{"id":5200,"date":"2010-11-09T00:15:29","date_gmt":"2010-11-09T00:15:29","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5200"},"modified":"2022-01-12T01:16:26","modified_gmt":"2022-01-12T01:16:26","slug":"dois-vectores","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5200","title":{"rendered":"Dois vetores"},"content":{"rendered":"<p><ul id='GTTabs_ul_5200' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5200' class='GTTabs_curr'><a  id=\"5200_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5200' ><a  id=\"5200_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_5200'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere os vetores $\\vec{u}(2,-1,3)$\u00a0 e $\\vec{v}(-4,2,5)$.<\/p>\n<ol>\n<li>Os vetores ${\\vec{u}}$ \u00a0e ${\\vec{v}}$ \u00a0s\u00e3o colineares? Porqu\u00ea?<\/li>\n<li>Determine os n\u00fameros reais <strong>a<\/strong> e <strong>b<\/strong>, para que o vetor $\\vec{w}(a,-6,b)$ \u00a0e o vetor ${\\vec{v}}$\u00a0 sejam colineares.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5200' onClick='GTTabs_show(1,5200)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5200'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Os vetores ${\\vec{u}}$\u00a0 e ${\\vec{v}}$\u00a0 s\u00e3o colineares se e s\u00f3 se $\\exists k\\in \\mathbb{R}:\\vec{u}=k\\vec{v}$ (1).<br \/>\nOra,<br \/>\n$\\begin{array}{*{35}{l}} \u00a0\u00a0 \\vec{u}=k\\vec{v} &amp; \\Leftrightarrow\u00a0 &amp; (2,-1,3)=k(-4,2,5)\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 2=-4k\u00a0 \\\\ \u00a0\u00a0 -1=2k\u00a0 \\\\ \u00a0\u00a0 3=5k\u00a0 \\\\ \\end{array} \\right.\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 k=-\\frac{1}{2}\u00a0 \\\\ \u00a0\u00a0 k=-\\frac{1}{2}\u00a0 \\\\ \u00a0\u00a0 k=\\frac{3}{5}\u00a0 \\\\ \\end{array} \\right.\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; k\\in \\left\\{ {} \\right\\}\u00a0 \\\\ \\end{array}$<br \/>\nPortanto, os vetores ${\\vec{u}}$ \u00a0e ${\\vec{v}}$ \u00a0n\u00e3o s\u00e3o colineares.<\/p>\n<p><strong>ALTERNATIVA<\/strong>:<br \/>\nPara que a condi\u00e7\u00e3o (1) se verifique \u00e9 necess\u00e1rio que os vetores possuam as coordenadas diretamente proporcionais. Ora, $\\frac{2}{-4}=\\frac{-1}{2}\\ne \\frac{3}{5}$. Logo, os vetores considerados n\u00e3o s\u00e3o colineares.<\/p>\n<\/li>\n<li>Ora,<br \/>\n$\\begin{array}{*{35}{l}} \u00a0\u00a0 \\vec{w}=k\\vec{v} &amp; \\Leftrightarrow\u00a0 &amp; (a,-6,b)=k(-4,2,5)\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 a=-4k\u00a0 \\\\ \u00a0\u00a0 -6=2k\u00a0 \\\\ \u00a0\u00a0 b=5k\u00a0 \\\\ \\end{array} \\right.\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 k=-3\u00a0 \\\\ \u00a0\u00a0 a=12\u00a0 \\\\ \u00a0\u00a0 b=-15\u00a0 \\\\ \\end{array} \\right.\u00a0 \\\\ \\end{array}$<\/p>\n<p>Portanto, os vetores considerados s\u00e3o colineares para $a=12\\wedge b=-15$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5200' onClick='GTTabs_show(0,5200)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere os vetores $\\vec{u}(2,-1,3)$\u00a0 e $\\vec{v}(-4,2,5)$. Os vetores ${\\vec{u}}$ \u00a0e ${\\vec{v}}$ \u00a0s\u00e3o colineares? Porqu\u00ea? Determine os n\u00fameros reais a e b, para que o vetor $\\vec{w}(a,-6,b)$ \u00a0e o vetor ${\\vec{v}}$\u00a0 sejam&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19377,"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-5200","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":1064,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Dois_vetores.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5200","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=5200"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5200\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19377"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5200"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5200"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5200"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5200"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}