{"id":402,"date":"2023-06-19T11:39:22","date_gmt":"2023-06-19T11:39:22","guid":{"rendered":"https:\/\/physigeek.com\/pt\/vetor-de-posicao\/"},"modified":"2023-06-19T11:39:22","modified_gmt":"2023-06-19T11:39:22","slug":"vetor-de-posicao","status":"publish","type":"post","link":"https:\/\/physigeek.com\/pt\/vetor-de-posicao\/","title":{"rendered":"Vetor de posi\u00e7\u00e3o"},"content":{"rendered":"<p>Este artigo explica o que \u00e9 o vetor de posi\u00e7\u00e3o na f\u00edsica. Assim, voc\u00ea descobrir\u00e1 quais s\u00e3o as propriedades do vetor posi\u00e7\u00e3o, como calcular o vetor posi\u00e7\u00e3o de um ponto e, tamb\u00e9m, um exemplo concreto passo a passo. <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%C2%BFQue-es-un-vector-de-posicion\"><\/span> O que \u00e9 um vetor de posi\u00e7\u00e3o?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Um <strong>vetor de posi\u00e7\u00e3o<\/strong> , ou simplesmente <strong>um vetor de posi\u00e7\u00e3o<\/strong> , \u00e9 um vetor que descreve a posi\u00e7\u00e3o de um ponto em rela\u00e7\u00e3o a um sistema de refer\u00eancia. Ou seja, o vetor posi\u00e7\u00e3o \u00e9 usado para indicar a posi\u00e7\u00e3o de um ponto em um sistema de coordenadas.<\/p>\n<p> Matematicamente, o vetor posi\u00e7\u00e3o de um ponto \u00e9 definido como o vetor que vai da origem das coordenadas at\u00e9 aquele ponto. Portanto, o vetor posi\u00e7\u00e3o de um ponto \u00e9 calculado subtraindo as coordenadas desse ponto menos as coordenadas de origem.<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-94a2838c46b47b0de3fbc3ab9548365c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{r_{p}}=PO\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: -6px;\"><\/p>\n<\/p>\n<p> Em geral, o vetor posi\u00e7\u00e3o \u00e9 expresso pelos vetores unit\u00e1rios<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-0b35266aff72392f18054a3ee0726b72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{i}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"><\/p>\n<p> ,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-e64160920c9450edf06e8f621fe04ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{j}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-493ea824d13cdf2c289249ee201f0079_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{k}[\/latex ] , qui correspondent respectivement aux coordonn\u00e9es des axes X, Y et Z. Par exemple, si les coordonn\u00e9es cart\u00e9siennes d'un point sont (3,4,5), le vecteur position de ce point est r=3i+4j+5k. \n\n<figure class=&quot;wp-block-image aligncenter size-full is-resized&quot;><img decoding=&quot;async&quot; loading=&quot;lazy&quot; src=&quot;https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/vecteur-de-position.png&quot; alt=&quot;vecteur de position&quot; class=&quot;wp-image-7644&quot; width=&quot;374&quot; height=&quot;308&quot; srcset=&quot;https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/vecteur-de-position-300x247.png 300w, https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/vecteur-de-position.png 697w&quot; sizes=&quot;(max-width: 300px) 100vw, 300px&quot;><\/figure>\n<h2 class=&quot;wp-block-heading&quot;><span class=&quot;ez-toc-section&quot; id=&quot;Caracteristicas-del-vector-de-posicion&quot;><\/span> Caract\u00e9ristiques du vecteur de position<span class=&quot;ez-toc-section-end&quot;><\/span><\/h2>\n<p> Maintenant que nous connaissons la d\u00e9finition du vecteur position, voyons quelles sont ses caract\u00e9ristiques.<\/p>\n<ul style=&quot;color:#4fd12f; font-weight: bold;&quot;>\n<li style=&quot;margin-bottom:20px&quot;> <span style=&quot;color:#101010;font-weight: normal;&quot;>Le vecteur position d&#8217;un point est d\u00e9fini comme la diff\u00e9rence entre les coordonn\u00e9es de ce point et l&#8217;origine des coordonn\u00e9es. Par cons\u00e9quent, la formule pour calculer le vecteur position d&#8217;un point est la suivante :<\/span><\/li>\n<p> [latex]\\vv{r}=PO&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;438&#8243; width=&#8221;1546&#8243; style=&#8221;vertical-align: -5px;&#8221;><\/p>\n<\/p>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\">As coordenadas do vetor posi\u00e7\u00e3o de um ponto s\u00e3o expressas pelos vetores unit\u00e1rios\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-0b35266aff72392f18054a3ee0726b72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{i}\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"6\" style=\"vertical-align: 0px;\"><\/p>\n<p><\/span> ,<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-e64160920c9450edf06e8f621fe04ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{j}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"><\/p>\n<p> E<\/p>\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-9a41deb220b0257eea947e8a946f9a2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{k}[ \/latex], qui repr\u00e9sentent respectivement les directions des axes OX, OY et OZ.<\/span><\/li>\n<p> [latex]\\vv{r}=x\\vv{i}+y\\vv{j}+z\\vv{k}&#8221; title=&#8221;Rendered by QuickLaTeX.com&#8221; height=&#8221;41&#8243; width=&#8221;616&#8243; style=&#8221;vertical-align: -5px;&#8221;><\/p>\n<p> <span style=\"color:#101010;font-weight: normal;\">A dire\u00e7\u00e3o do vetor posi\u00e7\u00e3o \u00e9 a linha que liga a origem da marca ao ponto considerado.<\/span><\/li>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\">A dire\u00e7\u00e3o do vetor posi\u00e7\u00e3o \u00e9 da origem ao ponto de estudo.<\/span><\/li>\n<li style=\"margin-bottom:20px\"> <span style=\"color:#101010;font-weight: normal;\">A magnitude do vetor posi\u00e7\u00e3o de um ponto \u00e9 a dist\u00e2ncia entre o ponto e a origem das coordenadas. A norma do vetor posi\u00e7\u00e3o \u00e9 igual \u00e0 raiz quadrada da soma dos quadrados de suas coordenadas.<\/span><\/li>\n<p style=\"text-align:center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-4e00e2651473f9b281caa2edb84af61b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"|\\vv{r}|=\\sqrt{x^2+y^2+z^2}\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"153\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Observe que o vetor posi\u00e7\u00e3o s\u00f3 ter\u00e1 duas coordenadas (x,y) se trabalharmos no plano. Por outro lado, se trabalharmos no espa\u00e7o, o vetor posi\u00e7\u00e3o ter\u00e1 tr\u00eas coordenadas (x,y,z). <\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Ejercicio-resuelto-del-vector-de-posicion\"><\/span> Exerc\u00edcio de vetor de posi\u00e7\u00e3o resolvido<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> Para entender bem o conceito, segue abaixo um exerc\u00edcio resolvido sobre como o vetor posi\u00e7\u00e3o \u00e9 calculado.<\/p>\n<ul>\n<li> O vetor posi\u00e7\u00e3o versus tempo de um corpo \u00e9 definido pela express\u00e3o abaixo. Calcule o vetor posi\u00e7\u00e3o do corpo no tempo t=3 s e seu m\u00f3dulo.<\/li>\n<\/ul>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-03e3237c194135f352830e0617712fdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\vv{r}(t)=4t\\vv{i}+2t^2\\vv{j}+5 \\vv{k}\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"166\" style=\"vertical-align: -5px;\"><\/p>\n<\/p>\n<p> Para encontrar o vetor posi\u00e7\u00e3o no instante t=3 s voc\u00ea deve substituir o par\u00e2metro t pelo seu valor e fazer os c\u00e1lculos:<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-dec24ff071b35d8e731e7a0a66d6c0e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}\\vv{r}(3)&amp;=4\\cdot 3\\vv{i}+2\\cdot 3^2\\vv{j}+5 \\vv{k}\\\\[2ex ]\\vv{r}(3)&amp;=12\\vv{i}+18\\vv{j}+5\\vv{k}\\end{aligned}\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"199\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<p> Em seguida, produzimos o m\u00f3dulo do vetor posi\u00e7\u00e3o calculando a raiz quadrada da soma dos quadrados das coordenadas: <\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-8c09dc9fc6db4d03b23adfc1b44c8168_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\begin{aligned}|\\vv{r}(3)|&amp;=\\sqrt{12^2+18^2+5^2}\\\\[2ex]|\\vv{r}(3)| &amp;=\\sqrt{493}\\end{align\u00e9}\" title=\"Rendered by QuickLaTeX.com\" height=\"69\" width=\"191\" style=\"vertical-align: 0px;\"><\/p>\n<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Vector-de-posicion-y-desplazamiento\"><\/span> Vetor de posi\u00e7\u00e3o e deslocamento<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p> O <strong>deslocamento<\/strong> de um ponto \u00e9 definido como a dist\u00e2ncia entre o ponto final e o ponto inicial. Portanto, o vetor deslocamento \u00e9 o vetor obtido pela subtra\u00e7\u00e3o do vetor posi\u00e7\u00e3o final menos o vetor posi\u00e7\u00e3o inicial.<\/p>\n<p class=\"has-text-align-center\">\n<p class=\"has-text-align-center\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/ql-cache\/quicklatex.com-c75a87e8719ce716eb129379b32a9966_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"\\Delta \\vv{r}=\\vv{r_2}-\\vv{r_1}\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"98\" style=\"vertical-align: -3px;\"><\/p>\n<\/p>\n<p> Portanto, a <strong>diferen\u00e7a entre o vetor posi\u00e7\u00e3o e o deslocamento<\/strong> \u00e9 que o vetor posi\u00e7\u00e3o indica a posi\u00e7\u00e3o de um corpo em um determinado momento, por outro lado, o vetor deslocamento indica a dist\u00e2ncia entre duas posi\u00e7\u00f5es diferentes de um corpo em movimento. <\/p>\n<figure class=\"wp-block-image aligncenter size-full is-resized\"><img decoding=\"async\" loading=\"lazy\" src=\"https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/vecteur-de-position-et-de-deplacement.png\" alt=\"\" class=\"wp-image-7663\" width=\"324\" height=\"263\" srcset=\"https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/vecteur-de-position-et-de-deplacement-300x243.png 300w, https:\/\/physigeek.com\/wp-content\/uploads\/2023\/09\/vecteur-de-position-et-de-deplacement.png 650w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\"><\/figure>\n<p> Da mesma forma, o movimento do percurso deve ser diferenciado. Caminho refere-se ao comprimento total do caminho percorrido, enquanto deslocamento refere-se \u00e0 dist\u00e2ncia da posi\u00e7\u00e3o final at\u00e9 a posi\u00e7\u00e3o final. O caminho pode, portanto, ser maior que o deslocamento.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Este artigo explica o que \u00e9 o vetor de posi\u00e7\u00e3o na f\u00edsica. Assim, voc\u00ea descobrir\u00e1 quais s\u00e3o as propriedades do vetor posi\u00e7\u00e3o, como calcular o vetor posi\u00e7\u00e3o de um ponto e, tamb\u00e9m, um exemplo concreto passo a passo. O que \u00e9 um vetor de posi\u00e7\u00e3o? Um vetor de posi\u00e7\u00e3o , ou simplesmente um vetor de &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/physigeek.com\/pt\/vetor-de-posicao\/\"> <span class=\"screen-reader-text\">Vetor de posi\u00e7\u00e3o<\/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":[3],"tags":[],"class_list":["post-402","post","type-post","status-publish","format-standard","hentry","category-cinematografico"],"yoast_head":"<!-- This site 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