{"id":265,"date":"2023-07-10T04:04:14","date_gmt":"2023-07-10T04:04:14","guid":{"rendered":"https:\/\/mathority.org\/pt\/geometria-plana\/"},"modified":"2023-07-10T04:04:14","modified_gmt":"2023-07-10T04:04:14","slug":"geometria-plana","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/geometria-plana\/","title":{"rendered":"O plano (geometria)"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 uma explica\u00e7\u00e3o do que \u00e9 um plano, como ele \u00e9 calculado e todas as suas propriedades. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de planos, quais s\u00e3o as posi\u00e7\u00f5es relativas entre dois planos, como determinar o \u00e2ngulo entre 2 planos e, por fim, como expressar numericamente qualquer plano usando as equa\u00e7\u00f5es dos planos.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-un-plano\"><\/span> O que \u00e9 um plano?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Na geometria anal\u00edtica, a defini\u00e7\u00e3o do plano \u00e9 a seguinte:<\/p>\n<p> <strong>Um plano \u00e9 um objeto geom\u00e9trico que possui duas dimens\u00f5es (comprimento e largura).<\/strong> <\/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\/quel-est-l-avion.webp\" alt=\"qual \u00e9 o plano cartesiano\" class=\"wp-image-3433\" width=\"417\" height=\"203\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Portanto, um plano cont\u00e9m infinitas retas e infinitos pontos. Na representa\u00e7\u00e3o gr\u00e1fica acima voc\u00ea pode ver a diferen\u00e7a entre um plano, uma linha e um ponto. Voc\u00ea tamb\u00e9m pode verificar se a linha<\/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-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> e a dica<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"><\/p>\n<p> est\u00e3o contidos no 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> Como voc\u00ea pode ver no plano gr\u00e1fico, os planos geralmente s\u00e3o nomeados com letras gregas:<\/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-fdef793ea5450647d6139c80d45be77a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi, \\alpha, \\beta,...\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"71\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Um exemplo de plano que usamos muito em matem\u00e1tica \u00e9 o plano cartesiano. O plano cartesiano \u00e9 o plano definido pelo eixo das abscissas (eixo X) e pelo eixo das ordenadas (eixo Y). Um dos usos do plano cartesiano \u00e9 que ele \u00e9 usado para descrever a posi\u00e7\u00e3o de um objeto em um sistema de refer\u00eancia.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"determinacion-de-un-plano\"><\/span>Determinando um plano<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Agora que vimos o significado de plano, vamos ver como qualquer plano no espa\u00e7o tridimensional (em R3) pode ser determinado.<\/p>\n<p> Um plano \u00e9 inteiramente determinado pelos seguintes elementos geom\u00e9tricos:<\/p>\n<ul>\n<li> Tr\u00eas pontos n\u00e3o alinhados.<\/li>\n<li> Uma linha reta e um ponto externo.<\/li>\n<li> Duas linhas paralelas ou duas linhas que se cruzam.<\/li>\n<\/ul>\n<p> Em rela\u00e7\u00e3o ao \u00faltimo ponto, voc\u00ea provavelmente j\u00e1 sabe o que significa duas retas serem paralelas. Mas o significado das retas secantes \u00e9 menos conhecido, ent\u00e3o se voc\u00ea tiver alguma d\u00favida aqui, voc\u00ea pode conferir <a href=\"https:\/\/mathority.org\/pt\/exemplos-de-linhas-que-se-cruzam\/\">o que s\u00e3o retas secantes<\/a> .<\/p>\n<p> Portanto, se tivermos alguma das 3 condi\u00e7\u00f5es anteriores, significa que podemos tra\u00e7ar um plano.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-del-plano\"><\/span> propriedades do plano<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> O plano atende \u00e0s seguintes caracter\u00edsticas:<\/p>\n<ul>\n<li> Um plano cont\u00e9m uma infinidade de pontos.<\/li>\n<li> Um plano cont\u00e9m uma infinidade de linhas.<\/li>\n<li> Um plano \u00e9 ilimitado, ou seja, \u00e9 uma superf\u00edcie que se estende no espa\u00e7o sem limites.<\/li>\n<li> Dois planos que se cruzam determinam uma linha.<\/li>\n<li> Uma reta que tem um ponto em um plano n\u00e3o est\u00e1 necessariamente contida nele. Para que uma reta fa\u00e7a parte de um plano, ela deve ter pelo menos dois pontos no plano.<\/li>\n<li> Planos infinitos cruzam uma linha reta.<\/li>\n<li> Um semiplano \u00e9 cada uma das 2 partes em que um plano \u00e9 dividido quando \u00e9 cortado por uma de suas linhas. <\/li>\n<\/ul>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuaciones-del-plano\"><\/span> equa\u00e7\u00f5es planas<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Na geometria anal\u00edtica, a <strong>equa\u00e7\u00e3o de um plano<\/strong> \u00e9 uma equa\u00e7\u00e3o que permite que qualquer plano seja expresso matematicamente. Ent\u00e3o, para encontrar a equa\u00e7\u00e3o de um plano, voc\u00ea s\u00f3 precisa de um ponto e dois vetores linearmente independentes pertencentes a esse plano. <\/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 do plano xy on-line\" class=\"wp-image-2443\" width=\"404\" height=\"142\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Por\u00e9m, como vimos acima na explica\u00e7\u00e3o do conceito de plano, existem diversas formas de determinar um plano. Bem, da mesma forma, tamb\u00e9m existem diferentes maneiras de expressar analiticamente um plano.<\/p>\n<p> Assim todos os tipos de equa\u00e7\u00f5es do plano s\u00e3o: a <strong>equa\u00e7\u00e3o vetorial<\/strong> , as <strong>equa\u00e7\u00f5es param\u00e9tricas<\/strong> , a <strong>equa\u00e7\u00e3o impl\u00edcita (ou geral)<\/strong> e a <strong>equa\u00e7\u00e3o can\u00f4nica (ou segmental)<\/strong> do plano.<\/p>\n<p> A seguir veremos detalhadamente a explica\u00e7\u00e3o e f\u00f3rmula de todas as equa\u00e7\u00f5es do plano. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-vectorial-del-plano\"><\/span> Equa\u00e7\u00e3o vetorial do plano<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> 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> A <strong>f\u00f3rmula para a equa\u00e7\u00e3o vetorial de um plano<\/strong> \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-d9227901692832cb0c176a896d35e896_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      (x,y,z)=P+\\lambda \\vv{\\text{u}} + \\mu \\vv{\\text{v}} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/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-78b41d21b63c22ec05d3f93576a897e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(x,y,z)=(P_x,P_y,P_z)+\\lambda (\\text{u}_x,\\text{u}_y,\\text{u}_z) + \\mu (\\text{v}_x,\\text{v}_y,\\text{v}_z)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"398\" style=\"vertical-align: -6px;\"><\/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-2b5c45836864531b8e37025dabadd24a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" 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-461fe1a58a75801541487ddf10d32abd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -4px;\"><\/p>\n<p> s\u00e3o dois escalares, ou seja, dois n\u00fameros reais. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuaciones-parametricas-del-plano\"><\/span> Equa\u00e7\u00f5es param\u00e9tricas do plano<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A f\u00f3rmula para a <strong>equa\u00e7\u00e3o param\u00e9trica de um plano<\/strong> \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-e1791802331aa9973126b3d7c7f1b716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\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} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"313\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Ouro:<\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2b5c45836864531b8e37025dabadd24a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lambda\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" 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-461fe1a58a75801541487ddf10d32abd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\mu\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"11\" style=\"vertical-align: -4px;\"><\/p>\n<p> s\u00e3o dois escalares, ou seja, dois n\u00fameros reais.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0b77e9af6839a6bc60da39dd1798dd6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{u}_x,\\text{u}_y,\\text{u}_z\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"69\" style=\"vertical-align: -6px;\"><\/p>\n<p> s\u00e3o os componentes de um dos dois vetores norteadores do plano<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b8a3eef2109d6a80a337c88337a1443e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}=(\\text{u}_x,\\text{u}_y,\\text{u}_z).\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"121\" style=\"vertical-align: -6px;\"><\/p>\n<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f61b32275ccdbca7f8d5e0b3c750dd35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\text{v}_x,\\text{v}_y,\\text{v}_z\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"67\" style=\"vertical-align: -6px;\"><\/p>\n<p> s\u00e3o os componentes do outro vetor diretor do plano <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-75c6a57a037206319f16dec389993ded_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(\\text{v}_x,\\text{v}_y,\\text{v}_z).\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"119\" style=\"vertical-align: -6px;\"><\/p>\n<\/li>\n<\/ul>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-implicita-o-general-del-plano\"><\/span> Equa\u00e7\u00e3o impl\u00edcita ou geral do plano<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> 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> 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> 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-7dcacf16123986ecd33dace4f4411914_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\displaystyle Ax+By+Cz+D=0 \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Este tipo de equa\u00e7\u00e3o plana tamb\u00e9m \u00e9 chamada de equa\u00e7\u00e3o plana cartesiana. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ecuacion-canonica-o-segmentaria-del-plano\"><\/span> Equa\u00e7\u00e3o can\u00f4nica ou segmentar do plano<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>f\u00f3rmula para a equa\u00e7\u00e3o can\u00f4nica ou segmentar de um plano<\/strong> \u00e9 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-7c19853d465a703aa398bde04fa3222c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\displaystyle \\cfrac{x}{a}+\\cfrac{y}{b} + \\cfrac{z}{c} = 1  \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Ouro:<\/p>\n<ul>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5c53d6ebabdbcfa4e107550ea60b1b19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 o ponto de intersec\u00e7\u00e3o entre o plano e o eixo X.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f56d50c26583f9a035ff6b4e3c0ca5c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 o ponto de intersec\u00e7\u00e3o entre o plano e o eixo Y.<\/li>\n<li>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-41a04eeea923a1a0c28094a8a4680525_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"c\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00c9 aqui que o plano cruza o eixo Z. <\/li>\n<\/ul>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><\/figure>\n<\/div>\n<p> A equa\u00e7\u00e3o can\u00f4nica (ou equa\u00e7\u00e3o segmental) do plano tamb\u00e9m pode ser obtida a partir de sua equa\u00e7\u00e3o geral:<\/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> Primeiro, resolvemos o coeficiente D 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-7e6829185741f883a29bf004cbf570a7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"Ax+By+Cz=-D\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"166\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Em seguida, dividimos toda a equa\u00e7\u00e3o do plano pelo valor do par\u00e2metro D com sinal alterado:<\/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-04843e22b4176c0ce921483f93dffeab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{Ax+By+Cz}{-D}=\\cfrac{-D}{-D}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"176\" style=\"vertical-align: -12px;\"><\/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-278ce62f85ca44612254f48e96154726_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{Ax}{-D}+\\cfrac{By}{-D}+\\cfrac{Cz}{-D}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"166\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> E, utilizando as propriedades das fra\u00e7\u00f5es, chegamos \u00e0 seguinte express\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-a58254e773c7c14b5b337a4330997125_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{x}{-\\frac{D}{A}}+\\cfrac{y}{-\\frac{D}{A}}+\\cfrac{z}{-\\frac{D}{A}}=1\" title=\"Rendered by QuickLaTeX.com\" height=\"42\" width=\"167\" style=\"vertical-align: -20px;\"><\/p>\n<\/p>\n<p> Deduzimos, portanto, desta express\u00e3o as f\u00f3rmulas que permitem calcular diretamente os termos da equa\u00e7\u00e3o can\u00f4nica ou segmentar 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-86975df14352b1b0c2ca05d2daaf40f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a=-\\cfrac{D}{A} \\qquad b=-\\cfrac{D}{B} \\qquad c=-\\cfrac{D}{C}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"260\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> conseq\u00fcentemente, para poder formar esta variante das equa\u00e7\u00f5es do plano, os coeficientes A, B e C devem ser diferentes de zero, evitando assim indetermina\u00e7\u00f5es das fra\u00e7\u00f5es. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"posicion-relativa-de-dos-planos\"><\/span> Posi\u00e7\u00e3o relativa de dois planos<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Na geometria anal\u00edtica, existem apenas tr\u00eas posi\u00e7\u00f5es relativas poss\u00edveis entre dois planos: planos secantes, planos paralelos e planos coincidentes.<\/p>\n<ul>\n<li> <strong>Planos que se cruzam<\/strong> : Dois planos est\u00e3o se cruzando se eles se cruzarem apenas em uma linha.<\/li>\n<li> <strong>Planos paralelos<\/strong> : Dois planos s\u00e3o paralelos se n\u00e3o se cruzam em nenhum ponto.<\/li>\n<li> <strong>Planos Coincidentes<\/strong> : Dois planos s\u00e3o coincidentes se todos tiverem pontos em comum. <\/li>\n<\/ul>\n<div class=\"wp-block-columns is-layout-flex wp-container-30\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#ff6f00\"> <strong>planos que se cruzam<\/strong> <\/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\/plans-secants.webp\" alt=\"posi\u00e7\u00e3o relativa de dois planos que se cruzam\" class=\"wp-image-2814\" width=\"265\" height=\"258\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#ff6f00\"> <strong>planos paralelos<\/strong> <\/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\/plans-paralleles-1.webp\" alt=\"posi\u00e7\u00e3o relativa de dois planos paralelos\" class=\"wp-image-2815\" width=\"266\" height=\"166\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center has-text-color has-medium-font-size\" style=\"color:#ff6f00\"> <strong>planos coincidentes<\/strong> <\/p>\n<figure class=\"wp-block-image size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/deux-avions-coincidents.webp\" alt=\"posi\u00e7\u00e3o relativa de dois planos coincidentes\" class=\"wp-image-2820\" width=\"294\" height=\"83\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<\/div>\n<p> Al\u00e9m disso, se dois planos que se cruzam se cruzam em um \u00e2ngulo de 90\u00ba, s\u00e3o dois <strong>planos perpendiculares entre si<\/strong> .<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"angulo-entre-dos-planos\"><\/span> \u00c2ngulo entre dois planos<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> O \u00e2ngulo entre dois planos \u00e9 igual ao \u00e2ngulo formado pelos vetores normais desses planos. Portanto, <strong>para encontrar o \u00e2ngulo entre dois planos, calcula-se o \u00e2ngulo formado por seus vetores normais, uma vez que s\u00e3o equivalentes.<\/strong><\/p>\n<p> Ent\u00e3o, uma vez que sabemos exatamente em que consiste o \u00e2ngulo entre dois planos, vamos ver a f\u00f3rmula para calcular o \u00e2ngulo entre dois planos no espa\u00e7o, que \u00e9 deduzida da f\u00f3rmula do \u00e2ngulo entre dois vetores:<\/p>\n<p> Dada a equa\u00e7\u00e3o geral (ou impl\u00edcita) de dois planos diferentes:<\/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-dfa3d7e6f1ece8353327be7c9227d75b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_1 : \\ A_1x+B_1y+C_1z+D_1=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"249\" 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-2c3966346685421fe3e535cf57a5491d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\pi_2 : \\ A_2x+B_2y+C_2z+D_2=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"249\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> O vetor normal de cada 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-eb0ca06882e0d61d6f8134368946ef29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_1=(A_1,B_1,C_1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" 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-22fba6a063a544bdf257e64d8d139238_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{n}_2=(A_2,B_2,C_2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> E o \u00e2ngulo formado por esses dois planos \u00e9 determinado calculando o \u00e2ngulo formado por seus vetores normais usando a seguinte 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-a0329572a30e8d75bd3795469fe65493_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\definecolor{taronjaquadreejemplo}{HTML}{FF9800}  \\newtcbox{\\mymath}[1][]{%     nobeforeafter, math upper, tcbox raise base,     enhanced, colframe=taronjaquadreejemplo,      boxrule=1.1pt, boxsep=2mm,     #1} \\begin{empheq}[box={\\mymath[colback=white, shadow={2mm}{-2mm}{0mm}{taronjaquadreejemplo!20!white,} ]}]{equation*}      \\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{n}_1 \\cdot \\vv{n}_2\\rvert}{\\lvert \\vv{n}_1 \\rvert \\cdot \\lvert \\vv{n}_2 \\rvert} \\end{empheq}\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"329\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Obviamente, uma vez calculado o cosseno do \u00e2ngulo formado pelos dois planos a partir da f\u00f3rmula, devemos inverter o cosseno para encontrar o valor desse \u00e2ngulo.<\/p>\n<p> Por outro lado, quando os dois planos s\u00e3o perpendiculares ou paralelos, n\u00e3o \u00e9 necess\u00e1rio aplicar a f\u00f3rmula, pois o \u00e2ngulo entre os 2 planos pode ser determinado diretamente:<\/p>\n<ul>\n<li> O \u00e2ngulo entre dois planos paralelos \u00e9 0\u00ba, pois seus vetores normais t\u00eam a mesma dire\u00e7\u00e3o.<\/li>\n<li> O \u00e2ngulo entre dois planos perpendiculares \u00e9 90\u00ba, porque seus vetores normais tamb\u00e9m s\u00e3o perpendiculares (ou ortogonais) entre si e, portanto, formam um \u00e2ngulo reto.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 uma explica\u00e7\u00e3o do que \u00e9 um plano, como ele \u00e9 calculado e todas as suas propriedades. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de planos, quais s\u00e3o as posi\u00e7\u00f5es relativas entre dois planos, como determinar o \u00e2ngulo entre 2 planos e, por fim, como expressar numericamente qualquer plano usando as equa\u00e7\u00f5es dos &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/geometria-plana\/\"> <span class=\"screen-reader-text\">O plano (geometria)<\/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-265","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>O plano (geometria) - 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\/geometria-plana\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"O plano (geometria) - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea encontrar\u00e1 uma explica\u00e7\u00e3o do que \u00e9 um plano, como ele \u00e9 calculado e todas as suas propriedades. Al\u00e9m disso, voc\u00ea poder\u00e1 ver exemplos de planos, quais s\u00e3o as posi\u00e7\u00f5es relativas entre dois planos, como determinar o \u00e2ngulo entre 2 planos e, por fim, como expressar numericamente qualquer plano usando as equa\u00e7\u00f5es dos &hellip; O plano (geometria) Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/geometria-plana\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-10T04:04:14+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/quel-est-l-avion.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=\"6 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/geometria-plana\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/geometria-plana\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"O plano (geometria)\",\"datePublished\":\"2023-07-10T04:04:14+00:00\",\"dateModified\":\"2023-07-10T04:04:14+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/geometria-plana\/\"},\"wordCount\":1223,\"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\/geometria-plana\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/geometria-plana\/\",\"url\":\"https:\/\/mathority.org\/pt\/geometria-plana\/\",\"name\":\"O plano (geometria) - 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