{"id":244,"date":"2023-07-10T14:01:40","date_gmt":"2023-07-10T14:01:40","guid":{"rendered":"https:\/\/mathority.org\/pt\/equacao-geral-ou-cartesiana-implicita-do-plano\/"},"modified":"2023-07-10T14:01:40","modified_gmt":"2023-07-10T14:01:40","slug":"equacao-geral-ou-cartesiana-implicita-do-plano","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/equacao-geral-ou-cartesiana-implicita-do-plano\/","title":{"rendered":"Equa\u00e7\u00e3o impl\u00edcita, geral ou cartesiana do plano"},"content":{"rendered":"<p>Explica\u00e7\u00e3o de como a equa\u00e7\u00e3o plana impl\u00edcita (f\u00f3rmula), tamb\u00e9m conhecida como equa\u00e7\u00e3o geral ou cartesiana, \u00e9 calculada. Al\u00e9m disso, voc\u00ea descobrir\u00e1 como encontrar a equa\u00e7\u00e3o do plano a partir de seu vetor normal. E ainda mais, voc\u00ea poder\u00e1 ver exemplos e exerc\u00edcios resolvidos passo a passo. <\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-104\"><\/div>\n<\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-la-ecuacion-implicita-o-general-del-plano\"><\/span> Qual \u00e9 a equa\u00e7\u00e3o impl\u00edcita ou geral do plano? <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/div>\n<\/div>\n<p> Na geometria anal\u00edtica, a <strong>equa\u00e7\u00e3o impl\u00edcita de um plano<\/strong> , tamb\u00e9m chamada de equa\u00e7\u00e3o <strong>geral<\/strong> ou <strong>cartesiana<\/strong> do plano, \u00e9 uma equa\u00e7\u00e3o que permite expressar matematicamente qualquer plano. Para encontrar a equa\u00e7\u00e3o impl\u00edcita ou geral de um plano, precisamos de um ponto e de dois vetores linearmente independentes pertencentes a esse plano. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-de-la-ecuacion-implicita-o-general-del-plano\"><\/span> F\u00f3rmula da equa\u00e7\u00e3o impl\u00edcita ou geral do plano <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><\/div>\n<\/div>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> Considere um ponto e dois vetores de dire\u00e7\u00e3o de um plano:<\/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-cf5d4130501bb01b15aa80f8f80caf1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c} P(P_x,P_y,P_z) \\\\[2ex] \\vv{\\text{u}}=(\\text{u}_x,\\text{u}_y,\\text{u}_z)\\\\[2ex] \\vv{\\text{v}}=(\\text{v}_x,\\text{v}_y,\\text{v}_z)\\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"95\" width=\"116\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> A equa\u00e7\u00e3o impl\u00edcita, geral ou cartesiana de um plano \u00e9 obtida resolvendo o seguinte determinante e igualando o resultado a 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-68d67612dfa54d76666aa37b702a472f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"text-align:left;\"> Assim, a <strong>equa\u00e7\u00e3o impl\u00edcita ou geral do plano resultante<\/strong> ser\u00e1 a seguinte:<\/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-27e298e3103f917bd81b20315b6d9025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"183\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<\/div>\n<p> \u00c9 importante que os dois vetores da f\u00f3rmula sejam linearmente independentes um do outro, ou seja, devem ter dire\u00e7\u00f5es diferentes. E para que esta condi\u00e7\u00e3o seja satisfeita basta que os dois vetores n\u00e3o sejam paralelos. <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/equations-planes.webp\" alt=\"equa\u00e7\u00e3o impl\u00edcita ou geral ou cartesiana de pan xy em r3\" class=\"wp-image-2443\" width=\"393\" height=\"138\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Embora n\u00e3o seja necess\u00e1rio saber o motivo desta f\u00f3rmula, voc\u00ea pode ver sua demonstra\u00e7\u00e3o a seguir.<\/p>\n<p> Partindo das equa\u00e7\u00f5es param\u00e9tricas de um plano, passaremos para a equa\u00e7\u00e3o impl\u00edcita (ou geral) do plano:<\/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-46f87775f11f01a59c70aa3ee864aebe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases}x=P_x + \\lambda \\text{u}_x + \\mu \\text{v}_x \\\\[1.7ex] y=P_y + \\lambda \\text{u}_y + \\mu \\text{v}_y\\\\[1.7ex] z=P_z + \\lambda\\text{u}_z + \\mu \\text{v}_z \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"167\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Primeiro, passamos o termo independente de cada equa\u00e7\u00e3o param\u00e9trica para o outro lado da equa\u00e7\u00e3o:<\/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-0c2f3831ca03939d7e23d24c7d435337_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases}x-P_x= \\lambda \\text{u}_x + \\mu \\text{v}_x \\\\[1.7ex] y-P_y = \\lambda \\text{u}_y + \\mu \\text{v}_y\\\\[1.7ex] z-P_z = \\lambda\\text{u}_z + \\mu \\text{v}_z \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"167\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ou equivalente:<\/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-aca0b16ff9b92401181c2bdc5ba981bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{cases} \\lambda \\text{u}_x + \\mu \\text{v}_x =x-P_x\\\\[1.7ex]  \\lambda \\text{u}_y + \\mu \\text{v}_y=y-P_y \\\\[1.7ex]  \\lambda\\text{u}_z + \\mu \\text{v}_z =z-P_z\\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"107\" width=\"166\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Para que o sistema de equa\u00e7\u00f5es acima tenha uma solu\u00e7\u00e3o vi\u00e1vel, a classifica\u00e7\u00e3o da seguinte matriz deve ser igual a 2 (teorema de Rouche-Frobenius):<\/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-6f802b760ba5ab681afd0f02c83eddb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{pmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z\\end{pmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"142\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, se o contradom\u00ednio da matriz anterior deve ser dois, o determinante 3&#215;3 deve necessariamente 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-917f1770ff2a17897e5df76998ec3519_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E resolvendo este determinante, obtemos a equa\u00e7\u00e3o geral, impl\u00edcita ou cartesiana de um plano:<\/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-27e298e3103f917bd81b20315b6d9025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"183\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Assim, acabamos de ver a equa\u00e7\u00e3o impl\u00edcita (ou geral) e as equa\u00e7\u00f5es param\u00e9tricas do plano, por\u00e9m, existem ainda mais maneiras de expressar um plano analiticamente, como a equa\u00e7\u00e3o vetorial e a equa\u00e7\u00e3o can\u00f4nica. Voc\u00ea pode ver a f\u00f3rmula e a explica\u00e7\u00e3o de todas as <a href=\"https:\/\/mathority.org\/pt\/equacoes-planas-no-espaco\/\">equa\u00e7\u00f5es do plano<\/a> neste link. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-como-hallar-la-ecuacion-implicita-o-general-del-plano\"><\/span> Exemplo de como encontrar a equa\u00e7\u00e3o impl\u00edcita ou geral do plano <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-109\"><\/div>\n<\/div>\n<p> Vejamos como determinar a equa\u00e7\u00e3o impl\u00edcita (ou geral ou cartesiana) de um plano atrav\u00e9s de um exemplo:<\/p>\n<ul>\n<li> Encontre a equa\u00e7\u00e3o impl\u00edcita ou geral do plano que passa pelo ponto\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2118d788ea1b10a70b36f284857de70e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3,1,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<p> e cont\u00e9m os vetores<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-ef85b7409667d52ad3c9f8981dad5f7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}=(2,0,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -5px;\"><\/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-a50e73125c9dcf4a9c58d316fbf850ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(4,-1,2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<p> Para calcular a equa\u00e7\u00e3o geral ou impl\u00edcita do plano \u00e9 necess\u00e1rio resolver o seguinte determinante formado pelos dois vetores, as vari\u00e1veis e as coordenadas do ponto:<\/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-917f1770ff2a17897e5df76998ec3519_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ent\u00e3o, substitu\u00edmos os vetores e o ponto na f\u00f3rmula:<\/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-aa25223c3a00e31f89043a3500d32c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}2 &amp; 4 &amp; x-3 \\\\[1.1ex]0 &amp; -1 &amp; y-1 \\\\[1.1ex]3&amp; 2 &amp; z-(-1) \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"173\" 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-7886f1c2758c204802b96f44acc8a7cd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}2 &amp; 4 &amp; x-3 \\\\[1.1ex]0 &amp; -1 &amp; y-1 \\\\[1.1ex]3&amp; 2 &amp; z+1 \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"147\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> E agora resolvemos o determinante de ordem 3, por exemplo com a regra de Sarrus ou por cofatores (ou deputados):<\/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-3d8c2ae50c69efa75c6f439ec502a6d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2(z+1)+12(y-1)+3(x-3)-4(y-1) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"370\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Agora operamos e agrupamos os termos: <\/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-e9cfe2d167a92bc1e64737ce9e7a5ed2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3(x-3)+8(y-1) -2(z+1) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"264\" 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-3cf3da637bf3a8d04fb2b8d791d78a9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x-9+8y-8 -2z-2 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"223\" style=\"vertical-align: -4px;\"><\/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-65167b5921553dd9e11f9ac9bb80864e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x+8y-2z-19 = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"170\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Portanto, a equa\u00e7\u00e3o impl\u00edcita ou geral do plano \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-152a3c9fd09fb72b6269201f02fb8303_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{3x+8y-2z-19 = 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"170\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"calcular-la-ecuacion-implicita-o-general-de-un-plano-a-partir-de-su-vector-normal\"><\/span> Calcule a equa\u00e7\u00e3o impl\u00edcita ou geral de um plano a partir de seu vetor normal<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Um problema muito t\u00edpico nas equa\u00e7\u00f5es de um plano \u00e9 descobrir como \u00e9 a equa\u00e7\u00e3o de um determinado plano, dado um ponto e seu vetor normal (ou perpendicular). Ent\u00e3o, vamos ver como funciona.<\/p>\n<p> Mas primeiro voc\u00ea deve saber que <strong>as componentes X, Y, Z do vetor normal a um plano coincidem <strong>respectivamente<\/strong><\/strong> <strong>com os coeficientes A, B, C da equa\u00e7\u00e3o impl\u00edcita (ou geral) desse plano.<\/strong><\/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-27f3ee5d7e81864550f3b86fdd53e89d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\color{orange} \\boxed{ \\color{black} \\quad \\pi : \\ Ax+By+C+D = 0 \\quad \\iff \\quad \\vv{n} = (A,B,C) \\quad \\vphantom{\\Bigl(}}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"540\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Ouro<\/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-10affe1faee06a5faa4ef6d9c0473b1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 o vetor ortogonal ao plano<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-26622dd58bf71cd1b543c3d83233c561_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"15\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Uma vez conhecida a rela\u00e7\u00e3o anterior, vamos ver um exemplo de resolu\u00e7\u00e3o deste tipo de problemas de equa\u00e7\u00f5es planas:<\/p>\n<ul>\n<li> Determine a equa\u00e7\u00e3o impl\u00edcita ou geral do plano que passa pelo ponto\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6df0e548515bb2b24f352853a2614015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,0,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<p> e um de seus vetores normais \u00e9<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-44298f830c420011a4326017f5fd7cfb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(3,-1,2) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"108\" style=\"vertical-align: -5px;\"><\/p>\n<\/li>\n<\/ul>\n<p> A f\u00f3rmula para a equa\u00e7\u00e3o impl\u00edcita, geral ou cartesiana de um plano \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-27e298e3103f917bd81b20315b6d9025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"183\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Assim, a partir do vetor normal, podemos encontrar os coeficientes A, B e C porque s\u00e3o equivalentes \u00e0s componentes do seu vetor normal:<\/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-596b3fdc65160234c06b0d28aebea74f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(3,-1,2) \\ \\longrightarrow \\ 3x-1y+2z+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"322\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Embora precisemos apenas encontrar o par\u00e2metro D. Para isso, substitu\u00edmos as coordenadas do ponto que pertence ao plano na equa\u00e7\u00e3o: <\/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-6df0e548515bb2b24f352853a2614015_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(1,0,-2)\" 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-09a0b099394ba4634fb6d7aad3a627e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3\\cdot 1-0+2\\cdot (-2)+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"210\" 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-b90d57a1708925626a300c7e5db673ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3-4+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"109\" style=\"vertical-align: -2px;\"><\/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-d2dfb99a77bfcee1b72540db9cf579a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"91\" style=\"vertical-align: -2px;\"><\/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-16ace3f6683252e5630a1091bbc0404e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D=1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Portanto, a equa\u00e7\u00e3o impl\u00edcita ou geral do plano \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-cde6db2f935eb7d0ad39141f86aab013_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{3x-y+2z+1 = 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"153\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-la-ecuacion-implicita-o-general-del-plano\"><\/span> Problemas resolvidos da equa\u00e7\u00e3o impl\u00edcita ou geral do plano<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Encontre a equa\u00e7\u00e3o impl\u00edcita ou geral do plano que passa pelo ponto<\/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-b61f338575699a918d594595ddc6fb02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-2,1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<p> e cont\u00e9m os vetores<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5150cc543174c7c079a38b016685eb3b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}=(4,1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -5px;\"><\/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-0664398c6f1938e4ec5efaae48ab1c70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(5,3,-1).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"107\" 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 calcular a equa\u00e7\u00e3o geral ou impl\u00edcita do plano \u00e9 necess\u00e1rio resolver o seguinte determinante formado pelos dois vetores, pelas tr\u00eas vari\u00e1veis e pelas coordenadas do ponto:<\/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-917f1770ff2a17897e5df76998ec3519_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Ent\u00e3o, substitu\u00edmos os vetores e o ponto na f\u00f3rmula:<\/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-02e103601cd9992a8a8c087d016a08c1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}4 &amp; 5 &amp; x+2 \\\\[1.1ex]1 &amp; 3 &amp; y-1 \\\\[1.1ex]3&amp; 1 &amp; z+1 \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"133\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E agora resolvemos o determinante da matriz 3\u00d73 com o m\u00e9todo de sua escolha:<\/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-ec35e71b9cca25aa9907c97da2ea2e2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"12(z+1)+15(y-1)+1(x+2)-9(x+2)-4(y-1)-5(z+1) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"534\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Por fim, realizamos as opera\u00e7\u00f5es e agrupamos termos semelhantes: <\/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-fea5513ce57c88cac89a695b47b7a0c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-8(x+2)+11(y-1)+7(z+1) = 0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" 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-b965060b2baad7f1b7fa66e0555a23f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-8x-16+11y-11+7z+7=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"262\" style=\"vertical-align: -4px;\"><\/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-52635b24c2d396ff9a47fbd210c56bc4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-8x+11y+7z-20= 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"192\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, a equa\u00e7\u00e3o impl\u00edcita ou geral do plano \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-fc97887dad7e25ff823eee155fb58358_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{-8x+11y+7z-20 = 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"192\" style=\"vertical-align: -4px;\"><\/p>\n<\/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 o ponto<\/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-10de27107c1cf63dc889433e271d4a78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1,5,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"><\/p>\n<p> pertence ao seguinte plano: <\/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-54a545de55c14f27d77bfda0188789a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi : \\ 2x+y+6z-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"184\" style=\"vertical-align: -4px;\"><\/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 o ponto esteja no plano, sua equa\u00e7\u00e3o deve ser verificada. Portanto, precisamos substituir as coordenadas cartesianas do ponto na equa\u00e7\u00e3o do plano e verificar se a equa\u00e7\u00e3o \u00e9 cumprida: <\/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-25ef56f6218537e6592c6ce17e0c3cb0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2x+y+6z-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"153\" style=\"vertical-align: -4px;\"><\/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-10de27107c1cf63dc889433e271d4a78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(-1,5,-3)\" 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-b30e787cb2dfac5e87b0758b866e10a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\cdot (-1)+5+6\\cdot (-3)-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"232\" 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-66acebc4d56a2fb70fe23b2ead99dcc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-2+5-18-5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"155\" style=\"vertical-align: -2px;\"><\/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-64706704bc6f3ef930293722159a8861_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-20\\neq 0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"63\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O ponto n\u00e3o respeita a equa\u00e7\u00e3o do plano, <strong>portanto n\u00e3o faz parte deste plano.<\/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> Encontre a equa\u00e7\u00e3o impl\u00edcita (ou geral) do plano que cont\u00e9m os tr\u00eas pontos a seguir: <\/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-c3003c4a812ae18bebda7f61c0bbe5f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(5,-1,-2) \\qquad B(2,1,3) \\qquad C(4,1,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"322\" 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 encontrar a equa\u00e7\u00e3o impl\u00edcita do plano, precisamos encontrar dois vetores linearmente independentes que se ligam no plano. E, para isso, podemos calcular dois vetores que s\u00e3o definidos pelos 3 pontos: <\/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-f96add9ffd85a60c66b7b40a65192537_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B - A = (2,1,3) - (5,-1,-2) = (-3,2,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"379\" 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-09c96a38b66a9b0108da71a2c7ea0b2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AC} = C - A = (4,1,-2) - (5,-1,-2) = (-1,2,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"392\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> As coordenadas dos dois vetores encontrados n\u00e3o s\u00e3o proporcionais, portanto s\u00e3o efetivamente linearmente independentes um do outro.<\/p>\n<p class=\"has-text-align-left\"> Agora j\u00e1 conhecemos dois vetores dire\u00e7\u00f5es e um ponto do plano, ent\u00e3o j\u00e1 podemos aplicar a f\u00f3rmula da equa\u00e7\u00e3o geral do plano:<\/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-917f1770ff2a17897e5df76998ec3519_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}\\text{u}_x &amp; \\text{v}_x &amp; x-P_x \\\\[1.1ex]\\text{u}_y &amp; \\text{v}_y &amp; y-P_y \\\\[1.1ex]\\text{u}_z &amp; \\text{v}_z &amp; z-P_z \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"163\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Substitu\u00edmos os vetores e um dos tr\u00eas pontos na f\u00f3rmula:<\/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-383093e607bc8ecc5f99e1815242b22a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle\\begin{vmatrix}-3 &amp; -1 &amp; x-5 \\\\[1.1ex]2 &amp; 2 &amp; y+1 \\\\[1.1ex]5&amp; 0 &amp; z+2 \\end{vmatrix} =0\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"161\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, finalmente, resolvemos o determinante: <\/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-a3d7401055c280280af3a3d0486cdb86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-6(z+2)-5(y+1)-10(x-5)+2(z+2)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"370\" 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-4e355f06e9bd200804e0a37e059a389c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-10(x-5)-5(y+1)-4(z+2)=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"286\" 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-0f005d4a2fdd910e746e7231a1c83f61_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-10x+50-5y-5-4z-8=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"253\" style=\"vertical-align: -4px;\"><\/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-7653b13f379b9d38230f457c59d1a568_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-10x-5y-4z+37=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"192\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resumindo, a equa\u00e7\u00e3o impl\u00edcita, geral ou cartesiana do plano em quest\u00e3o \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-ac148d30ada4a3e256874712b4b196db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{-10x-5y-4z+37=0}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"192\" style=\"vertical-align: -4px;\"><\/p>\n<\/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<div class=\"wp-block-group\">\n<div class=\"wp-block-group__inner-container is-layout-flow\">\n<p> Calcula a equa\u00e7\u00e3o impl\u00edcita ou geral do plano no espa\u00e7o que passa pelo ponto<\/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-e42bc8fa114f50a19858a526eabb6e30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3,4,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"83\" style=\"vertical-align: -5px;\"><\/p>\n<p> e um de seus vetores normais \u00e9 <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b007924d3ec5c7cd5de5d1d46cc86711_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(5,-2,-3) .\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"122\" 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\"> A f\u00f3rmula para a equa\u00e7\u00e3o impl\u00edcita, geral ou cartesiana de um plano \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-27e298e3103f917bd81b20315b6d9025_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"183\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Bom, a partir do vetor normal podemos encontrar os coeficientes A, B e C, pois s\u00e3o respectivamente iguais \u00e0s componentes do vetor normal:<\/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-8f4b71cc8d8c1610a5d5706ac44d1ad3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}=(5,-2,-3) \\ \\longrightarrow \\ 5x-2y-3z+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"336\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, s\u00f3 precisamos encontrar o par\u00e2metro D. Para isso, substitu\u00edmos as coordenadas do ponto que pertence ao plano na equa\u00e7\u00e3o: <\/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-e42bc8fa114f50a19858a526eabb6e30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3,4,-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-720c689bc073e6c0f865bf406d92cbba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"5\\cdot 3-2\\cdot 4-3\\cdot (-3)+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"232\" 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-3edf41b26a3b9c8bdead74d05767ec60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"15-8+9+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"147\" style=\"vertical-align: -2px;\"><\/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-0db457697a3637eb92bff90460d8e98f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"16+D=0\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"86\" style=\"vertical-align: -2px;\"><\/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-a2efb3d90e15e4474268d6de0570da4b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"D=-16\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"71\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Concluindo, a equa\u00e7\u00e3o impl\u00edcita ou geral do plano \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-ee0c114a24764dd24df9d58aab7155ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{5x-2y-3z-16 = 0}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"170\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Explica\u00e7\u00e3o de como a equa\u00e7\u00e3o plana impl\u00edcita (f\u00f3rmula), tamb\u00e9m conhecida como equa\u00e7\u00e3o geral ou cartesiana, \u00e9 calculada. Al\u00e9m disso, voc\u00ea descobrir\u00e1 como encontrar a equa\u00e7\u00e3o do plano a partir de seu vetor normal. E ainda mais, voc\u00ea poder\u00e1 ver exemplos e exerc\u00edcios resolvidos passo a passo. Qual \u00e9 a equa\u00e7\u00e3o impl\u00edcita ou geral do plano? &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/equacao-geral-ou-cartesiana-implicita-do-plano\/\"> <span class=\"screen-reader-text\">Equa\u00e7\u00e3o impl\u00edcita, geral ou cartesiana do plano<\/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-244","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>Equa\u00e7\u00e3o impl\u00edcita, geral ou cartesiana do plano - Mathoridade<\/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\/equacao-geral-ou-cartesiana-implicita-do-plano\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Equa\u00e7\u00e3o impl\u00edcita, geral ou cartesiana do plano - Mathoridade\" \/>\n<meta property=\"og:description\" content=\"Explica\u00e7\u00e3o de como a equa\u00e7\u00e3o plana impl\u00edcita (f\u00f3rmula), tamb\u00e9m conhecida como equa\u00e7\u00e3o geral ou cartesiana, \u00e9 calculada. 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