{"id":209,"date":"2023-07-11T21:43:45","date_gmt":"2023-07-11T21:43:45","guid":{"rendered":"https:\/\/mathority.org\/pt\/multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar\/"},"modified":"2023-07-11T21:43:45","modified_gmt":"2023-07-11T21:43:45","slug":"multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar","status":"publish","type":"post","link":"https:\/\/mathority.org\/pt\/multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar\/","title":{"rendered":"Multiplica\u00e7\u00e3o de um vetor por um n\u00famero"},"content":{"rendered":"<p>Esta p\u00e1gina explica como multiplicar um vetor por um n\u00famero real (ou escalar) numericamente e graficamente. Al\u00e9m disso, voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos do produto de um vetor por um escalar. Por fim, tamb\u00e9m s\u00e3o explicadas as propriedades deste tipo de opera\u00e7\u00e3o com vetores. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%c2%bfcomo-se-multiplica-un-vector-por-un-numero-real\"><\/span> Como voc\u00ea multiplica um vetor por um n\u00famero real? <span class=\"ez-toc-section-end\"><\/span><\/h2>\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\"> Para calcular numericamente o produto de um vetor e um n\u00famero (ou escalar), cada componente do vetor deve ser multiplicado pelo n\u00famero.<\/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-d4b2f8c9cdb09377a66fbce8392c30ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} = (\\text{u}_x,\\text{u}_y)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"91\" style=\"vertical-align: -6px;\"><\/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-abb3b5391be9c0e4d871ff45367ca838_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k\\cdot \\vv{\\text{u}} =(k\\cdot \\text{u}_x \\ , \\ k\\cdot \\text{u}_y)\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"170\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<\/div>\n<p> Assim, o resultado da multiplica\u00e7\u00e3o de um vetor por um n\u00famero d\u00e1 origem a um novo vetor com as seguintes caracter\u00edsticas:<\/p>\n<ul 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;\">O resultado do produto de um vetor por um escalar produz um novo vetor com a mesma dire\u00e7\u00e3o do vetor original.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Al\u00e9m disso, o novo vetor ter\u00e1 a mesma dire\u00e7\u00e3o se o n\u00famero for positivo.<\/span><\/li>\n<li style=\"margin-bottom:18px\"> <span style=\"color:#000000;font-weight: normal;\">Ou ter\u00e1 o significado oposto se o n\u00famero for negativo.<\/span><\/li>\n<li> <span style=\"color:#000000;font-weight: normal;\">A magnitude do vetor resultante \u00e9 equivalente \u00e0 magnitude do vetor original vezes o escalar.<\/span><\/li>\n<\/ul>\n<p> No gr\u00e1fico a seguir voc\u00ea pode ver como a dire\u00e7\u00e3o do vetor \u00e9 mantida independentemente do sinal do escalar. Por outro lado, a dire\u00e7\u00e3o do vetor depende do sinal do n\u00famero que ele multiplica. <\/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\/multiplication-ou-produit-dun-nombre-ou-dun-scalaire-par-un-vecteur.webp\" alt=\"multiplica\u00e7\u00e3o ou produto de um n\u00famero ou escalar por um vetor\" class=\"wp-image-283\" width=\"305\" height=\"206\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Al\u00e9m disso, no gr\u00e1fico a seguir v\u00ea-se claramente que a magnitude do vetor produto resultante \u00e9 igual \u00e0 magnitude do vetor original multiplicado pelo escalar. <\/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\/numero-de-produit-de-multiplication-de-module-par-vecteur.webp\" alt=\"Como evolui o m\u00f3dulo de uma multiplica\u00e7\u00e3o ou produto de um n\u00famero por um vetor?\" class=\"wp-image-290\" width=\"255\" height=\"313\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<p> Obviamente, se multiplicarmos o vetor por um n\u00famero maior que 1, o resultado ser\u00e1 um vetor de maior comprimento (de maior m\u00f3dulo). Por outro lado, se multiplicarmos o vetor por um n\u00famero menor que 1, o resultado ser\u00e1 um vetor de comprimento menor (m\u00f3dulo menor).<\/p>\n<p> <strong>Nota:<\/strong> N\u00e3o confunda o produto de um vetor e um escalar com o <a href=\"https:\/\/mathority.org\/pt\/calcular-o-produto-escalar-entre-dois-vetores-exemplos-exercicios-resolvidos\/\">produto escalar de vetores<\/a> . Embora tenham um nome semelhante, s\u00e3o dois conceitos completamente diferentes. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejemplo-del-producto-de-un-vector-por-un-escalar\"><\/span> Exemplo de produto de um vetor por um escalar<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> A seguir, veremos um exemplo num\u00e9rico de como \u00e9 calculado o produto de um vetor e um n\u00famero:<\/p>\n<ul>\n<li> Multiplique o seguinte vetor por 4:<\/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-0a3b02659b94a332ef3cd2cc62769d94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{\\text{u}} =(2,-3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" 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-998ae86a47fde44f6006a0744f03c562_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot \\vv{\\text{u}} =(4 \\cdot 2 \\ , \\ 4 \\cdot (-3)) = \\bm{(8,-12)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"263\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Como voc\u00ea viu, esse tipo de opera\u00e7\u00e3o vetorial n\u00e3o \u00e9 muito complicada, pois n\u00e3o \u00e9 necess\u00e1rio fazer muitos c\u00e1lculos.<\/p>\n<p> No entanto, existem opera\u00e7\u00f5es vetoriais mais complicadas, como adi\u00e7\u00e3o e subtra\u00e7\u00e3o de vetores. Se voc\u00ea j\u00e1 entendeu como calcular o produto de um vetor e um escalar, recomendamos que passe para o pr\u00f3ximo n\u00edvel e veja como resolver <a href=\"https:\/\/mathority.org\/pt\/soma-de-vetores-exemplos-resolvidos-graficamente-numericamente-exercicios-adicionar\/\">adi\u00e7\u00e3o<\/a> e <a href=\"https:\/\/mathority.org\/pt\/subtrair-vetores-numericamente-exemplos-graficos-exercicio-resolvido-subtrair\/\">subtra\u00e7\u00e3o<\/a> vetorial, pois s\u00e3o opera\u00e7\u00f5es um pouco mais dif\u00edceis e, na verdade, eles s\u00e3o muito mais usados (s\u00e3o mais importantes). <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"propiedades-de-la-multiplicacion-de-un-vector-por-un-numero\"><\/span> Propriedades de multiplicar um vetor por um n\u00famero<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> O produto de um vetor por um n\u00famero possui as seguintes propriedades:<\/p>\n<ul>\n<li> <strong>Propriedade associativa<\/strong> : Quando o vetor \u00e9 multiplicado por mais de um n\u00famero, a ordem das multiplica\u00e7\u00f5es n\u00e3o importa.<\/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-a7cc3b0e34695b9960f32199046e868e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k \\cdot (k' \\cdot \\vv{\\text{u}}) =(k\\cdot k') \\cdot \\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"171\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Propriedade distributiva<\/strong> em rela\u00e7\u00e3o \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-a8b55a3b25ad66154d6e8bb18fc70da7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k \\cdot (\\vv{\\text{u}}+  \\vv{\\text{v}})=k\\cdot \\vv{\\text{u}} + k\\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" 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-086f60151d160ac3b6d4f5add59434b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"k \\cdot (\\vv{\\text{u}}- \\vv{\\text{v}})=k\\cdot \\vv{\\text{u}} - k\\cdot \\vv{\\text{v}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"187\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Propriedade distributiva<\/strong> em rela\u00e7\u00e3o \u00e0 adi\u00e7\u00e3o de escalares:<\/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-7354955effb7b59f41793537f4517851_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(k+k') \\cdot \\vv{\\text{u}} =k\\cdot \\vv{\\text{u}} + k'\\cdot \\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"196\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<ul>\n<li> <strong>Elemento neutro<\/strong> : Obviamente, qualquer vetor multiplicado por 1 d\u00e1 o pr\u00f3prio 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-154c7987de1e55e762dcf229eae9fd23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"1\\cdot \\vv{\\text{u}} = \\vv{\\text{u}}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"64\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"ejercicios-resueltos-de-multiplicacion-de-un-vector-por-un-escalar\"><\/span> Problemas resolvidos de multiplica\u00e7\u00e3o de um vetor por um escalar<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<h3 class=\"wp-block-heading\"> Exerc\u00edcio 1<\/h3>\n<p> Calcule analiticamente o resultado do produto do seguinte vetor por 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-8b5a56f394f2b178a0175a53e6a55a1a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a}=(-2,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"85\" 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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Para encontrar o produto, voc\u00ea deve multiplicar cada coordenada do vetor por 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-4e0b8a6167489c27e91bef24f222e3af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"3\\cdot \\vv{a}=(3\\cdot (-2), 3 \\cdot 5) = \\bm{(-6,15)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"250\" 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> Multiplique o seguinte vetor por 6 e encontre 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-85db994a57a1325a456909a04efa2e23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a}=(3,-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" 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 solu\u00e7\u00e3o<\/strong><\/div>\n<\/div>\n<p class=\"has-text-align-left\"> Primeiro multiplicamos o vetor pelo escalar:<\/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-7c6c6b2d9291528aff357e7e94daf221_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6\\cdot \\vv{a}=(6\\cdot 3, 6 \\cdot (-4)) = (18,-24)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"259\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Existem agora duas maneiras de calcular a magnitude do vetor obtido. A primeira \u00e9 encontrar a magnitude do vetor original e depois multiplic\u00e1-la por 6: <\/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-7a7bc760b63c8efa2c36d2d49802000c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert \\vv{a} \\rvert =\\sqrt{3^2+(-4)^2} = \\sqrt{25} =5\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"229\" style=\"vertical-align: -6px;\"><\/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-32f0e3a2cb32142aa26a1fbb0f3bf9df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"6\\cdot \\lvert \\vv{a} \\rvert =6\\cdot 5 =\\bm{30}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"137\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E a segunda forma \u00e9 calcular diretamente a magnitude do vetor obtido na multiplica\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-06f0793bfdae917904c47247b9b1f08f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\lvert 6\\cdot \\vv{a} \\rvert =\\sqrt{18^2+(-24)^2} = \\sqrt{900} =\\bm{30}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"287\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Assim, com ambos os procedimentos mostra-se que o resultado n\u00e3o depende do m\u00e9todo pelo qual o m\u00f3dulo \u00e9 calculado.<\/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> 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-cf8d63b14f91ee738fe5107ab2a05779_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a}=(4,-1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Calcule as seguintes opera\u00e7\u00f5es algebricamente: <\/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-5557b66d322653674e2266800c73c958_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\cdot \\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" 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-a2f511cc4014aae8cf90bef442f30bdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3\\cdot \\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"43\" 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-d09a188710cecb56283c4a2dea7a97e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{2} \\cdot \\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"33\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-7c4742e8a37f45e358d2e7740c5aadef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot \\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> A seguir, determine se os vetores resultantes t\u00eam a mesma dire\u00e7\u00e3o e sentido do vetor original e ordene-os do mais curto para o mais longo. <\/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\"> Primeiro calculamos as multiplica\u00e7\u00f5es: <\/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-6b92c621a30ebf15a0628cba3e0f6be5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\cdot \\vv{a}=(2\\cdot 4, 2 \\cdot (-1)) = (8,-2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"241\" 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-71e1eb36b88b2ca47d69f0bfe4d1a33b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3\\cdot \\vv{a}=(-3\\cdot 4, -3 \\cdot (-1)) = (-12,3)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"290\" 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-d9866b27859b66475a36d182919ec4e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{2}\\cdot \\vv{a}=\\left(\\frac{1}{2}\\cdot 4, \\frac{1}{2} \\cdot (-1)\\right) = \\left(2,-\\frac{1}{2}\\right)\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"277\" 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-6904a22660dc2e21ab3f775e1453535c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot \\vv{a}=(4\\cdot 4, 4 \\cdot (-1)) = (16,-4)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"250\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Portanto, os vetores multiplicados por n\u00fameros positivos t\u00eam a mesma dire\u00e7\u00e3o e sentido do vetor original. E os vetores multiplicados por n\u00fameros negativos t\u00eam a mesma dire\u00e7\u00e3o, mas a dire\u00e7\u00e3o oposta ao vetor original.<\/p>\n<p class=\"has-text-align-center\"> Vetores da mesma dire\u00e7\u00e3o e da mesma dire\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-50dd29950a1b957a78493d3de5b4bf5b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle 2\\vv{a}, \\ 4\\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"50\" style=\"vertical-align: -4px;\"><\/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-9fac7295c812fa36c4fcba97e316b5f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{1}{2}\\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"20\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"> Vetores com a mesma dire\u00e7\u00e3o, mas com significados 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-d1b95a4d066d0dd0da8cdf0373d19c8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3\\vv{a}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Finalmente, devemos ordenar os vetores de acordo com o seu comprimento, ou equivalentemente, o seu m\u00f3dulo. O vetor de maior comprimento (ou maior m\u00f3dulo) ser\u00e1 aquele que foi multiplicado por um n\u00famero maior (em valor absoluto), e o vetor de menor comprimento (ou menor m\u00f3dulo) ser\u00e1 aquele que foi multiplicado por um menor n\u00famero (em valor absoluto). Ent\u00e3o a ordem dos comprimentos \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-b1e7cf58774569c9f128f49948c9fead_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\begin{vmatrix} \\displaystyle\\frac{1}{2}\\vv{a}\\end{vmatrix} <\\lvert 2\\vv{a}\\rvert < \\lvert -3\\vv{a}\\rvert < \\lvert 4\\vv{a}\\rvert\" title=\"Rendered by QuickLaTeX.com\" height=\"44\" width=\"199\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Observe que o comprimento ou m\u00f3dulo n\u00e3o depende do sinal do escalar que \u00e9 multiplicado, pois a dire\u00e7\u00e3o do vetor n\u00e3o modifica seu m\u00f3dulo.<\/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> Considere 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-740cd5f2540cb57342f82c58e1076cbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{a} =(7,-2) \\qquad \\vv{b} =(-3,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"205\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Calcule a seguinte opera\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-5abd316200fbcf062072dfdf51266cda_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2 \\vv{a} - 3 \\vv{b}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" 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\"> Primeiro resolvemos as multiplica\u00e7\u00f5es de vetores por n\u00fameros: <\/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-08a07c35741b9e7427990abb5bb89ef9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2 \\cdot \\vv{a} - 3 \\cdot \\vv{b}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"82\" 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-5fa31543d7c2a129c47b030cfa5fc4e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2 \\cdot (7,-2) - 3 \\cdot(-3,5)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"171\" 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-d7624e801962ac397ca316243c4e3420_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(14,-4) - (-9,15)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> E ent\u00e3o subtra\u00edmos 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-72660b240eab9459d0b0aefb1d10d728_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(14-(-9),-4-15)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"158\" 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-fe8b5f018f4a24a925bfce0c2c997dae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"(23,-19)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" 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 5<\/h3>\n<p> Execute as seguintes multiplica\u00e7\u00f5es de vetores por escalares e represente graficamente os resultados: <\/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-b1c2fa6e2c114b3bbce76ca987020ad9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\cdot (1,2)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" 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-1adddd551d57bc95b951c6573080090a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3\\cdot (-1,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"87\" 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-33be3689306562b8a4ae74d0970fc1f3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot (2,1)\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"60\" 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-bd3d3bba47ccc84d7616e38ca863b22d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{3}{2}\\cdot (-2,4)\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"76\" style=\"vertical-align: -12px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-8fe6e18de5ed93f74fa207a3f9ecc2c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -\\frac{1}{2}\\cdot (8,6)\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"77\" 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\"> Primeiro multiplicamos os vetores pelos escalares reais: <\/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-6846258bc5f69aaea3f86991095d3199_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"2\\cdot (1,2) = (2\\cdot 1,2\\cdot 2)=\\bm{(2,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"230\" 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-605e263cab7e34f01290372f0e2666c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"-3\\cdot (-1,1) = (-3\\cdot (-1),-3\\cdot 1)=\\bm{(3,-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"325\" 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-b8b4cf7b606673d72ae9cdd0004f0757_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"4\\cdot (2,1) = (4\\cdot 2,4\\cdot 1)=\\bm{(8,4)}\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"230\" 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-51454cc5c3239bb4552918b1a693b499_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle \\frac{3}{2}\\cdot (-2,4)= \\left(\\frac{3}{2}\\cdot (-2), \\frac{3}{2}\\cdot 4 \\right) = \\bm{(-3,6)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"307\" 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-8fbbfff835499e861d11de5da6714d10_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\displaystyle -\\frac{1}{2}\\cdot (8,6)= \\left(-\\frac{1}{2}\\cdot 8, -\\frac{1}{2}\\cdot 6 \\right) = \\bm{(-4,-3)}\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"322\" style=\"vertical-align: -17px;\"><\/p>\n<\/p>\n<p class=\"has-text-align-left\"> Finalmente, uma vez calculados os vetores, n\u00f3s os representamos no gr\u00e1fico: <\/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\/exercices-et-problemes-resolus-de-multiplication-dun-vecteur-par-un-nombre.webp\" alt=\"exerc\u00edcios e problemas resolvidos de multiplica\u00e7\u00e3o de um vetor por um n\u00famero\" class=\"wp-image-327\" width=\"469\" height=\"370\" srcset=\"\" sizes=\"auto, \"><\/figure>\n<\/div>\n<div class=\"wp-block-otfm-box-spoiler-end otfm-sp_end\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Esta p\u00e1gina explica como multiplicar um vetor por um n\u00famero real (ou escalar) numericamente e graficamente. Al\u00e9m disso, voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos do produto de um vetor por um escalar. Por fim, tamb\u00e9m s\u00e3o explicadas as propriedades deste tipo de opera\u00e7\u00e3o com vetores. Como voc\u00ea multiplica um vetor por um n\u00famero real? &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathority.org\/pt\/multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar\/\"> <span class=\"screen-reader-text\">Multiplica\u00e7\u00e3o de um vetor por um n\u00famero<\/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-209","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>Multiplica\u00e7\u00e3o de um vetor por um n\u00famero - 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\/multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar\/\" \/>\n<meta property=\"og:locale\" content=\"pt_BR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Multiplica\u00e7\u00e3o de um vetor por um n\u00famero - Mathority\" \/>\n<meta property=\"og:description\" content=\"Esta p\u00e1gina explica como multiplicar um vetor por um n\u00famero real (ou escalar) numericamente e graficamente. Al\u00e9m disso, voc\u00ea tamb\u00e9m encontrar\u00e1 exemplos e exerc\u00edcios resolvidos do produto de um vetor por um escalar. Por fim, tamb\u00e9m s\u00e3o explicadas as propriedades deste tipo de opera\u00e7\u00e3o com vetores. Como voc\u00ea multiplica um vetor por um n\u00famero real? &hellip; Multiplica\u00e7\u00e3o de um vetor por um n\u00famero Leia mais &raquo;\" \/>\n<meta property=\"og:url\" content=\"https:\/\/mathority.org\/pt\/multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar\/\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-11T21:43:45+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/mathority.org\/wp-content\/ql-cache\/quicklatex.com-d4b2f8c9cdb09377a66fbce8392c30ec_l3.png\" \/>\n<meta name=\"author\" content=\"Equipe Mathoridade\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"Equipe Mathoridade\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. tempo de leitura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/mathority.org\/pt\/multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/mathority.org\/pt\/multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar\/\"},\"author\":{\"name\":\"Equipe Mathoridade\",\"@id\":\"https:\/\/mathority.org\/pt\/#\/schema\/person\/26defeb7b79f5baaedafa33a1ac6ac00\"},\"headline\":\"Multiplica\u00e7\u00e3o de um vetor por um n\u00famero\",\"datePublished\":\"2023-07-11T21:43:45+00:00\",\"dateModified\":\"2023-07-11T21:43:45+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/mathority.org\/pt\/multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar\/\"},\"wordCount\":121,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/mathority.org\/pt\/#organization\"},\"articleSection\":[\"Vetores\"],\"inLanguage\":\"pt-BR\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/mathority.org\/pt\/multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/mathority.org\/pt\/multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar\/\",\"url\":\"https:\/\/mathority.org\/pt\/multiplicacao-do-produto-de-um-vetor-por-um-numero-real-escalar\/\",\"name\":\"Multiplica\u00e7\u00e3o de um vetor por um n\u00famero - 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