{"id":5215,"date":"2010-11-09T00:55:42","date_gmt":"2010-11-09T00:55:42","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5215"},"modified":"2022-01-12T01:22:58","modified_gmt":"2022-01-12T01:22:58","slug":"tres-vectores","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5215","title":{"rendered":"Tr\u00eas vetores"},"content":{"rendered":"<p><ul id='GTTabs_ul_5215' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5215' class='GTTabs_curr'><a  id=\"5215_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5215' ><a  id=\"5215_1\" onMouseOver=\"GTTabsShowLinks('Resolu\u00e7\u00e3o'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Resolu\u00e7\u00e3o<\/a><\/li>\n<\/ul>\n\n<div class='GTTabs_divs GTTabs_curr_div' id='GTTabs_0_5215'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere os vetores $\\vec{u}(-3,1,2)$, $\\vec{v}(4,-2,5)$ e $\\vec{w}(2,3,-1)$.<\/p>\n<ol>\n<li>Calcule os n\u00fameros reais a e b, para que o vetor $\\vec{v}-\\vec{w}$ e o vetor $(a,b,3-a)$ sejam colineares.<\/li>\n<li>Calcule os n\u00fameros reais $\\alpha $ e $\\beta $, para que o vetor $\\alpha \\vec{u}+\\beta \\vec{v}$ seja igual ao vetor de coordenadas $(-10,4,-1)$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5215' onClick='GTTabs_show(1,5215)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5215'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora, $\\vec{v}-\\vec{w}=(4-2,-2-3,5+1)=(2,-5,6)$.<br \/>\nPara que os vetores considerados sejam colineares, as suas coordenadas t\u00eam de ser diretamente proporcionais.<br \/>\nAssim, temos:\u00a0\\[\\frac{a}{2}=\\frac{b}{-5}=\\frac{3-a}{6}\\Leftrightarrow \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\frac{a}{2}=\\frac{b}{-5}\u00a0 \\\\<br \/>\n\\frac{a}{2}=\\frac{3-a}{6}\u00a0 \\\\<br \/>\n\\end{array} \\right.\\Leftrightarrow \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n6a=6-2a\u00a0 \\\\<br \/>\nb=-\\frac{5}{2}a\u00a0 \\\\<br \/>\n\\end{array} \\right.\\Leftrightarrow \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na=\\frac{3}{4}\u00a0 \\\\<br \/>\nb=-\\frac{15}{8}\u00a0 \\\\<br \/>\n\\end{array} \\right.\\]<br \/>\n\u00ad<\/li>\n<li>Ora, $\\alpha \\vec{u}+\\beta \\vec{v}=(-3\\alpha ,\\alpha ,2\\alpha )+(4\\beta ,-2\\beta ,5\\beta )=(-3\\alpha +4\\beta ,\\alpha -2\\beta ,2\\alpha +5\\beta )$.<br \/>\nPara que os vetores considerados sejam iguais, as suas coordenadas t\u00eam de ser iguais.<br \/>\nAssim, temos: \\[\\alpha \\vec{u}+\\beta \\vec{v}=(-10,4,-1)\\Leftrightarrow \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n-3\\alpha +4\\beta =-10\u00a0 \\\\<br \/>\n\\alpha -2\\beta =4\u00a0 \\\\<br \/>\n2\\alpha +5\\beta =-1\u00a0 \\\\<br \/>\n\\end{array} \\right.\\Leftrightarrow \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\alpha =2\\beta +4\u00a0 \\\\<br \/>\n-6\\beta -12+4\\beta =-10\u00a0 \\\\<br \/>\n4\\beta +8+5\\beta =-1\u00a0 \\\\<br \/>\n\\end{array} \\right.\\Leftrightarrow \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\beta =-1\u00a0 \\\\<br \/>\n\\beta =-1\u00a0 \\\\<br \/>\n\\alpha =2\u00a0 \\\\<br \/>\n\\end{array} \\right.\\Leftrightarrow \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\alpha =2\u00a0 \\\\<br \/>\n\\beta =-1\u00a0 \\\\<br \/>\n\\end{array} \\right.\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5215' onClick='GTTabs_show(0,5215)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere os vetores $\\vec{u}(-3,1,2)$, $\\vec{v}(4,-2,5)$ e $\\vec{w}(2,3,-1)$. Calcule os n\u00fameros reais a e b, para que o vetor $\\vec{v}-\\vec{w}$ e o vetor $(a,b,3-a)$ sejam colineares. Calcule os n\u00fameros reais $\\alpha $&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19378,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,111],"series":[],"class_list":["post-5215","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-11--ano","category-aplicando","category-geometria-analitica","tag-11-o-ano","tag-geometria","tag-vectores"],"views":1528,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Tres_vetores.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5215","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5215"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5215\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19378"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5215"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5215"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5215"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5215"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}