{"id":228,"date":"2023-07-10T21:50:05","date_gmt":"2023-07-10T21:50:05","guid":{"rendered":"https:\/\/mathority.org\/pt\/posicao-relativa-de-duas-linhas-no-plano-cartesiano-2d-r2-exemplos\/"},"modified":"2023-07-10T21:50:05","modified_gmt":"2023-07-10T21:50:05","slug":"posicao-relativa-de-duas-linhas-no-plano-cartesiano-2d-r2-exemplos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/posicao-relativa-de-duas-linhas-no-plano-cartesiano-2d-r2-exemplos\/","title":{"rendered":"Posi\u00e7\u00e3o relativa de duas retas no plano"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o dos diferentes m\u00e9todos que existem para determinar a posi\u00e7\u00e3o relativa de duas retas no plano (em R2). Al\u00e9m disso, voc\u00ea ver\u00e1 v\u00e1rios exemplos e poder\u00e1 praticar com 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%bfcual-es-la-posicion-relativa-de-dos-rectas-en-el-plano\"><\/span> Qual \u00e9 a posi\u00e7\u00e3o relativa de duas retas no 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> Antes de olhar para as posi\u00e7\u00f5es relativas entre duas retas no plano, obviamente voc\u00ea precisa saber exatamente o que \u00e9 uma reta, voc\u00ea pode encontr\u00e1-la na <a href=\"https:\/\/mathority.org\/pt\/definicao-de-tipos-de-caracteristicas-de-linha-exemplos-de-linha-reta\/\">defini\u00e7\u00e3o de reta<\/a> .<\/p>\n<p> Assim, ao trabalhar em duas dimens\u00f5es (em R2), existem 3 tipos de posi\u00e7\u00f5es relativas poss\u00edveis entre duas linhas: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-136\">\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>linhas 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\/angles-droits-secants.webp\" alt=\"posi\u00e7\u00e3o relativa de duas linhas que se cruzam\" class=\"wp-image-1644\" width=\"205\" height=\"192\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Duas linhas que se cruzam t\u00eam apenas um ponto em comum. <\/p>\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><strong>Linhas paralelas<\/strong><\/strong> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/droites-paralleles-a-langle.webp\" alt=\"posi\u00e7\u00e3o relativa de linhas paralelas\" width=\"209\" height=\"189\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Duas retas s\u00e3o paralelas se n\u00e3o t\u00eam ponto comum. Isto \u00e9, se eles nunca se cruzarem. <\/p>\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>linhas coincidentes<\/strong> <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/angle-coincident-lignes.webp\" alt=\"posi\u00e7\u00e3o relativa de linhas coincidentes\" width=\"189\" height=\"168\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Duas retas s\u00e3o iguais se todos os seus pontos forem comuns. <\/p>\n<\/div>\n<\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-106\"><\/div>\n<\/div>\n<p> Por outro lado, o \u00e2ngulo entre duas linhas no plano tamb\u00e9m depende da sua posi\u00e7\u00e3o relativa:<\/p>\n<ul>\n<li> As linhas que se cruzam se cruzam em um \u00e2ngulo entre 0\u00ba (n\u00e3o inclu\u00eddo) e 90\u00ba (inclusive). Al\u00e9m disso, se formarem apenas um \u00e2ngulo reto de 90\u00ba, isso significa que as duas linhas s\u00e3o perpendiculares.<\/li>\n<li> Retas paralelas formam um \u00e2ngulo de 0\u00ba, pois possuem a mesma dire\u00e7\u00e3o.<\/li>\n<li> E, pelo mesmo motivo, as retas coincidentes tamb\u00e9m fazem um \u00e2ngulo de 0\u00ba entre elas.<\/li>\n<\/ul>\n<p> Se voc\u00ea quiser saber como o \u00e2ngulo entre duas linhas \u00e9 calculado, voc\u00ea pode verificar a <a href=\"https:\/\/mathority.org\/pt\/angulo-entre-duas-linhas-exemplos-de-formulas-exercicios-resolvidos-declives-vetor-diretor\/\">f\u00f3rmula do \u00e2ngulo entre duas linhas<\/a> . Aqui voc\u00ea encontrar\u00e1 uma explica\u00e7\u00e3o detalhada de como determinar o \u00e2ngulo entre duas retas, al\u00e9m de diversos exemplos e at\u00e9 exerc\u00edcios resolvidos para que voc\u00ea possa praticar e compreender totalmente o conceito. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"como-hallar-la-posicion-relativa-de-dos-rectas-en-el-plano\"><\/span>Como encontrar a posi\u00e7\u00e3o relativa de duas retas no 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> Conhecer a posi\u00e7\u00e3o relativa entre duas linhas no espa\u00e7o bidimensional depende de como as linhas s\u00e3o expressas:<\/p>\n<ul>\n<li> <strong>Vetores de dire\u00e7\u00e3o de linha:<\/strong> se duas linhas tiverem um vetor de dire\u00e7\u00e3o diferente, elas dever\u00e3o se cruzar. Por outro lado, se as coordenadas dos seus vetores de dire\u00e7\u00e3o forem iguais ou proporcionais, podem ser paralelos ou coincidentes (\u00e9 necess\u00e1rio verificar se possuem um ponto comum).<\/li>\n<\/ul>\n<ul>\n<li> <strong>Equa\u00e7\u00e3o expl\u00edcita:<\/strong> quando duas retas t\u00eam inclina\u00e7\u00f5es diferentes\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cbf220caa0234311cbdde7c24a842b1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(m)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"28\" style=\"vertical-align: -5px;\"><\/p>\n<p> eles secam Pelo contr\u00e1rio, se as linhas t\u00eam a mesma inclina\u00e7\u00e3o, mas uma ordem diferente na origem<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-9098105a2659ae64387c39c113fab615_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(n)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"23\" style=\"vertical-align: -5px;\"><\/p>\n<p> eles s\u00e3o paralelos. Finalmente, duas retas s\u00e3o confundidas quando originalmente t\u00eam inclina\u00e7\u00f5es e ordenadas iguais.<\/li>\n<\/ul>\n<ul>\n<li> <strong>Equa\u00e7\u00e3o geral (ou impl\u00edcita):<\/strong> duas retas com coeficientes n\u00e3o proporcionais A e B sempre se cruzar\u00e3o. Por\u00e9m, ser\u00e3o paralelos quando estes dois par\u00e2metros forem proporcionais entre si, mas n\u00e3o ao coeficiente C. E, finalmente, quando os tr\u00eas termos forem proporcionais, isso implica que as retas est\u00e3o confusas.<\/li>\n<\/ul>\n<p> Se voc\u00ea tiver alguma d\u00favida sobre as equa\u00e7\u00f5es da reta acima, pode consultar a explica\u00e7\u00e3o das <a href=\"https:\/\/mathority.org\/pt\/equacoes-de-linha-todas-as-formulas-exemplos-exercicios-resolvidos\/\">equa\u00e7\u00f5es da reta no plano<\/a> . Aqui voc\u00ea encontrar\u00e1 a f\u00f3rmula de todas as equa\u00e7\u00f5es lineares, como s\u00e3o calculadas, exemplos e exerc\u00edcios resolvidos de equa\u00e7\u00f5es lineares.<\/p>\n<p> Na tabela a seguir voc\u00ea tem um resumo das propriedades anteriores: <\/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\/position-relative-de-deux-droites-dans-le-plan-1.webp\" alt=\"posi\u00e7\u00e3o relativa de duas linhas no plano\" class=\"wp-image-1718\" width=\"675\" height=\"403\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> A seguir, veremos dois exemplos de como determinar a posi\u00e7\u00e3o relativa entre duas linhas:<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-1\"><\/span> Exemplo 1<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li> Encontre a posi\u00e7\u00e3o relativa entre as duas linhas a seguir definidas na forma de uma equa\u00e7\u00e3o expl\u00edcita:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-da7047627c75c1e890b8dcd638f77ed6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\ y=3x+2 \\qquad \\qquad s: \\ y=3x-4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"294\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> As duas linhas t\u00eam a mesma inclina\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-873fd0b2da446bff0d72d56424846ad5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_r = m_s = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"102\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p> Mas eles t\u00eam computadores diferentes na origem:<\/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-dd55fdc8ff8e806fd683fad5ee625dc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"n_r =2\\neq n_s=-4\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"139\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Portanto, como t\u00eam a mesma inclina\u00e7\u00e3o, mas intercepta\u00e7\u00f5es diferentes, <strong>as retas s\u00e3o paralelas<\/strong> .<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-2\"><\/span> Exemplo 2<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li> Determine a posi\u00e7\u00e3o relativa entre as duas linhas a seguir expressas com sua equa\u00e7\u00e3o impl\u00edcita (ou geral):<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c917d50b692b0f59a5846479d74b6e19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\ 4x+2y-6=0 \\qquad \\qquad s: \\ -2x-y+3=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"378\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Ambas as retas s\u00e3o expressas como uma equa\u00e7\u00e3o expl\u00edcita, portanto precisamos ver se algum de seus coeficientes \u00e9 proporcional:<\/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-cce65e5ed8b8969e9349325d13a192ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{4}{-2}=\\cfrac{2}{-1} = \\cfrac{-6}{3} = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"173\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Os 3 termos das retas s\u00e3o proporcionais, portanto <strong>as retas coincidem<\/strong> . <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"determinar-la-posicion-relativa-de-dos-rectas-en-el-plano-con-un-sistema-de-ecuaciones\"><\/span> Determine a posi\u00e7\u00e3o relativa de duas retas no plano com um sistema de equa\u00e7\u00f5es<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Outra forma de saber a posi\u00e7\u00e3o relativa entre duas retas \u00e9 analisar o sistema de equa\u00e7\u00f5es formado pelas equa\u00e7\u00f5es das retas:<\/p>\n<ul>\n<li> Se o <strong>sistema tiver uma solu\u00e7\u00e3o \u00fanica<\/strong> , as linhas se cruzam. Al\u00e9m disso, o ponto de intersec\u00e7\u00e3o das duas retas \u00e9 a solu\u00e7\u00e3o do sistema.<\/li>\n<li> Se for um <strong>sistema sem solu\u00e7\u00e3o<\/strong> , isso indica que as retas n\u00e3o possuem pontos em comum e, portanto, s\u00e3o retas paralelas.<\/li>\n<li> Se o <strong>sistema tiver infinitas solu\u00e7\u00f5es<\/strong> , isso significa que as retas possuem todos os pontos em comum e, portanto, s\u00e3o retas que se cruzam.<\/li>\n<\/ul>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-3\"><\/span> Exemplo 3<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li> Calcule a posi\u00e7\u00e3o relativa das duas linhas a seguir usando um sistema de equa\u00e7\u00f5es:<\/li>\n<\/ul>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a333dd2d8363e9caf5bfcb7fcc4b307a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\ 3x+4y+5=0 \\qquad \\qquad s: \\ 5x+y-3=0\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"364\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Para encontrar a posi\u00e7\u00e3o relativa das duas retas, precisamos resolver o seguinte sistema de equa\u00e7\u00f5es lineares formado pelas duas retas:<\/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-243b1e787e6532fbafbfca53d934f4ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{l} 3x+4y+5=0\\\\[2ex] 5x+y-3=0\\end{array}\\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"143\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Neste caso, resolveremos o sistema com o m\u00e9todo de substitui\u00e7\u00e3o. Iremos, portanto, isolar a vari\u00e1vel<\/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-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> da segunda equa\u00e7\u00e3o e substitua-a na primeira 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-ad3456c92c838f40d60afdb45e1eb2f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{l} 3x+4y+5=0\\\\[2ex] 5x+y-3=0\\end{array}\\right\\} \\begin{array}{l} \\\\[2ex] \\longrightarrow \\ y=3-5x \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"279\" 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-e878bb5de718162dd547d349041c5a17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x+4(3-5x)+5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"176\" 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-b44339147102246aa99ce775ae1f4415_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x+12-20x+5=0\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"171\" 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-e51763736cd6ea39f0cf08f9c8d067f9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3x-20x=-12-5\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"153\" 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-55630337168997a6a2a76aeaef1ad669_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-17x=-17\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"96\" 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-50540499fd4810f10aa8d48376dc8370_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{-17}{-17} = 1\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"107\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> E quando soubermos quanto vale 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-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> Substitu\u00edmos seu valor na express\u00e3o encontrada para <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-526e84c5b87e970b9045246e059785fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" 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-b53c99a78c482f7c71588f8e0d769e9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=3-5x \\ \\xrightarrow{x \\ = \\ 1} \\ y = 3 -5\\cdot 1\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"245\" 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-9985617b47dec6ac0faa8d98665cd8b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 3 -5 = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"118\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> Obtivemos, portanto, apenas uma solu\u00e7\u00e3o do sistema de equa\u00e7\u00f5es composto pelas duas retas, portanto <strong>as duas retas se cruzam<\/strong> . E o ponto onde eles se cruzam \u00e9 a solu\u00e7\u00e3o do sistema, ou seja, 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-d9dbd776d7ff2a9a72963326b278e12b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(1,-2).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"57\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-posicion-relativa-de-dos-rectas-en-el-plano\"><\/span> Problemas resolvidos de posi\u00e7\u00e3o relativa de duas retas no plano<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Determine se as seguintes linhas se cruzam, s\u00e3o paralelas ou coincidem: <\/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-d8857a4b8101efdfb88107fae20a16bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\ 3x-y+4=0 \\qquad \\qquad s: \\ 9x-3y+3=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"364\" 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\"> Ambas as retas s\u00e3o expressas como uma equa\u00e7\u00e3o impl\u00edcita (ou geral), portanto precisamos ver se algum de seus coeficientes \u00e9 proporcional:<\/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-1fd9a2de250f3a5cb1a7025e30eb00b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{3}{9}=\\cfrac{-1}{-3} \\neq \\cfrac{4}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"99\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Apenas os coeficientes A e B das retas s\u00e3o proporcionais entre si, e n\u00e3o ao coeficiente C. Portanto <strong>as duas retas s\u00e3o paralelas<\/strong> .<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 2<\/h3>\n<p> Encontre a posi\u00e7\u00e3o relativa entre as duas linhas a seguir expressas como equa\u00e7\u00f5es param\u00e9tricas: <\/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-bafb951a2141722b0bbb7a1681f506ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ \\begin{cases} x=4-5t \\\\[2ex] y= 1+3t \\end{cases}\\qquad \\qquad s: \\ \\begin{cases} x=-2t \\\\[2ex] y=6+9t \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"341\" style=\"vertical-align: 0px;\"><\/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\"> Poder\u00edamos resolver o sistema de equa\u00e7\u00f5es formado pelas duas retas para encontrar sua posi\u00e7\u00e3o relativa. Por\u00e9m, por estarem na forma de equa\u00e7\u00f5es param\u00e9tricas, pode-se facilmente encontrar seus vetores de dire\u00e7\u00e3o e se n\u00e3o forem proporcionais significa que as retas se cruzam. E neste caso, n\u00e3o gastaremos tanto tempo resolvendo todo um sistema de equa\u00e7\u00f5es.<\/p>\n<p class=\"has-text-align-left\"> Para que as coordenadas cartesianas do vetor de dire\u00e7\u00e3o de cada linha sejam os n\u00fameros na frente 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-40f8b062c79839dcf7f2885a9e1469e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"t:\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" 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-2428dfdbfe571940b2c02da2581f3c83_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{r} =(-5,3) \\qquad \\qquad \\vv{s}=(-2,9)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Uma vez conhecidos os vetores de dire\u00e7\u00e3o, verificamos sua proporcionalidade:<\/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-38d5cefd7229992008378fea28376585_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-5}{-2} \\neq \\cfrac{3}{9}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"65\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Os vetores de dire\u00e7\u00e3o n\u00e3o s\u00e3o proporcionais, portanto <strong>as linhas se cruzam<\/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> Indique se as seguintes linhas se cruzam, s\u00e3o paralelas ou coincidentes e tamb\u00e9m encontre um ponto de intersec\u00e7\u00e3o entre elas (se aplic\u00e1vel). <\/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-9e1c8f120a7d86580808425ce4452bbc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\ y=4x-5 \\qquad \\qquad s: \\ y=-2x+7\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"308\" 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\"> As duas retas s\u00e3o definidas por sua equa\u00e7\u00e3o expl\u00edcita e t\u00eam inclina\u00e7\u00f5es 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-5f8ff0ff270ad51df36633f3a7da08f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_r =4 \\neq m_s = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"147\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Como possuem inclina\u00e7\u00f5es diferentes, <strong>as linhas se cruzam<\/strong> .<\/p>\n<p class=\"has-text-align-left\"> Portanto, como as retas se cruzam, elas ter\u00e3o 1 ponto em comum e para calcul\u00e1-lo devemos resolver o sistema de equa\u00e7\u00f5es formado pelas duas retas:<\/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-b3923ff74a214543ddd2cc44a42e3813_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left.\\begin{array}{l} y=4x-5\\\\[2ex] y=-2x+7\\end{array}\\right\\}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"117\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Neste caso resolveremos o sistema com o m\u00e9todo de equaliza\u00e7\u00e3o porque ambos<\/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-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> j\u00e1 foram exclu\u00eddos: <\/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-1043bcfc74c9c720f4d738403a0a5de9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" 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-30260dd1609f72d1b91e2431be1c4fec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4x-5=-2x+7\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"137\" 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-13a0532699efab3ce3fa0565ddd295f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4x+2x=5+7\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"123\" 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-20e2c2528083b16701309669036284a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6x=12\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"60\" 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-54e0507e23104e014591f7bb9d0b9e02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x=\\cfrac{12}{6} = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"85\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E uma vez que temos 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-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> substitu\u00edmos seu valor em qualquer express\u00e3o de<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0af556714940c351c933bba8cf840796_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> para saber quanto vale: <\/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-fa1fe37c29ef252a2729d235c875f0ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=4x-5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" 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-e2858c9e13db1ef7d1777f6ee50501ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y = 4\\cdot 2 -5 = 8 -5 = 3\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"190\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, o ponto de intersec\u00e7\u00e3o das duas retas \u00e9 o resultado do sistema: <\/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-786ded60a053402b1fedabd27477fa77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{(2,3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/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<p> Calcule o valor das inc\u00f3gnitas<\/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-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> E<\/p>\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> de modo que as duas linhas a seguir sejam paralelas: <\/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-22406447457807bd671683c76d717494_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\ 2x-4y+6=0 \\qquad \\qquad s: \\ x+ay+b=0\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"363\" 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\"> As linhas s\u00e3o descritas em forma de equa\u00e7\u00e3o geral (ou impl\u00edcita). Portanto, para que as duas retas sejam paralelas, seus coeficientes A e B devem ser proporcionais, ou seja, deve-se cumprir a seguinte 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-426cf79f7967a39d05301b57327425b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2}{1} = \\cfrac{-4}{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"65\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Devemos, portanto, resolver a equa\u00e7\u00e3o anterior para obter o valor da inc\u00f3gnita<\/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-2ebc59bdf10d3d739bfa532b65c85287_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"13\" style=\"vertical-align: 0px;\"><\/p>\n<p> Para fazer isso, multiplicamos as fra\u00e7\u00f5es transversalmente: <\/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-778399973685974e5fab6ef86a4ff316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2 \\cdot a = -4 \\cdot 1\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"98\" 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-fa72e494965cde5a39eef1003b449a40_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"a = \\cfrac{-4}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"66\" 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-115c97df060d69970606b1ae494bb4b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{a=-2}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"55\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Por outro lado, para que as retas sejam paralelas seus termos independentes n\u00e3o podem ser proporcionais aos demais coeficientes:<\/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-0eee380a4c98c2379e3b88fb5c00038e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{2}{1} \\neq \\cfrac{6}{b}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"43\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, como antes, resolvemos a desigualdade multiplicando as fra\u00e7\u00f5es transversalmente: <\/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-e8f6fbac3664ac51fabbd88c63a6265c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2 \\cdot b \\neq 6 \\cdot 1\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"83\" 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-0d382f6f8b7a55e13812975129a29da1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"b \\neq \\cfrac{6}{2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"42\" 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-b74bf2f543632f7b7adec4ebd1f59b0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{b\\neq 3}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Em suma, para que as duas linhas sejam paralelas<\/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-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> deve ser 2 e<\/p>\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> pode ser qualquer n\u00famero real, exceto 3.<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 5<\/h3>\n<p> Encontre a equa\u00e7\u00e3o expl\u00edcita da reta paralela \u00e0 reta<\/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 o que acontece al\u00e9m do ponto<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f3beed6e0b45d9e34dfb895ea2711797_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3,-1).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/p>\n<p> ser direto <\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c5986f031932e6b3512dc564514c34b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r:\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" 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-731b64c6b0926aa016c615182da3d7d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r: \\; y=2x+5\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"110\" 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 a linha seja paralela \u00e0 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-42ca8c420951296e93092e708435813a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r,\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: -4px;\"><\/p>\n<p> ambos devem ter a mesma inclina\u00e7\u00e3o. e a inclina\u00e7\u00e3o da linha<\/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> \u00e9 2:<\/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-f3439899d4781058b8eb19021ac25e27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m = 2\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, a equa\u00e7\u00e3o da reta que precisamos encontrar ser\u00e1:<\/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-1bbbc8cea82d1688f21bcabc8ef7fa3e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y=2x+n\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E uma vez que conhecemos a inclina\u00e7\u00e3o da reta, podemos calcular a intercepta\u00e7\u00e3o y substituindo o ponto que pertence \u00e0 reta na equa\u00e7\u00e3o da reta: <\/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-e83b6065f321642320b736c8b866043c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(3,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" 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-826526b2aef06a1f04f17c54f8db7369_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y= 2x+n \\ \\xrightarrow{x=3 \\ ; \\ y=-1} \\ -1=2\\cdot 3 +n\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"303\" 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-0aba6175b45d89611019ec5f89ebd82f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1=6+ n\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"87\" 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-afeec8d1085931aed83404a8a7ba45dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1-6= n\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"87\" 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-35b29a0fad2239bd507e2d044f846092_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-7= n\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, a equa\u00e7\u00e3o expl\u00edcita da reta \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-fd88ac415e522e13e3437bb0e3da393a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{y=2x-7}\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"83\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<p> Se voc\u00ea chegou at\u00e9 aqui, significa que j\u00e1 dominou as posi\u00e7\u00f5es relativas entre duas linhas no plano. Bom trabalho!<\/p>\n<p> Mas uma coisa que muitos se perguntam \u00e9&#8230; e qual \u00e9 a utilidade de saber a posi\u00e7\u00e3o relativa entre duas linhas?<\/p>\n<p> Pois bem, uma das aplica\u00e7\u00f5es da posi\u00e7\u00e3o relativa entre linhas \u00e9 poder saber a dist\u00e2ncia entre 2 linhas, j\u00e1 que o c\u00e1lculo da dist\u00e2ncia entre duas linhas depende da sua posi\u00e7\u00e3o relativa:<\/p>\n<ul>\n<li> Se as linhas se cruzarem ou coincidirem, a dist\u00e2ncia ser\u00e1 zero.<\/li>\n<li> Por outro lado, quando as retas s\u00e3o paralelas, deve ser aplicada uma f\u00f3rmula espec\u00edfica. Se voc\u00ea estiver mais interessado, pode conferir como \u00e9 calculada a <a href=\"https:\/\/mathority.org\/pt\/distancia-entre-duas-linhas-paralelas-exemplos-de-formula-de-exercicios-resolvidos\/\">dist\u00e2ncia entre duas retas paralelas<\/a> .<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o dos diferentes m\u00e9todos que existem para determinar a posi\u00e7\u00e3o relativa de duas retas no plano (em R2). Al\u00e9m disso, voc\u00ea ver\u00e1 v\u00e1rios exemplos e poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo. Qual \u00e9 a posi\u00e7\u00e3o relativa de duas retas no plano? Antes de olhar para as posi\u00e7\u00f5es relativas &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/posicao-relativa-de-duas-linhas-no-plano-cartesiano-2d-r2-exemplos\/\"> <span class=\"screen-reader-text\">Posi\u00e7\u00e3o relativa de duas retas no 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-228","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>Posi\u00e7\u00e3o relativa de duas retas no plano \u2013 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\/posicao-relativa-de-duas-linhas-no-plano-cartesiano-2d-r2-exemplos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Posi\u00e7\u00e3o relativa de duas retas no plano \u2013 Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o dos diferentes m\u00e9todos que existem para determinar a posi\u00e7\u00e3o relativa de duas retas no plano (em R2). Al\u00e9m disso, voc\u00ea ver\u00e1 v\u00e1rios exemplos e poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo. Qual \u00e9 a posi\u00e7\u00e3o relativa de duas retas no plano? Antes de olhar para as posi\u00e7\u00f5es relativas &hellip; Posi\u00e7\u00e3o relativa de duas retas no plano Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/posicao-relativa-de-duas-linhas-no-plano-cartesiano-2d-r2-exemplos\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-10T21:50:05+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/angles-droits-secants.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=\"8 minutos\" \/>\n<script type=\"application\/ld+json\" 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