto de pontos \u00e9 coplanar \u00e9 <strong><a href=\"https:\/\/mathority.org\/pt\/equacoes-planas-no-espaco\/\">encontrar a equa\u00e7\u00e3o do plano<\/a><\/strong> formado por 3 pontos do conjunto, e se os demais pontos satisfazem esta equa\u00e7\u00e3o, ent\u00e3o isso significa que todos os pontos do conjunto s\u00e3o coplanares.<\/li>\n<\/ul>\n<p> Embora dependa do problema, recomendamos utilizar o primeiro dos dois m\u00e9todos, pois \u00e9 muito mais simples e r\u00e1pido verificar se os vetores s\u00e3o coplanares do que calcular a equa\u00e7\u00e3o de um plano. Mas, obviamente, use o que preferir. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-puntos-coplanarios\"><\/span> Problemas de pontos coplanares resolvidos<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 pontos 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-9a6d39406adb983eaad3c40c2e5d352c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(3,5,1)\" 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-a2db4553e8942cf714a98783699f44f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(0,-2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" 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-93a412e1bb305cd785882fd481852c9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C(4,-1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" 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\"> Neste caso n\u00e3o \u00e9 necess\u00e1rio fazer nenhum c\u00e1lculo porque <strong>3 pontos s\u00e3o sempre coplanares<\/strong> , sejam eles quais forem.<\/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 quatro pontos 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-adbc97b4e3f0c30a1cd47fb6eea0b566_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(1,2,0)\" 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-d5ca7779c360bcb8ad9cf4d6b6a455a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(-1,3,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" 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-8910b93724e6f59b070ee26673c9426a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C(1,0,-1)\" 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-6d40d5bb66afe9a8e8445cdb8b7cac0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D(2,5,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" 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-e7922b55a4e968ce9ef945ddf4b0973e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B- A = (-1,3,4)-(1,2,0) = (-2,1,4)\" 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-8dfa4c2463f3a69890706b0387f1fbc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C- A = (1,0,-1)-(1,2,0) = (0,-2,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"378\" 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-e5617a7317e2c18fcb39b3be782e26c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AD} = D- A = (2,5,-3)-(1,2,0) = (1,3,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"367\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Vamos agora construir a matriz formada pelos 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-873ec750665fdf215a648778d6fcf042_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} -2&amp;1&amp;4 \\\\[1.1ex]0&amp;-2&amp;-1 \\\\[1.1ex] 1&amp;3&amp;-3\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"164\" 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 anterior deve ser igual a 2. E, para isso, o determinante de toda a matriz 3\u00d73 deve ser igual a zero: <\/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-18eb85dd3870c1e7c865705200cf6414_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-3eadb62f19c07b8a646fe9b2a00f9003_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} -2&amp;1&amp;4 \\\\[1.1ex]0&amp;-2&amp;-1 \\\\[1.1ex] 1&amp;3&amp;-3\\end{vmatrix} =-11\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"198\" 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-466d66279a42e8aeb422d568e1547ef9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"rg(A)=3\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Por\u00e9m, o determinante de toda a matriz \u00e9 diferente de zero, ent\u00e3o o posto da matriz \u00e9 3 e, portanto <strong>, os 4 pontos n\u00e3o 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 3<\/h3>\n<p> Descubra se os cinco pontos 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-083776221bb4167cd960c67da2435d9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(3,1,1)\" 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-1a43aebd7e3c897898050e7440b0fe9c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"B(0,0,2)\" 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-59e69260b8e3baacdfd4f673aeb87fba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"C(4,-2,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" 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-e862d9c0d7c715b2268b594c199f506a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D(-1,1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"84\" 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-f361ff1c29cf0a1dba88da8bdb1fb448_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"E(2,4,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"69\" 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 todos os cinco pontos sejam coplanares, os vetores por eles definidos 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-9b6a321df8b7b3e37dc31bf0cb9542dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B- A = (0,0,2)-(3,1,1) = (-3,-1,1)\" 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-611bc1c76bcadfe27e2a793128a5a3c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C- A = (4,-2,-1)-(3,1,1) = (1,-3,-2)\" 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-0ff805267450b865998e62bbc20111b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AD} = D- A = (-1,1,3)-(3,1,1) = (-4,0,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"366\" 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-494b794700c9c99091abf0fd59523dd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AE} = E- A = (2,4,3)-(3,1,1) = (-1,3,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"350\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Vamos agora construir a matriz composta pelos 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-5d571ea172dbf5e6b7d85da6db466eb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle A= \\begin{pmatrix} -3&amp;-1&amp;1 \\\\[1.1ex]1&amp;-3&amp;-2 \\\\[1.1ex] -4&amp;0&amp;2\\\\[1.1ex] -1&amp;3&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\"> Para que os vetores resultantes sejam coplanares, o posto da matriz anterior deve ser igual a 2. Calculamos, portanto, o posto da matriz de vetores por determinantes para verificar se eles 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-18eb85dd3870c1e7c865705200cf6414_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-bec15883d9e3aa5a401afa41cd3992ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} -3&amp;-1&amp;1 \\\\[1.1ex]1&amp;-3&amp;-2 \\\\[1.1ex] -4&amp;0&amp;2\\end{vmatrix} = 0 \\qquad \\begin{vmatrix} -3&amp;-1&amp;1 \\\\[1.1ex]1&amp;-3&amp;-2 \\\\[1.1ex] -1&amp;3&amp;2\\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"326\" 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-cd2f8cfa577e91b74c69e50cc36dfd9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} -3&amp;-1&amp;1 \\\\[1.1ex] -4&amp;0&amp;2\\\\[1.1ex] -1&amp;3&amp;2\\end{vmatrix} = 0 \\qquad \\begin{vmatrix} 1&amp;-3&amp;-2 \\\\[1.1ex] -4&amp;0&amp;2\\\\[1.1ex] -1&amp;3&amp;2\\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"312\" 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-d551dddceb916cc6cf8bf4137903da1d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} -3&amp;-1 \\\\[1.1ex] 1&amp;-3\\end{vmatrix} =10\\neq  0\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"145\" 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-6865ac4ae80c8b5a6eff791a9da5a937_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 \u00e9 equivalente a 2, portanto os vetores s\u00e3o coplanares e portanto <strong>os 5 pontos tamb\u00e9m 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<p> Por fim, se este artigo foi \u00fatil para voc\u00ea, provavelmente voc\u00ea tamb\u00e9m est\u00e1 interessado em como <a href=\"https:\/\/mathority.org\/pt\/formula-para-a-distancia-entre-dois-pontos-exemplos-de-geometria-e-exercicios-resolvidos\/\">\u00e9 calculada a dist\u00e2ncia entre dois pontos (f\u00f3rmula)<\/a> , j\u00e1 que \u00e0s vezes em problemas de geometria anal\u00edtica nos perguntam qual \u00e9 a dist\u00e2ncia entre dois pontos. Na p\u00e1gina do link voc\u00ea encontrar\u00e1 uma explica\u00e7\u00e3o bem detalhada, al\u00e9m de exemplos e exerc\u00edcios resolvidos passo a passo.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea descobrir\u00e1 o que s\u00e3o pontos coplanares (ou coplanares) e como saber se determinados pontos s\u00e3o coplanares ou n\u00e3o. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e praticar com exerc\u00edcios de pontos coplanares resolvidos. O que s\u00e3o pontos coplanares? Na geometria anal\u00edtica, o significado dos pontos coplanares (ou coplanares) \u00e9 o seguinte: Pontos coplanares &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Pontos 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":[20],"tags":[],"class_list":["post-259","post","type-post","status-publish","format-standard","hentry","category-pontos-retas-e-planos"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Pontos 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\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Pontos coplanares (ou coplanares) - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea descobrir\u00e1 o que s\u00e3o pontos coplanares (ou coplanares) e como saber se determinados pontos s\u00e3o coplanares ou n\u00e3o. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e praticar com exerc\u00edcios de pontos coplanares resolvidos. O que s\u00e3o pontos coplanares? Na geometria anal\u00edtica, o significado dos pontos coplanares (ou coplanares) \u00e9 o seguinte: Pontos coplanares &hellip; Pontos coplanares (ou coplanares) Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-10T06:51:49+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/points-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\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Pontos coplanares (ou coplanares)\",\"datePublished\":\"2023-07-10T06:51:49+00:00\",\"dateModified\":\"2023-07-10T06:51:49+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/\"},\"wordCount\":816,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Pontos, retas e planos\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/\",\"url\":\"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/\",\"name\":\"Pontos coplanares (ou coplanares) - 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Mathority","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/","og_locale":"pt_BR","og_type":"article","og_title":"Pontos coplanares (ou coplanares) - Mathority","og_description":"Nesta p\u00e1gina voc\u00ea descobrir\u00e1 o que s\u00e3o pontos coplanares (ou coplanares) e como saber se determinados pontos s\u00e3o coplanares ou n\u00e3o. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos e praticar com exerc\u00edcios de pontos coplanares resolvidos. O que s\u00e3o pontos coplanares? Na geometria anal\u00edtica, o significado dos pontos coplanares (ou coplanares) \u00e9 o seguinte: Pontos coplanares &hellip; Pontos coplanares (ou coplanares) Leia mais &raquo;","og_url":"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/","article_published_time":"2023-07-10T06:51:49+00:00","og_image":[{"url":"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/points-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\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/#article","isPartOf":{"@id":"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/"},"author":{"name":"Equipe Mathoridade","@id":"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00"},"headline":"Pontos coplanares (ou coplanares)","datePublished":"2023-07-10T06:51:49+00:00","dateModified":"2023-07-10T06:51:49+00:00","mainEntityOfPage":{"@id":"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/"},"wordCount":816,"commentCount":0,"publisher":{"@id":"https:\/\/mathority.org\/pt\/#organization"},"articleSection":["Pontos, retas e planos"],"inLanguage":"pt-BR","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/","url":"https:\/\/mathority.org\/pt\/exemplos-de-pontos-coplanares-ou-coplanares-exercicios-resolvidos\/","name":"Pontos coplanares (ou coplanares) - 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