{"id":71,"date":"2023-09-16T13:03:21","date_gmt":"2023-09-16T13:03:21","guid":{"rendered":"https:\/\/mathority.org\/pt\/produto-vetorial-de-dois-vetores-exemplos-de-formulas-cruzadas-exercicios-resolvidos\/"},"modified":"2023-09-16T13:03:21","modified_gmt":"2023-09-16T13:03:21","slug":"produto-vetorial-de-dois-vetores-exemplos-de-formulas-cruzadas-exercicios-resolvidos","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/produto-vetorial-de-dois-vetores-exemplos-de-formulas-cruzadas-exercicios-resolvidos\/","title":{"rendered":"Produto vetorial de dois vetores (ou produto vetorial)"},"content":{"rendered":"<p>Esta p\u00e1gina explica o que \u00e9 o produto vetorial de dois vetores e como ele \u00e9 calculado. Voc\u00ea tamb\u00e9m ver\u00e1 como encontrar a dire\u00e7\u00e3o e dire\u00e7\u00e3o do produto vetorial usando a regra da m\u00e3o direita (ou saca-rolhas). E mais, voc\u00ea encontrar\u00e1 as utiliza\u00e7\u00f5es deste tipo de opera\u00e7\u00e3o, al\u00e9m de exemplos, exerc\u00edcios e problemas resolvidos passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfque-es-el-producto-vectorial-de-dos-vectores\"><\/span> Qual \u00e9 o produto vetorial de dois vetores?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Em matem\u00e1tica, o <strong>produto vetorial<\/strong> \u00e9 uma opera\u00e7\u00e3o entre dois vetores no espa\u00e7o tridimensional (em R3). O resultado desta opera\u00e7\u00e3o vetorial \u00e9 um vetor com dire\u00e7\u00e3o perpendicular aos dois vetores multiplicados, e com m\u00f3dulo igual ao produto dos m\u00f3dulos dos vetores multiplicadores pelo seno do \u00e2ngulo que eles formam. Em outras palavras, sua f\u00f3rmula \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-625267c9d98347748a771c7cec9bfcec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\times \\vv{\\text{v}}\\rvert = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}}\\rvert \\cdot \\text{sen}(\\alpha)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"186\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como voc\u00ea pode ver na f\u00f3rmula anterior, o produto vetorial \u00e9 denotado<\/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-850177c0489097bd9409ba9b13b07506_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\bm{\\times}\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"10\" style=\"vertical-align: 0px;\"><\/p>\n<p> , por isso tamb\u00e9m \u00e9 chamado de <strong>produto vetorial.<\/strong> \u00c0s vezes tamb\u00e9m \u00e9 chamado de produto vetorial de Gibbs, j\u00e1 que foi ele quem o inventou. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/quel-est-le-produit-vectoriel-de-deux-vecteurs-.webp\" alt=\"\" class=\"wp-image-3883\" width=\"307\" height=\"310\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> Como voc\u00ea pode ver na representa\u00e7\u00e3o gr\u00e1fica anterior, o produto vetorial \u00e9 perpendicular aos dois vetores que eles multiplicam e, portanto, \u00e9 normal ao plano que os cont\u00e9m. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"formula-para-calcular-el-producto-vectorial-de-dos-vectores\"><\/span> F\u00f3rmula para calcular o produto vetorial de dois vetores<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Se conhecermos as coordenadas cartesianas dos vetores, a maneira mais simples de calcular seu produto vetorial \u00e9 resolver um determinante 3&#215;3. Veja como \u00e9 feito: <\/p>\n<div style=\"background-color:#FFCC8080;padding-top: 20px; padding-bottom: 0.5px; padding-right: 30px; padding-left: 30px; border: 2px solid #FFB74D; border-radius:20px;\">\n<p style=\"text-align:left\"> Considere quaisquer 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-581394386a4c68ca2bfa92fb4e2445ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (\\text{u}_x,\\text{u}_y,\\text{u}_z) \\qquad \\vv{\\text{v}}= (\\text{v}_x,\\text{v}_y,\\text{v}_z)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"267\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Seu produto vetorial \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-56551111a4f5a18a4609772ebaeaf919_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\begin{vmatrix} \\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] \\text{u}_x &amp; \\text{u}_y &amp; \\text{u}_z \\\\[1.1ex] \\text{v}_x &amp;\\text{v}_y&amp;\\text{v}_z \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"159\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p style=\"text-align:left\"> Onde os 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-899f7cb82c85508ac2129e2393976f80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{i}, \\vv{j},\\vv{k}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"38\" style=\"vertical-align: -4px;\"><\/p>\n<p> Estes s\u00e3o os vetores unit\u00e1rios nas dire\u00e7\u00f5es dos eixos X, Y e Z, respectivamente.<\/p>\n<\/div>\n<p> Vejamos um exemplo de como calcular o produto vetorial entre 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-89a657062237b32001dad723a07ad2ae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (3,1,0) \\qquad \\vv{\\text{v}}= (2,1,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"227\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para determinar o produto vetorial entre os vetores, devemos fazer o seguinte determinante de ordem 3:<\/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-abc77b698bf6f4fddec1ab2dcc8b07f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\begin{vmatrix} \\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] 3&amp; 1 &amp; 0 \\\\[1.1ex] 2 &amp;1&amp;-1 \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"145\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Neste caso, resolveremos o determinante por adjuvantes ou cofatores (a regra de Sarrus tamb\u00e9m poderia ser usada):<\/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-eeeac04b3f0edd64e5413629051551fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\vv{\\text{u}} \\times \\vv{\\text{v}}=\\begin{vmatrix} \\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] 3&amp; 1 &amp; 0 \\\\[1.1ex] 2 &amp;1&amp;-1 \\end{vmatrix} &amp; = \\vv{i}\\begin{vmatrix} 1 &amp; 0 \\\\[1.1ex] 1&amp;-1 \\end{vmatrix} -\\vv{j}\\begin{vmatrix}  3&amp;  0 \\\\[1.1ex] 2 &amp;-1 \\end{vmatrix}+\\vv{z}\\begin{vmatrix}3&amp; 1 \\\\[1.1ex] 2 &amp;1 \\end{vmatrix}  \\\\[2ex] &amp; = -\\vv{i}+3\\vv{j}+\\vv{z}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"411\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> O resultado do produto vetorial dos dois vetores \u00e9, portanto: <\/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-625f3af33d3cd3b9991682014c911024_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\bm{(-1,3,1)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"determinar-la-direccion-y-el-sentido-del-producto-vectorial\"><\/span> Determina a dire\u00e7\u00e3o e dire\u00e7\u00e3o do produto vetorial<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> \u00c0s vezes n\u00e3o precisamos conhecer as componentes do vetor resultante do produto vetorial, mas basta encontrar seu m\u00f3dulo, sua dire\u00e7\u00e3o e seu sentido. Isso acontece frequentemente na f\u00edsica, especialmente no c\u00e1lculo de for\u00e7as.<\/p>\n<p> Assim, existem diversas regras para encontrar a dire\u00e7\u00e3o e sentido do produto vetorial, as mais conhecidas s\u00e3o a <strong>regra da m\u00e3o direita<\/strong> , seja com tr\u00eas dedos ou com a m\u00e3o inteira, e a <strong>regra do saca-rolhas (ou do parafuso)<\/strong> . Voc\u00ea pode usar qualquer uma delas, ent\u00e3o n\u00e3o precisa conhecer todas, ainda vamos te explicar as tr\u00eas regras para que voc\u00ea fique com a que mais gosta. \ud83d\ude09<\/p>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"regla-de-la-mano-derecha-3-dedos\"><\/span> Regra da m\u00e3o direita (3 dedos)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A vers\u00e3o de tr\u00eas dedos da regra ou lei da m\u00e3o direita envolve a execu\u00e7\u00e3o das seguintes etapas:<\/p>\n<ol style=\"color:#ff6f00; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#000000;font-weight: normal;\">Coloque o dedo indicador da m\u00e3o direita em dire\u00e7\u00e3o ao primeiro vetor do produto vetorial\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a13b5c3a9f833efc0ddf4caf790f0483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"27\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Coloque o dedo m\u00e9dio (ou dedo m\u00e9dio) da m\u00e3o direita em dire\u00e7\u00e3o ao segundo vetor do produto vetorial\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-acab6198e5d0337e7d0e9ed7814c16d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{v}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"26\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">A posi\u00e7\u00e3o do polegar resultante indica a dire\u00e7\u00e3o e dire\u00e7\u00e3o do produto vetorial\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2fce88450a631ec24e37f45befae0675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}\\times\\vv{\\text{v}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> <\/li>\n<\/ol>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/regle-ou-loi-de-la-main-droite.webp\" alt=\"\" class=\"wp-image-892\" width=\"392\" height=\"353\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"regla-de-la-mano-derecha-palma-de-la-mano\"><\/span> Regra da m\u00e3o direita (palma da m\u00e3o)<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A vers\u00e3o palmar da regra ou lei da m\u00e3o direita \u00e9 muito semelhante \u00e0 regra anterior. Para aplic\u00e1-lo, voc\u00ea deve seguir os seguintes passos:<\/p>\n<ol style=\"color:#ff6f00; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#000000;font-weight: normal;\">Coloque sua m\u00e3o direita apontando com os dedos na mesma dire\u00e7\u00e3o do primeiro vetor do produto vetorial\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a13b5c3a9f833efc0ddf4caf790f0483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"27\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Feche a m\u00e3o direita movendo os dedos em dire\u00e7\u00e3o ao segundo vetor do produto vetorial\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-acab6198e5d0337e7d0e9ed7814c16d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{v}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"26\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> Voc\u00ea precisa fechar a m\u00e3o do lado onde o \u00e2ngulo (ou dist\u00e2ncia) entre os vetores \u00e9 menor.<\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">A posi\u00e7\u00e3o resultante do polegar determina a dire\u00e7\u00e3o do produto vetorial\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2fce88450a631ec24e37f45befae0675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}\\times\\vv{\\text{v}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> <\/li>\n<\/ol>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/vecteur-produit-regle-de-droite.webp\" alt=\"produto de vetor de r\u00e9gua da m\u00e3o direita\" class=\"wp-image-898\" width=\"373\" height=\"342\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"regla-del-sacacorchos\"><\/span> regra saca-rolhas<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p> A <strong>regra do saca-rolhas ou parafuso<\/strong> \u00e9 semelhante \u00e0 regra da m\u00e3o direita usando toda a palma da m\u00e3o. O procedimento \u00e9 o seguinte:<\/p>\n<ol style=\"color:#ff6f00; font-weight: bold;>\n<li><span style=\" color:#262626;font-weight:=\"\" normal;\"=\"\">\n<li style=\"margin-bottom:18px\"><span style=\"color:#000000;font-weight: normal;\">Usando sua imagina\u00e7\u00e3o, coloque um saca-rolhas (ou parafuso) com a al\u00e7a apontando na mesma dire\u00e7\u00e3o do primeiro vetor do produto vetorial.\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-a13b5c3a9f833efc0ddf4caf790f0483_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"27\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Em seguida, gire o saca-rolhas em dire\u00e7\u00e3o ao segundo vetor do produto vetorial\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-4c87e248254db96e4d2c996a62911e87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{v}})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"21\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> como se voc\u00ea fosse coloc\u00e1-lo em uma rolha. Voc\u00ea precisa girar o saca-rolhas para o lado onde a dist\u00e2ncia entre os vetores \u00e9 menor.<\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">A dire\u00e7\u00e3o para a qual aponta a espiral do saca-rolhas ser\u00e1 a dire\u00e7\u00e3o e a dire\u00e7\u00e3o do produto vetorial\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-2fce88450a631ec24e37f45befae0675_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(\\vv{\\text{u}}\\times\\vv{\\text{v}}).\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -5px;\"><\/p>\n<p><\/span> <\/li>\n<\/ol>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/regle-du-tire-bouchon-ou-de-la-vis.webp\" alt=\"saca-rolhas ou r\u00e9gua de parafuso\" class=\"wp-image-902\" width=\"327\" height=\"500\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-del-producto-vectorial-de-dos-vectores\"><\/span> Propriedades do produto vetorial de dois vetores<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> O produto vetorial de dois vetores tem as seguintes caracter\u00edsticas:<\/p>\n<ul>\n<li> <strong>Propriedade anticomutativa:<\/strong> a ordem dos vetores envolvidos no produto vetorial n\u00e3o \u00e9 indiferente, pois o sinal varia em fun\u00e7\u00e3o dela.<\/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-da6aeea2768e41a7e3630dc83ff1e31b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\\times\\vv{\\text{v}} = - \\vv{\\text{v}}\\times\\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"9\" width=\"119\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Propriedade distributiva<\/strong> relativa \u00e0 adi\u00e7\u00e3o e subtra\u00e7\u00e3o de vetores:<\/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-bfddca49d7ded207f392f54341fff56d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\\times(\\vv{\\text{v}}+\\vv{\\text{w}}) = \\vv{\\text{u}}\\times\\text{v}}+\\vv{\\text{u}}\\times \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"220\" 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-68e08abafcbf4b05614819ca1a364a47_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}\\times(\\vv{\\text{v}}-\\vv{\\text{w}}) = \\vv{\\text{u}}\\times\\text{v}}-\\vv{\\text{u}}\\times \\vv{\\text{w}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"220\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Propriedade homog\u00eanea<\/strong> : multiplicar um vetor do produto vetorial por um escalar (um n\u00famero real) equivale a multiplicar o resultado do produto vetorial pelo referido escalar.<\/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-2ee3552bb6b72db4d30b4aa9f73e99a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k \\cdot (\\vv{\\text{u}}\\times\\vv{\\text{v}}) =  (k\\cdot \\vv{\\text{u}})\\times\\vv{\\text{v}}=\\vv{\\text{u}}\\times(k\\cdot\\vv{\\text{v}})\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"278\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> O vetor resultante do produto vetorial \u00e9 <strong>perpendicular<\/strong> aos dois vetores envolvidos na opera\u00e7\u00e3o.<\/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-6a96345e09a0fdb952557c9138c72ac4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{array}{c} \\vv{\\text{u}} \\perp (\\vv{\\text{u}}\\times\\vv{\\text{v}}) \\\\[2ex] \\vv{\\text{v}} \\perp (\\vv{\\text{u}}\\times\\vv{\\text{v}}) \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"57\" width=\"87\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> Al\u00e9m disso, se os dois vetores forem ortogonais, as seguintes equa\u00e7\u00f5es ser\u00e3o satisfeitas:<\/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-d49d463798c6381c9a8c065417ee3dbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\perp \\vv{\\text{v}} \\ \\longrightarrow \\ \\begin{cases} \\vv{\\text{u}} \\cdot (\\vv{\\text{u}}\\times\\vv{\\text{v}})=0 \\\\[2ex] \\vv{\\text{v}} \\cdot (\\vv{\\text{u}}\\times\\vv{\\text{v}})=0 \\end{array}\" title=\"Rendered by QuickLaTeX.com\" height=\"65\" width=\"220\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<ul>\n<li> O produto vetorial de dois <strong>vetores paralelos<\/strong> \u00e9 igual ao vetor zero (ou zero).<\/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-5a2e4e335c132d024b2932139a51f101_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\ || \\ \\vv{\\text{v}} \\ \\longrightarrow \\ \\vv{\\text{u}}\\times\\vv{\\text{v}}=0\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"165\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> Se n\u00e3o conhecemos o \u00e2ngulo formado por dois vetores, o m\u00f3dulo do seu produto vetorial tamb\u00e9m pode ser calculado usando a seguinte express\u00e3o: <\/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-f27e24e2a7310fb1e727891c17de911a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}}\\times\\vv{\\text{v}} \\rvert = \\sqrt{ \\lvert \\vv{\\text{u}}\\rvert ^2 \\cdot \\lvert \\vv{\\text{v}} \\rvert ^2 - (\\vv{\\text{u}}\\cdot \\vv{\\text{v}})^2 \\vphantom{\\frac{1}{2}}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"233\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"calcular-el-area-de-un-paralelogramo-o-un-triangulo-mediante-el-producto-vectorial\"><\/span> Calcule a \u00e1rea de um paralelogramo ou tri\u00e2ngulo usando o produto vetorial<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Geometricamente, o m\u00f3dulo do produto vetorial de dois vetores coincide com a \u00e1rea do paralelogramo que tem esses dois vetores como lados. Portanto, <strong>o produto vetorial pode ser usado para calcular a \u00e1rea de um paralelogramo.<\/strong> <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/produit-vectoriel-de-deux-vecteurs-dans-lespace.webp\" alt=\"produto vetorial de dois vetores no espa\u00e7o\" class=\"wp-image-929\" width=\"300\" height=\"173\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> Al\u00e9m disso, a diagonal de um paralelogramo o divide em dois tri\u00e2ngulos, ou em outras palavras, um tri\u00e2ngulo \u00e9 metade de um paralelogramo. Assim, a <strong>\u00e1rea de um tri\u00e2ngulo<\/strong> \u00e9 metade do m\u00f3dulo do produto vetorial tomando dois de seus lados como vetores. <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/produit-vectoriel-de-deux-vecteurs-dans-r2.webp\" alt=\"produto vetorial de dois vetores em r2\" class=\"wp-image-927\" width=\"300\" height=\"173\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> Lembre-se de que o m\u00f3dulo de um vetor em um espa\u00e7o tridimensional \u00e9 a raiz da soma dos quadrados de suas coordenadas:<\/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-8cf7995798482007ea32b809e80a4062_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{v}} \\rvert =  \\sqrt{\\vphantom{\\frac{1}{2}} \\text{v}_x^2+\\text{v}_y^2+\\text{v}_z^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"155\" style=\"vertical-align: -11px;\"><\/p>\n<\/p>\n<p> Estas s\u00e3o duas das aplica\u00e7\u00f5es do produto vetorial de dois vetores no campo da matem\u00e1tica. Por\u00e9m, ainda tem outros usos, por exemplo, na f\u00edsica \u00e9 usado para calcular o campo magn\u00e9tico. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-productos-vectoriales-de-vectores\"><\/span> Exerc\u00edcios resolvidos sobre produtos vetoriais de vetores<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Calcule o produto vetorial entre os dois vetores 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-a1c06e6d75f54661416056d31d409d8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (-1,4,2) \\qquad \\vv{\\text{v}}= (0,-2,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para determinar o produto vetorial entre os vetores, devemos resolver o seguinte determinante de dimens\u00e3o 3\u00d73:<\/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-44db63ee02936f6e5f21891c3e412fb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\begin{vmatrix} \\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] -1&amp; 4 &amp; 2 \\\\[1.1ex] 0 &amp;-2&amp;1  \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"160\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Neste caso, resolveremos o determinante por adjuvantes ou cofatores (mas a regra de Sarrus tamb\u00e9m poderia ser usada):<\/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-fe298c37814c92498e4fd8ade0620951_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\begin{vmatrix}\\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] -1&amp; 4 &amp; 2 \\\\[1.1ex] 0 &amp;-2&amp;1\\end{vmatrix} &amp; = \\vv{i}\\begin{vmatrix} 4 &amp; 2 \\\\[1.1ex]-2&amp;1\\end{vmatrix} -\\vv{j}\\begin{vmatrix}  -1&amp; 2 \\\\[1.1ex] 0 &amp;1\\end{vmatrix}+\\vv{z}\\begin{vmatrix}-1&amp; 4 \\\\[1.1ex] 0 &amp;-2\\end{vmatrix}  \\\\[2ex] &amp; = 8\\vv{i}+\\vv{j}+2\\vv{z}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"387\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O resultado do produto vetorial dos dois vetores \u00e9, portanto: <\/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-a1efe3e9fce5a83e193da24a5ff60835_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\bm{(8,1,2)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" 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 produto vetorial entre os dois vetores 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-210c7ec7b29e06198c491e83e7825a42_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (3,-2,4) \\qquad \\vv{\\text{v}}= (1,5,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para encontrar o produto vetorial entre os dois vetores, devemos resolver o seguinte determinante 3\u00d73:<\/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-6a23d8e45f9065f70c576e6b8db02465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\begin{vmatrix} \\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] 3&amp; -2 &amp; 4 \\\\[1.1ex] 1 &amp;5&amp;-3  \\end{vmatrix}\" title=\"Rendered by QuickLaTeX.com\" height=\"86\" width=\"159\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Neste caso, resolveremos o determinante por adjuntos ou cofatores (embora a regra de Sarrus possa ser usada de forma intercambi\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-02ffb40666893faa7677234065f3f85f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\begin{vmatrix}\\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex] 3&amp; -2 &amp; 4 \\\\[1.1ex] 1 &amp;5&amp;-3\\end{vmatrix} &amp; = \\vv{i}\\begin{vmatrix} -2 &amp; 4 \\\\[1.1ex] 5&amp;-3\\end{vmatrix} -\\vv{j}\\begin{vmatrix}  3&amp; 4 \\\\[1.1ex]  1&amp;-3\\end{vmatrix}+\\vv{z}\\begin{vmatrix}3&amp; -2  \\\\[1.1ex] 1 &amp;5\\end{vmatrix}  \\\\[2ex] &amp; = -14\\vv{i}+13\\vv{j}+17\\vv{z}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"386\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> O resultado do produto vetorial entre os dois vetores \u00e9, portanto: <\/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-a8f7c253fdd7c5be3635a42a3826acfc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} \\times \\vv{\\text{v}}=\\bm{(-14,13,17)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"160\" 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> Conhecendo os m\u00f3dulos de dois vetores e o \u00e2ngulo que eles formam:<\/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-b632ab1324e7ece2d4c1f5c54249b425_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"|\\vv{\\text{u}}|= 5 \\qquad |\\vv{\\text{v}}|= 6 \\qquad \\alpha = 30\u00ba\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"226\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Determine a magnitude do produto vetorial dos dois vetores. <\/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 a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Podemos calcular facilmente o m\u00f3dulo do produto vetorial entre os dois vetores aplicando a 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-06dcff41e0dcf31152f0047507056f24_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\lvert \\vv{\\text{u}} \\times \\vv{\\text{v}}\\rvert &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}}\\rvert \\cdot \\text{sen}(\\alpha) \\\\[2ex] &amp; = 5 \\cdot 6 \\cdot \\text{sen}(30\u00ba) \\\\[2ex] &amp;= 30 \\cdot 0,5 \\\\[2ex] &amp;= \\bm{15} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"140\" width=\"186\" style=\"vertical-align: 0px;\"><\/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> Dos seguintes vetores contidos no plano da tela: <\/p>\n<figure class=\"wp-block-image aligncenter size-large is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/uploads\/2023\/07\/produit-vectoriel-en-ligne-de-deux-vecteurs.webp\" alt=\"produto vetorial de dois vetores alinhados\" class=\"wp-image-938\" width=\"151\" height=\"231\" srcset=\"\" sizes=\"auto, \" data-src=\"\"><\/figure>\n<p> Calcule a magnitude, dire\u00e7\u00e3o e sentido do vetor resultante da seguinte opera\u00e7\u00e3o vetorial: <\/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-facaf3cf3a645b725e61c9fcb195e53c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{\\text{u}} \\times \\vv{\\text{v}}\\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Os dois vetores s\u00e3o perpendiculares, ent\u00e3o a norma do produto vetorial 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-d8f675fe7eb44c050c508c4771c0a439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\lvert \\vv{\\text{u}} \\times \\vv{\\text{v}}\\rvert &amp; = \\lvert \\vv{\\text{u}} \\rvert \\cdot \\lvert \\vv{\\text{v}}\\rvert \\cdot \\text{sen}(\\alpha) \\\\[2ex] &amp; = 3 \\cdot 4 \\cdot \\text{sen}(90\u00ba) \\\\[2ex] &amp;= 12 \\cdot 1 \\\\[2ex] &amp;= \\bm{12} \\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"140\" width=\"186\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Por outro lado, o vetor resultante do produto vetorial \u00e9 perpendicular aos dois vetores que participam da opera\u00e7\u00e3o, <strong>sua dire\u00e7\u00e3o ser\u00e1 portanto perpendicular \u00e0 tela.<\/strong><\/p>\n<p class=\"has-text-align-left\"> E finalmente, utilizando a regra da reta (ou saca-rolhas), podemos deduzir que <strong>a dire\u00e7\u00e3o do vetor resultante ser\u00e1 para o interior da tela.<\/strong><\/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> Calcule a \u00e1rea do paralelogramo que tem os seguintes vetores como dois de seus lados: <\/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-43326a03cb829fb91d8c265bddf92b8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}}= (2,3,-2) \\qquad \\vv{\\text{v}}= (5,0,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> A \u00e1rea de um paralelogramo coincide com o m\u00f3dulo do produto vetorial dos vetores que o formam. Portanto, calculamos o produto vetorial dos 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-0e7c1825be82d94c4eae49c73f509858_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\vv{\\text{u}}\\times \\vv{\\text{v}} = \\begin{vmatrix}\\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex]2&amp; 3 &amp; -2 \\\\[1.1ex] 5 &amp;0&amp;-1\\end{vmatrix} &amp; = \\vv{i}\\begin{vmatrix} 3 &amp; -2 \\\\[1.1ex] 0&amp;-1\\end{vmatrix} -\\vv{j}\\begin{vmatrix}  2&amp; -2 \\\\[1.1ex] 5 &amp;-1\\end{vmatrix}+\\vv{z}\\begin{vmatrix}2&amp; 3  \\\\[1.1ex] 5 &amp;0\\end{vmatrix}  \\\\[2ex] &amp; = -3\\vv{i}-8\\vv{j}-15\\vv{z}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"411\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E ent\u00e3o seu m\u00f3dulo: <\/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-164e46102f5d27babfec98f25d479fab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A=\\lvert\\vv{\\text{u}}\\times \\vv{\\text{v}} \\rvert = \\sqrt{\\vphantom{\\frac{1}{2}} (-3)^2+(-8)^2+(-15)^2}=\\bm{17,26} \\ \\mathbf{u}\\bm{^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"404\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n<h3 class=\"wp-block-heading\">Exerc\u00edcio 6<\/h3>\n<p> Encontre a \u00e1rea do tri\u00e2ngulo cujos v\u00e9rtices s\u00e3o os seguintes pontos: <\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-dc08f9ac339b7ac3787a449cf5558c68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A(2,1,0) \\qquad B(4,0,3)\\qquad C(-1,2,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"294\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<div class=\"wp-block-otfm-box-spoiler-start otfm-sp__wrapper otfm-sp__box js-otfm-sp-box__closed otfm-sp__E4F0FE\" role=\"button\" tabindex=\"0\" aria-expanded=\"false\" data-otfm-spc=\"#E4F0FE\" style=\"text-align:center\">\n<div class=\"otfm-sp__title\"> <strong>Veja a solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiramente devemos calcular os vetores que formam os lados do tri\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-f7661fc5ae3b35c76e0fe98203258962_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{AB} = B- A = (4,0,3)-(2,1,0) = (2,-1,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"351\" 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-5d3f55a240188777fdbb3d74cbf4f61f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{BC} =C- B =(-1,2,3)- (4,0,3) = (-5,2,0)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"366\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> A \u00e1rea de um tri\u00e2ngulo \u00e9 metade da magnitude do produto vetorial dos vetores que o formam. Portanto, calculamos o produto vetorial dos 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-42a0ae5858bcb681ee92ec1ed67424c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned} \\vv{\\text{u}}\\times \\vv{\\text{v}} = \\begin{vmatrix}\\vv{i}&amp; \\vv{j}&amp; \\vv{k} \\\\[1.1ex]2&amp; -1 &amp; 3 \\\\[1.1ex] -5 &amp;2&amp;0\\end{vmatrix} &amp; = \\vv{i}\\begin{vmatrix} -1 &amp; 3 \\\\[1.1ex] 2&amp;0\\end{vmatrix} -\\vv{j}\\begin{vmatrix}  2&amp;  3 \\\\[1.1ex] -5 &amp;0\\end{vmatrix}+\\vv{z}\\begin{vmatrix}2&amp; -1  \\\\[1.1ex] -5 &amp;2\\end{vmatrix}  \\\\[2ex] &amp; = -6\\vv{i}-15\\vv{j}-\\vv{z}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"128\" width=\"453\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Depois do seu m\u00f3dulo:<\/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-4a52c4a997a5b68d67b9ae52ca599661_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert\\vv{\\text{u}}\\times \\vv{\\text{v}} \\rvert = \\sqrt{\\vphantom{\\frac{1}{2}} (-6)^2+(-15)^2+(-1)^2}=16,19\" title=\"Rendered by QuickLaTeX.com\" height=\"32\" width=\"342\" style=\"vertical-align: -10px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E por fim, a \u00e1rea do tri\u00e2ngulo ser\u00e1 metade do m\u00f3dulo: <\/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-b56f201265ba609f4b7c75f486120fc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"A=\\cfrac{1}{2}\\cdot  \\lvert\\vv{\\text{u}}\\times \\vv{\\text{v}} \\rvert = \\cfrac{1}{2}\\cdot 16,19=\\bm{8,09} \\ \\mathbf{u}\\bm{^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"284\" 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>Esta p\u00e1gina explica o que \u00e9 o produto vetorial de dois vetores e como ele \u00e9 calculado. Voc\u00ea tamb\u00e9m ver\u00e1 como encontrar a dire\u00e7\u00e3o e dire\u00e7\u00e3o do produto vetorial usando a regra da m\u00e3o direita (ou saca-rolhas). E mais, voc\u00ea encontrar\u00e1 as utiliza\u00e7\u00f5es deste tipo de opera\u00e7\u00e3o, al\u00e9m de exemplos, exerc\u00edcios e problemas resolvidos passo &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/produto-vetorial-de-dois-vetores-exemplos-de-formulas-cruzadas-exercicios-resolvidos\/\"> <span class=\"screen-reader-text\">Produto vetorial de dois vetores (ou produto vetorial)<\/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-71","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>\u25b7 Calcule o produto vetorial de dois vetores (exemplos)<\/title>\n<meta name=\"description\" content=\"Explica\u00e7\u00e3o de como calcular o produto vetorial de dois vetores. 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