{"id":218,"date":"2023-07-11T05:13:14","date_gmt":"2023-07-11T05:13:14","guid":{"rendered":"https:\/\/mathority.org\/pt\/vetores-paralelos\/"},"modified":"2023-07-11T05:13:14","modified_gmt":"2023-07-11T05:13:14","slug":"vetores-paralelos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/vetores-paralelos\/","title":{"rendered":"Vetores paralelos"},"content":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 tudo sobre vetores paralelos: o que significam, quando dois vetores s\u00e3o paralelos, como encontrar um vetor paralelo a outro vetor, as propriedades deste tipo de vetor,\u2026 Al\u00e9m disso, voc\u00ea poder\u00e1 ver v\u00e1rios exemplos e exerc\u00edcios de vetores paralelos resolvidos. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-son-los-vectores-paralelos\"><\/span> O que s\u00e3o vetores paralelos? <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p class=\"has-background\" style=\"background-color:#ffe0aa\"> <strong>Vetores paralelos<\/strong> s\u00e3o vetores que possuem a mesma dire\u00e7\u00e3o. Em outras palavras, dois vetores s\u00e3o paralelos se estiverem contidos em duas retas paralelas. Portanto, dois vetores paralelos formam um \u00e2ngulo entre eles de 0 ou 180 graus.<\/p>\n<p> Por exemplo, os tr\u00eas vetores a seguir s\u00e3o paralelos: <\/p>\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/que-sont-deux-vecteurs-paralleles.webp\" alt=\"Quais s\u00e3o dois vetores paralelos?\" class=\"wp-image-1210\" width=\"172\" height=\"179\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Al\u00e9m disso, o paralelismo de dois vetores depende apenas da sua dire\u00e7\u00e3o. Ou seja, dois vetores ser\u00e3o paralelos se coincidirem em dire\u00e7\u00e3o, quer tenham a mesma dire\u00e7\u00e3o ou dire\u00e7\u00e3o oposta. E o mesmo acontece com o m\u00f3dulo (ou magnitude), dois vetores podem ter m\u00f3dulos diferentes e ser paralelos.<\/p>\n<p> Por outro lado, quando dois vetores t\u00eam a mesma dire\u00e7\u00e3o, mas opostas, eles s\u00e3o chamados <strong>de vetores antiparalelos<\/strong> . <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcomo-se-sabe-cuando-dos-vectores-son-paralelos\"><\/span> Como voc\u00ea sabe se dois vetores s\u00e3o paralelos? <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p class=\"has-background\" style=\"background-color:#ffe0aa\"> <strong>Dois vetores s\u00e3o paralelos quando s\u00e3o proporcionais.<\/strong> Portanto, para saber se dois vetores s\u00e3o paralelos, precisamos determinar se as suas respectivas componentes s\u00e3o proporcionais ou n\u00e3o.<\/p>\n<p> Veremos como saber se dois vetores s\u00e3o paralelos atrav\u00e9s de dois exerc\u00edcios resolvidos diferentes, um com vetores com 2 coordenadas e outro com vetores com 3 coordenadas. <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-vectores-paralelos-en-el-plano-en-r2\"><\/span> Exemplo de vetores paralelos ao plano (em R2)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li> Determine se os dois vetores a seguir s\u00e3o paralelos:<\/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-c173e0154210a443e280bb34e4966353_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}=(-2,4) \\qquad\\vv{\\text{v}}=(1,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"208\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para saber se s\u00e3o realmente vetores paralelos, devemos verificar se suas coordenadas cartesianas s\u00e3o proporcionais:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-b970d4aa09dea3c37f9ae2a3586bc175_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{-2}{1} = \\cfrac{4}{-2} = -2\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"124\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> Dividir as componentes X e as componentes Y entre elas d\u00e1 o mesmo resultado (-2), portanto os dois vetores s\u00e3o proporcionais e portanto tamb\u00e9m <strong>paralelos<\/strong> .<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-c66764b9ce657ef712e852b979d40918_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\parallel \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"38\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Observe que em matem\u00e1tica, quando dois elementos geom\u00e9tricos s\u00e3o paralelos, isso \u00e9 indicado por duas barras verticais (II). <\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-de-vectores-paralelos-en-el-espacio-en-r3\"><\/span> Exemplo de vetores paralelos no espa\u00e7o (em R3)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li> Descubra se a condi\u00e7\u00e3o de paralelismo \u00e9 satisfeita nos dois vetores 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-08ef4919348b52567179757f34e1bad3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}=(1,3,-2) \\qquad\\vv{\\text{v}}=(2,6,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"227\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para determinar se s\u00e3o de fato vetores paralelos, devemos verificar se as coordenadas dos vetores s\u00e3o proporcionais:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4c13df7c3c1c0daf95938c3b2e391ce4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\cfrac{1}{2} = \\cfrac{3}{6} = 0,5  \\neq \\cfrac{-2}{4} = -0,5\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"211\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p> As componentes X e as componentes Y dos vetores s\u00e3o proporcionais entre si porque ao dividi-las obtemos o mesmo resultado, por outro lado, n\u00e3o s\u00e3o proporcionais \u00e0 componente Z. Portanto, os vetores n\u00e3o s\u00e3o proporcionais a todos e, portanto, <strong>n\u00e3o s\u00e3o paralelos<\/strong> . <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-0369ae5c4f9eba22ba004b8ad22a3791_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\ \\cancel{\\parallel} \\ \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"40\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcomo-calcular-un-vector-paralelo\"><\/span> Como calcular um vetor paralelo? <span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p class=\"has-background\" style=\"background-color:#ffe0aa\"> Para encontrar um vetor paralelo a outro vetor, basta <strong>multiplic\u00e1-lo por um escalar<\/strong> (um n\u00famero real) diferente de zero (0). H\u00e1, portanto, um n\u00famero infinito de vetores paralelos entre si, pois o vetor pode ser multiplicado por um n\u00famero infinito de n\u00fameros.<\/p>\n<p> Por exemplo, calcularemos v\u00e1rios vetores paralelos do seguinte vetor:<\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-217fe4e1853355088713c8976a72dfdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}}=(2,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> O resultado de todos os produtos a seguir s\u00e3o vetores paralelos ao vetor anterior: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-f43bc21e50858797698ead97f18ab01e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\vv{\\text{v}}=(4,8)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"80\" 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-4e5cf4e24b83fa659231fe8849030ac0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3\\vv{\\text{v}}=(6,12)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" 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-7df6eabe318b8ed9a2e9fd7a6e968e09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-1\\vv{\\text{v}}=(-2,-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" 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-089421914e4d52862a277469fd332a22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{2}\\vv{\\text{v}}=(1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"82\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-de-los-vectores-paralelos\"><\/span> Propriedades de vetores paralelos<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Vetores paralelos t\u00eam as seguintes caracter\u00edsticas:<\/p>\n<ul>\n<li> <strong>Propriedade reflexiva<\/strong> : cada vetor \u00e9 paralelo a si mesmo.<\/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-4be4d866adc6357e0dd4119dfc7fad9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{v}} \\parallel  \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"37\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Propriedade sim\u00e9trica<\/strong> : se um vetor \u00e9 paralelo a outro, esse vetor tamb\u00e9m \u00e9 paralelo ao primeiro. Esta propriedade tamb\u00e9m \u00e9 possu\u00edda por <a href=\"https:\/\/mathority.org\/pt\/vetores-perpendiculares-ortogonais\/\">vetores perpendiculares<\/a> .<\/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-1af75669cfe20e558447253ac3bd5ff1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\parallel  \\vv{\\text{v}} \\ \\longrightarrow \\ \\vv{\\text{v}} \\parallel \\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Propriedade transitiva<\/strong> : se um vetor \u00e9 paralelo a outro vetor, e este segundo vetor \u00e9 paralelo a um terceiro vetor, o primeiro vetor tamb\u00e9m \u00e9 paralelo ao terceiro vetor.<\/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-97b12a1e00fb21369eea8ce80b3e1c72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\left. \\begin{array}{c} \\vv{\\text{u}} \\parallel  \\vv{\\text{v}} \\\\[2ex] \\vv{\\text{v}} \\parallel  \\vv{\\text{w}} \\end{array} \\right\\} \\longrightarrow \\ \\vv{\\text{u}} \\parallel  \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"151\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> O produto escalar de dois vetores paralelos \u00e9 igual ao produto de seus m\u00f3dulos. Voc\u00ea pode verificar por que isso est\u00e1 acontecendo em particular nas <a href=\"https:\/\/mathority.org\/pt\/calcular-o-produto-escalar-entre-dois-vetores-exemplos-exercicios-resolvidos\/\">propriedades do produto escalar<\/a> .<\/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-af3ec9660e232f8b69bdc0f0aa194027_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\parallel \\vv{\\text{v}} \\ \\longrightarrow \\ \\vv{\\text{u}} \\cdot \\vv{\\text{v}}= \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}} \\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"194\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Dois vetores paralelos s\u00e3o sempre linearmente dependentes. Este conceito \u00e9 muito importante, portanto, se voc\u00ea n\u00e3o o conhece, pode consultar <a href=\"https:\/\/mathority.org\/pt\/vetores-independentes-e-linearmente-dependentes-independencia-dependencia-linear\/\">o que s\u00e3o dois vetores linearmente dependentes<\/a> .<\/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-1a467a484d13c5204dd6aba944f29ebd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\parallel \\vv{\\text{v}} \\ \\longrightarrow \\ LD\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -5px;\"><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Nesta p\u00e1gina voc\u00ea encontrar\u00e1 tudo sobre vetores paralelos: o que significam, quando dois vetores s\u00e3o paralelos, como encontrar um vetor paralelo a outro vetor, as propriedades deste tipo de vetor,\u2026 Al\u00e9m disso, voc\u00ea poder\u00e1 ver v\u00e1rios exemplos e exerc\u00edcios de vetores paralelos resolvidos. O que s\u00e3o vetores paralelos? Vetores paralelos s\u00e3o vetores que possuem a &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/vetores-paralelos\/\"> <span class=\"screen-reader-text\">Vetores paralelos<\/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":[27],"tags":[],"class_list":["post-218","post","type-post","status-publish","format-standard","hentry","category-vetores"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.2 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Vetores paralelos - 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\/vetores-paralelos\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Vetores paralelos - Mathority\" \/>\n<meta property=\"og:description\" content=\"Nesta p\u00e1gina voc\u00ea encontrar\u00e1 tudo sobre vetores paralelos: o que significam, quando dois vetores s\u00e3o paralelos, como encontrar um vetor paralelo a outro vetor, as propriedades deste tipo de vetor,\u2026 Al\u00e9m disso, voc\u00ea poder\u00e1 ver v\u00e1rios exemplos e exerc\u00edcios de vetores paralelos resolvidos. 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