{"id":227,"date":"2023-07-10T22:13:46","date_gmt":"2023-07-10T22:13:46","guid":{"rendered":"https:\/\/mathority.org\/pt\/angulo-entre-duas-linhas-exemplos-de-formulas-exercicios-resolvidos-declives-vetor-diretor\/"},"modified":"2023-07-10T22:13:46","modified_gmt":"2023-07-10T22:13:46","slug":"angulo-entre-duas-linhas-exemplos-de-formulas-exercicios-resolvidos-declives-vetor-diretor","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/angulo-entre-duas-linhas-exemplos-de-formulas-exercicios-resolvidos-declives-vetor-diretor\/","title":{"rendered":"\u00c2ngulo entre duas linhas (f\u00f3rmula)"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o de como calcular o \u00e2ngulo entre duas retas (f\u00f3rmula). Voc\u00ea tamb\u00e9m poder\u00e1 ver diversos exemplos e, al\u00e9m disso, 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%bfque-es-el-angulo-entre-dos-rectas\"><\/span> Qual \u00e9 o \u00e2ngulo entre duas linhas? <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> <strong>O \u00e2ngulo entre duas linhas \u00e9 o menor \u00e2ngulo entre essas duas linhas.<\/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\/angle-entre-deux-lignes-1.webp\" alt=\"\u00e2ngulo entre duas linhas\" class=\"wp-image-1637\" width=\"225\" height=\"206\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> No plano existem quatro tipos de retas dependendo do \u00e2ngulo que formam entre elas: retas que se cruzam (entre 0\u00ba e 90\u00ba), retas perpendiculares (90\u00ba), retas paralelas (0\u00ba) e retas coincidentes (0\u00ba). <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-143\">\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=\"\u00e2ngulo entre duas linhas que se cruzam\" class=\"wp-image-1644\" width=\"205\" height=\"192\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> As linhas que se cruzam se cruzam em um \u00e2ngulo agudo entre 0\u00ba e 90\u00ba. <\/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 retas perpendiculares<\/strong><\/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\/lignes-perpendiculaires-a-90-degres.webp\" alt=\"\u00e2ngulo entre duas linhas perpendiculares\" class=\"wp-image-1884\" width=\"181\" height=\"207\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> As linhas perpendiculares se cruzam em um \u00e2ngulo reto de 90\u00ba. <\/p>\n<\/div>\n<\/div>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-105\"><\/div>\n<\/div>\n<div class=\"wp-block-columns is-layout-flex wp-container-146\">\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 paralelas<\/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\/droites-paralleles-a-langle.webp\" alt=\"\" class=\"wp-image-1643\" width=\"217\" height=\"195\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> As linhas paralelas nunca se tocam e formam um \u00e2ngulo de 0\u00ba entre elas. <\/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 size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/angle-coincident-lignes.webp\" alt=\"\" class=\"wp-image-1646\" width=\"189\" height=\"168\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Duas retas coincidentes possuem todos os pontos em comum e, portanto, existe sempre um \u00e2ngulo de 0\u00ba entre elas.<\/p>\n<\/div>\n<\/div>\n<p> Concluindo, o c\u00e1lculo do \u00e2ngulo entre duas retas paralelas, coincidentes ou perpendiculares \u00e9 imediato: as retas paralelas e as retas coincidentes formam um \u00e2ngulo de 0 graus, pois t\u00eam a mesma dire\u00e7\u00e3o, e as retas perpendiculares se cruzam com um \u00e2ngulo de 90 graus . Por outro lado, para encontrar o \u00e2ngulo entre duas retas que se cruzam, deve-se aplicar uma f\u00f3rmula (veremos a seguir). <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcomo-se-calcula-el-angulo-entre-dos-rectas\"><\/span> Como \u00e9 calculado o \u00e2ngulo entre duas linhas? <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<p> Existem duas maneiras de calcular o \u00e2ngulo entre duas linhas. O primeiro m\u00e9todo utiliza o <strong>vetor de dire\u00e7\u00e3o<\/strong> de cada reta e o segundo m\u00e9todo \u00e9 baseado na <strong>inclina\u00e7\u00e3o<\/strong> de cada reta.<\/p>\n<p> Nenhum procedimento \u00e9 melhor que o outro, na verdade ambos s\u00e3o bastante f\u00e1ceis, mas dependendo de como as linhas s\u00e3o expressas, um m\u00e9todo ou outro \u00e9 pr\u00e1tico. Portanto, recomendamos que voc\u00ea saiba usar ambos os m\u00e9todos matem\u00e1ticos. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"metodo-de-los-vectores-directores-de-las-rectas\"><\/span> M\u00e9todo de orienta\u00e7\u00e3o vetorial de linha<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A f\u00f3rmula para calcular o \u00e2ngulo entre duas linhas usando seus vetores de dire\u00e7\u00e3o \u00e9: <\/p>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> Dados os vetores de dire\u00e7\u00e3o de duas linhas 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-b626c82ac04d69ba3bcafb5fa87d7d00_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (\\text{u}_x,\\text{u}_y)\\qquad \\vv{\\text{v}} = (\\text{v}_x,\\text{v}_y)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"216\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> O \u00e2ngulo entre essas duas linhas pode ser calculado com 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-19eb97a6cf27fffc3ea832e388f924a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert \\vv{\\text{u}} \\cdot \\vv{\\text{v}}\\rvert}{\\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"127\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> 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-4501274336c637b37c6332eae5c6c229_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" 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-9a59cd4f2581db3318d38a2a77340a64_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{v}} \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"15\" style=\"vertical-align: -5px;\"><\/p>\n<p> s\u00e3o os m\u00f3dulos dos vetores<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-cac24ae79c1e4cbc459f01ed5e4f824e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-391ac2e3ba0b7f327ba5a0edc1ba162d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\"><\/p>\n<p> respectivamente.<\/p>\n<\/div>\n<p> Lembre-se de que a f\u00f3rmula para a magnitude de um vetor \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-0761a6a31d273eefccceb4aad7556a6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{\\text{v}} \\rvert = \\sqrt{ \\text{v}_x^2+\\text{v}_y^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"117\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p> Vamos ver como encontrar o \u00e2ngulo entre duas linhas com um exemplo:<\/p>\n<ul>\n<li> Calcule o \u00e2ngulo entre as duas linhas a seguir: <\/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-a336a6cbbd7581f1fb6481561aef1efc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ \\begin{cases} x=2-3t \\\\[2ex]y=1+4t \\end{cases} \\qquad s: \\ 2x-5y+7=0\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"334\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<div class=\"adsb30\" style=\" margin:12px; text-align:center\">\n<div id=\"ezoic-pub-ad-placeholder-109\"><\/div>\n<\/div>\n<p> Para calcular o \u00e2ngulo entre as duas retas, primeiro voc\u00ea deve encontrar o vetor de dire\u00e7\u00e3o de cada reta.<\/p>\n<p> o certo<\/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> \u00e9 expresso na forma de <a href=\"https:\/\/mathority.org\/pt\/formulas-de-equacoes-parametricas-de-uma-reta\/\">uma equa\u00e7\u00e3o param\u00e9trica<\/a> , portanto os componentes do vetor que marca sua dire\u00e7\u00e3o s\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-5d3e98a6c4a49b9b38e463795eb44b82_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{r} = (-3,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> e a lei<\/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-ea93feaa2c7157ec666d9a59c0f6a699_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  s\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 definido na forma de uma equa\u00e7\u00e3o impl\u00edcita (ou geral), ent\u00e3o as coordenadas de seu vetor de dire\u00e7\u00e3o s\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-25fa1b333fb55fd35e2ff773a99aab2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{s} = (-B,A)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"94\" 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-59f221044ed855cbee5120d8936cc247_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{s} = (5,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Agora que sabemos o vetor diretor de cada reta, podemos usar a f\u00f3rmula do \u00e2ngulo entre 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-790804eb21bd7b19771c5597b3cea577_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert\\vv{r} \\cdot \\vv{s}\\rvert}{\\lvert \\vv{r} \\rvert \\cdot \\lvert \\vv{s} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"125\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p> Portanto, determinamos a magnitude dos dois vetores:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-5630e1894a54b931779a240cce2b3460_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{r} \\rvert = \\sqrt{(-3)^2+4^2}= 5\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"172\" 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-3b8dca924b988372d9cc00e5a3e79041_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{s} \\rvert = \\sqrt{5^2+2^2}= \\sqrt{29}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"169\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Realizamos as opera\u00e7\u00f5es vetoriais da f\u00f3rmula do \u00e2ngulo:<\/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-790804eb21bd7b19771c5597b3cea577_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert\\vv{r} \\cdot \\vv{s}\\rvert}{\\lvert \\vv{r} \\rvert \\cdot \\lvert \\vv{s} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"125\" style=\"vertical-align: -17px;\"><\/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-1f1810380fc6ddab753a49fb43d8d136_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert(-3,4) \\cdot (5,2)\\rvert}{5 \\cdot \\sqrt{29}}= \\cfrac{\\lvert-3 \\cdot 5 + 4\\cdot 2\\rvert}{26,93} = \\cfrac{7}{26,93} = 0,26\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"449\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p> E, por fim, calculamos o \u00e2ngulo formado pelas duas retas com o inverso do cosseno:<\/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-839ce1333f41e5392ef7d2127853aae2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha= \\text{cos}^{-1}(0,26) = \\bm{74,93\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"193\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Lembre-se que voc\u00ea pode calcular o inverso do cosseno usando a calculadora com a tecla <\/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-8b70d14d21b828bcf46c4104f901c916_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\boxed{\\cos ^{-1}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"57\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"metodo-de-las-pendientes\"><\/span> m\u00e9todo de inclina\u00e7\u00e3o<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> Obviamente, para entender esse m\u00e9todo, voc\u00ea precisa conhecer a <a href=\"https:\/\/mathority.org\/pt\/formula-de-inclinacao-da-linha\/\">inclina\u00e7\u00e3o da reta<\/a> . Voc\u00ea pode revisar este conceito no link, onde encontrar\u00e1 uma explica\u00e7\u00e3o detalhada do que significa, como \u00e9 calculado, exemplos e exerc\u00edcios resolvidos da inclina\u00e7\u00e3o de uma reta.<\/p>\n<p> A f\u00f3rmula para calcular o \u00e2ngulo entre duas linhas a partir de suas inclina\u00e7\u00f5es \u00e9: <\/p>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 40px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> Ou duas linhas distintas:<\/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-a9768adb30eaa8e08b67c58e5c4921df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r_1 : \\ y=m_1 x+n_1 \\qquad r_2: \\ y=m_2 x+n_2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"321\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> O \u00e2ngulo entre essas duas linhas pode ser determinado com 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-82acfc9ae51ee3a469cfabc7024aa75c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{tg}(\\alpha) =\\begin{vmatrix} \\cfrac{m_2-m_1}{1+m_1\\cdot m_2} \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"166\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> 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-51921237944fd6e43f0640228a37376f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"22\" style=\"vertical-align: -3px;\"><\/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-f6dae86895dac0d4644151786b47c7ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"23\" style=\"vertical-align: -3px;\"><\/p>\n<p> s\u00e3o as inclina\u00e7\u00f5es das linhas<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6ce00e1b287bac058a29aa4a5cc2b715_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r_1\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"14\" style=\"vertical-align: -3px;\"><\/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-80681c4f8159fb897fed760530a2ef01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r_2\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"15\" style=\"vertical-align: -3px;\"><\/p>\n<p> respectivamente.<\/p>\n<\/div>\n<p> Vamos ver como calcular o \u00e2ngulo entre duas retas usando suas inclina\u00e7\u00f5es com um exemplo:<\/p>\n<ul>\n<li> Encontre o \u00e2ngulo entre as duas linhas a seguir:<\/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-37af9568ead27bf5cc0bedd4e23107b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ y=4x-2 \\qquad s: \\ y=-3x+1\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"272\" style=\"vertical-align: -4px;\"><\/p>\n<\/p>\n<p> A inclina\u00e7\u00e3o de cada linha \u00e9 o n\u00famero antes da 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-845f2902b8bebf60c3c7372a7fbe4d02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x:\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"19\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-6fd143f62c08661d4c17431b128bdcf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_r = 4\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"56\" style=\"vertical-align: -3px;\"><\/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-a6063973129bb0bac4b98714e474f8ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_s = -3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"69\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p> Portanto, o \u00e2ngulo entre as duas linhas pode ser encontrado aplicando a f\u00f3rmula da 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-551288f526b75201969ebf9117fc9b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{tg}(\\alpha) =\\begin{vmatrix} \\cfrac{m_s-m_r}{1+m_r\\cdot m_s} \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"165\" style=\"vertical-align: -17px;\"><\/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-639af21616a579864711c6c3466a5157_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{tg}(\\alpha) =\\begin{vmatrix} \\cfrac{-3-4}{1+4\\cdot (-3)} \\end{vmatrix}=\\begin{vmatrix} \\cfrac{-7}{-11} \\end{vmatrix} = 0,64\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"294\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p> E finalmente encontramos o \u00e2ngulo com o inverso da tangente:<\/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-caed83d6028d223b06c41f639e5323e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha= \\text{tg}^{-1}(0,64) = \\bm{32,62\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"184\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Lembre-se que voc\u00ea pode calcular o inverso da tangente usando a calculadora com a tecla<\/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-2be52d0cf4b9ef4f831429feec90b416_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\boxed{\\tan ^{-1}}.\" title=\"Rendered by QuickLaTeX.com\" height=\"28\" width=\"59\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p> Acabamos de ver um exemplo com os declives de duas retas expressos como uma equa\u00e7\u00e3o expl\u00edcita, mas se estivessem na forma de uma <a href=\"https:\/\/mathority.org\/pt\/equacao-ponto-inclinacao-da-formula-da-linha\/\">equa\u00e7\u00e3o de declive de ponto,<\/a> este mesmo procedimento teria que ser utilizado. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-angulos-entre-dos-rectas\"><\/span> Resolvendo problemas de \u00e2ngulo entre duas linhas<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Determine o \u00e2ngulo formado pelas duas retas 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-975bcacc5eecede0a2288a39eeb27a73_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ \\begin{cases} x=4+t \\\\[2ex]y=-3-2t \\end{cases} \\qquad s: \\ \\begin{cases} x=4t \\\\[2ex]y=-1-t \\end{cases}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"324\" 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\"> Neste caso, usaremos o m\u00e9todo do vetor de dire\u00e7\u00e3o. Portanto, devemos primeiro encontrar o vetor diretor de cada reta. Ambas as retas s\u00e3o expressas como equa\u00e7\u00f5es param\u00e9tricas, portanto os componentes de seus vetores de dire\u00e7\u00e3o s\u00e3o os termos 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-ac0191cf7cbec493c10a4fa8197e2a6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{r} = (1,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" 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-97225ee00d3957d5d85cdc93c8015ed4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{s} = (4,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Agora que sabemos o vetor diretor de cada reta, podemos usar a f\u00f3rmula do \u00e2ngulo entre 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-790804eb21bd7b19771c5597b3cea577_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert\\vv{r} \\cdot \\vv{s}\\rvert}{\\lvert \\vv{r} \\rvert \\cdot \\lvert \\vv{s} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"125\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, determinamos a magnitude dos dois vetores: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2e501df610a9ae606c598ec472017f78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{r} \\rvert = \\sqrt{1^2+(-2)^2}= \\sqrt{5}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"187\" 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-b9e23805a7bdb58bd0b4893d4b6e586a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{s} \\rvert = \\sqrt{4^2+(-1)^2}= \\sqrt{17}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"196\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Resolvemos o produto escalar entre os dois vetores do numerador e a multiplica\u00e7\u00e3o dos m\u00f3dulos do denominador: <\/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-790804eb21bd7b19771c5597b3cea577_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert\\vv{r} \\cdot \\vv{s}\\rvert}{\\lvert \\vv{r} \\rvert \\cdot \\lvert \\vv{s} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"125\" style=\"vertical-align: -17px;\"><\/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-e5d926125129db13c541515e1dd0beba_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert(1,-2) \\cdot (4,-1)\\rvert}{\\sqrt{5} \\cdot \\sqrt{17}}= \\cfrac{\\lvert 1 \\cdot 4 + (-2)\\cdot (-1)\\rvert}{9,22} = \\cfrac{6}{9,22} = 0,65\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"494\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, por fim, encontramos o \u00e2ngulo formado pelas duas retas fazendo o inverso do cosseno: <\/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-3ede88ebdcf81c8914fed546ba2a0d1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha= \\text{cos}^{-1}(0,65) = \\bm{49,40\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"193\" 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 2<\/h3>\n<p> Encontre o \u00e2ngulo entre as duas linhas 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-818fae5a2074424ec782243f26c5708c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ -3x+4y+1=0 \\qquad s: \\ \\cfrac{x-1}{6} = \\cfrac{y+5}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"39\" width=\"337\" style=\"vertical-align: -12px;\"><\/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\"> Resolveremos este problema usando o m\u00e9todo do vetor de dire\u00e7\u00e3o, ent\u00e3o primeiro precisamos encontrar o vetor de dire\u00e7\u00e3o de cada reta. o certo<\/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> \u00e9 expresso na forma de uma equa\u00e7\u00e3o geral (ou impl\u00edcita), tal que os componentes do vetor que marca sua dire\u00e7\u00e3o s\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-381327f58ef6c881ed34e78624c91b8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{r} = (-B,A)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"94\" 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-383a5264ab7ded87d5684560e6263e15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{r} = (-4,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"99\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> e a lei<\/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-ea93feaa2c7157ec666d9a59c0f6a699_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle  s\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e9 definido na forma de uma equa\u00e7\u00e3o cont\u00ednua, ent\u00e3o as coordenadas cartesianas de seu vetor de dire\u00e7\u00e3o s\u00e3o os n\u00fameros dos denominadores:<\/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-4ba6fe3a3d80f3a44c2c3a0c8345ffa4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{s} = (6,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"71\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Uma vez conhecido o vetor diretor de cada reta, podemos usar a f\u00f3rmula para o \u00e2ngulo entre 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-790804eb21bd7b19771c5597b3cea577_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert\\vv{r} \\cdot \\vv{s}\\rvert}{\\lvert \\vv{r} \\rvert \\cdot \\lvert \\vv{s} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"125\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, determinamos os m\u00f3dulos dos dois vetores: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-565377b63a2e28ce9613745bc0c0b756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{r} \\rvert = \\sqrt{(-4)^2+(-3)^2}= 5\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"199\" 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-65100d7dbdf97aa72e9212379ff54de8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\lvert \\vv{s} \\rvert = \\sqrt{6^2+3^2}= \\sqrt{45}\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"169\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Realizamos as opera\u00e7\u00f5es entre vetores da f\u00f3rmula do \u00e2ngulo: <\/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-790804eb21bd7b19771c5597b3cea577_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert\\vv{r} \\cdot \\vv{s}\\rvert}{\\lvert \\vv{r} \\rvert \\cdot \\lvert \\vv{s} \\rvert}\" title=\"Rendered by QuickLaTeX.com\" height=\"45\" width=\"125\" style=\"vertical-align: -17px;\"><\/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-a36b58fc65b59f20656acc68016020ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\cos(\\alpha) =\\cfrac{\\lvert(-4,-3) \\cdot (6,3)\\rvert}{5 \\cdot \\sqrt{45}}= \\cfrac{\\lvert -4 \\cdot 6 + (-3)\\cdot 3\\rvert}{33,54} = \\cfrac{33}{33,54} = 0,98\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"490\" style=\"vertical-align: -16px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E, por fim, calculamos o \u00e2ngulo formado pelas duas retas com o inverso do cosseno: <\/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-216ef184adb4e8e26ea4dba3a0d41a67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha= \\text{cos}^{-1}(0,98) = \\bm{10,30\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"193\" 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 3<\/h3>\n<p> Qual \u00e9 o \u00e2ngulo entre as duas linhas 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-3a559370fd832ad2f4707782cf40cb37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ y=-2x+9 \\qquad s: \\ y=5x-1\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"272\" 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\"> Neste caso, utilizaremos o m\u00e9todo das inclina\u00e7\u00f5es das retas para descobrir o \u00e2ngulo que elas formam, j\u00e1 que as retas est\u00e3o na forma de uma equa\u00e7\u00e3o expl\u00edcita.<\/p>\n<p class=\"has-text-align-left\"> A inclina\u00e7\u00e3o de cada linha \u00e9 o n\u00famero que acompanha a vari\u00e1vel independente <\/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-845f2902b8bebf60c3c7372a7fbe4d02_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"x:\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"19\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-42a36a89145f23919d8665908c3e2bc3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_r = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"69\" style=\"vertical-align: -3px;\"><\/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-42de98336b7cbc2dc475ea3037bebc55_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_s = 5\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, o \u00e2ngulo entre as duas linhas pode ser determinado aplicando a f\u00f3rmula da 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-551288f526b75201969ebf9117fc9b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{tg}(\\alpha) =\\begin{vmatrix} \\cfrac{m_s-m_r}{1+m_r\\cdot m_s} \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"165\" style=\"vertical-align: -17px;\"><\/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-172bd34124a3b9e86696158d992eebb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{tg}(\\alpha) =\\begin{vmatrix} \\cfrac{-2-5}{1+5\\cdot (-2)} \\end{vmatrix}=\\begin{vmatrix} \\cfrac{-7}{-9} \\end{vmatrix} = 0,78\" title=\"Rendered by QuickLaTeX.com\" height=\"46\" width=\"293\" style=\"vertical-align: -19px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E finalmente encontramos o \u00e2ngulo entre as duas retas invertendo a tangente: <\/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-6c84b4105875a851c576fb326e2ba6f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\alpha= \\text{tg}^{-1}(0,78) = \\bm{37,87\u00ba}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"185\" 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> Encontre a equa\u00e7\u00e3o da reta 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-6958f848b3f39930bc315b56f627f888_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(5,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"66\" style=\"vertical-align: -5px;\"><\/p>\n<p> e faz um \u00e2ngulo de 45\u00ba com a linha<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-aa03a29f511592c1a1ecc8b306b0cf0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"r.\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"12\" style=\"vertical-align: 0px;\"><\/p>\n<p> Seja dita 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-34edcc0a8f3b1c557be083882ab8b7e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle r: \\ y=2x+4\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"112\" 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 resolver o problema, ligaremos<\/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-ae1901659f469e6be883797bfd30f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\"><\/p>\n<p> \u00e0 direita que vamos calcular. Al\u00e9m disso, usaremos o m\u00e9todo da inclina\u00e7\u00e3o porque conhecemos a inclina\u00e7\u00e3o da reta<\/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-9edd8ad155030a560ef8313513b5ac14_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"m_r=2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"55\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A partir da f\u00f3rmula do \u00e2ngulo entre duas retas (m\u00e9todo da inclina\u00e7\u00e3o) podemos obter o valor da inclina\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-580a84fe09f12aa20c352a8336880e41_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"s:\" 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-551288f526b75201969ebf9117fc9b1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{tg}(\\alpha) =\\begin{vmatrix} \\cfrac{m_s-m_r}{1+m_r\\cdot m_s} \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"165\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Substitu\u00edmos os valores conhecidos 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-ace7fdfc7474a43a6fad81e0185d0050_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\text{tg}(45\u00ba) =\\begin{vmatrix} \\cfrac{m_s-2}{1+2\\cdot m_s} \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"158\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E tentamos resolver a equa\u00e7\u00e3o resultante:<\/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-0f2ea68d68e24c3e30df526dfb88873c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 1 =\\begin{vmatrix} \\cfrac{m_s-2}{1+2m_s} \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"105\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O valor absoluto da equa\u00e7\u00e3o dificulta um pouco a solu\u00e7\u00e3o, pois \u00e9 necess\u00e1rio analisar tanto as op\u00e7\u00f5es positivas quanto as negativas: <\/p>\n<div class=\"wp-block-columns is-layout-flex wp-container-149\">\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7ad2ad90ee94f08746cea11db3a6917f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 1 =+\\cfrac{m_s-2}{1+2m_s}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"110\" style=\"vertical-align: -15px;\"><\/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-56c4a065c3919fbe023153fe2ba9133c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 1 \\cdot (1+2m_s)=m_s-2\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"173\" 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-d31fa5d4b608e062c0e15476b3f15e7f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 1+2m_s=m_s-2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"137\" style=\"vertical-align: -3px;\"><\/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-870381f6e915e33b32ad147c9a4de5fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2m_s-m_s=-2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"152\" style=\"vertical-align: -3px;\"><\/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-1d7c13f0d7d21af8c407f7f535e0d994_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m_s=-3\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"69\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<\/div>\n<div class=\"wp-block-column is-layout-flow\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4b9e6f5eb952e8ea9758af5497ed8cf2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 1 =-\\cfrac{m_s-2}{1+2m_s}\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"110\" style=\"vertical-align: -15px;\"><\/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-2b984c3bc74269597752a909f457c8ff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 1 \\cdot (1+2m_s)=-(m_s-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"200\" 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-df53d7b057ec0e6b6f7701fb5149cbe0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 1+2m_s=-m_s+2\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"151\" style=\"vertical-align: -3px;\"><\/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-9a96107cb59f52dcf9bb703e54c27757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2m_s+m_s=2-1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"138\" style=\"vertical-align: -3px;\"><\/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-067dac6d6aea31f65caccf5ed9c30052_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 3m_s=1\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"63\" style=\"vertical-align: -3px;\"><\/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-9003dbf8679e5b0aaa8c10777f6d38fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle m_s=\\cfrac{1}{3}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"57\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Temos, portanto, duas solu\u00e7\u00f5es poss\u00edveis: uma reta com inclina\u00e7\u00e3o -3 e outra reta com inclina\u00e7\u00e3o um ter\u00e7o.<\/p>\n<p class=\"has-text-align-left\"> A f\u00f3rmula para a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o de uma 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-3441e1da8c7da5805b1133af77b14f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"y-y_0=m(x-x_0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"149\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, uma vez conhecida a inclina\u00e7\u00e3o das duas retas poss\u00edveis, podemos escrever a equa\u00e7\u00e3o ponto-inclina\u00e7\u00e3o de cada reta com o ponto pelo qual elas devem passar de acordo com a afirma\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-c8726fb72614f7e8f7e546f9ac6995cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"P(5,-1):\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" 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-0a69157a1cf6a00b750b804590e63524_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle s: \\ y+1=-3(x-5) \\qquad \\qquad s': \\ y+1=\\cfrac{1}{3}(x-5)\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"402\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o de como calcular o \u00e2ngulo entre duas retas (f\u00f3rmula). Voc\u00ea tamb\u00e9m poder\u00e1 ver diversos exemplos e, al\u00e9m disso, poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo. Qual \u00e9 o \u00e2ngulo entre duas linhas? O \u00e2ngulo entre duas linhas \u00e9 o menor \u00e2ngulo entre essas duas linhas. No plano existem &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/angulo-entre-duas-linhas-exemplos-de-formulas-exercicios-resolvidos-declives-vetor-diretor\/\"> <span class=\"screen-reader-text\">\u00c2ngulo entre duas linhas (f\u00f3rmula)<\/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-227","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>\u00c2ngulo entre duas linhas (f\u00f3rmula) - 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\/angulo-entre-duas-linhas-exemplos-de-formulas-exercicios-resolvidos-declives-vetor-diretor\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"\u00c2ngulo entre duas linhas (f\u00f3rmula) - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea encontrar\u00e1 a explica\u00e7\u00e3o de como calcular o \u00e2ngulo entre duas retas (f\u00f3rmula). Voc\u00ea tamb\u00e9m poder\u00e1 ver diversos exemplos e, al\u00e9m disso, poder\u00e1 praticar com exerc\u00edcios resolvidos passo a passo. Qual \u00e9 o \u00e2ngulo entre duas linhas? O \u00e2ngulo entre duas linhas \u00e9 o menor \u00e2ngulo entre essas duas linhas. 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