{"id":5229,"date":"2010-11-09T01:41:51","date_gmt":"2010-11-09T01:41:51","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5229"},"modified":"2022-01-12T01:25:36","modified_gmt":"2022-01-12T01:25:36","slug":"rascunho-21","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5229","title":{"rendered":"Mais dois vetores"},"content":{"rendered":"<p><ul id='GTTabs_ul_5229' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5229' class='GTTabs_curr'><a  id=\"5229_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5229' ><a  id=\"5229_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_5229'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Os vetores $\\overrightarrow{u}$ e $\\overrightarrow{v}$ verificam a condi\u00e7\u00f5es, em $(O,\\overrightarrow{i},\\overrightarrow{j})$:<\/p>\n<ul>\n<li>$\\begin{matrix}<br \/>\n\\overrightarrow{u}.\\overrightarrow{i}=2 &amp; \\wedge\u00a0 &amp; \\overrightarrow{u}.\\overrightarrow{j}=-3\u00a0 \\\\<br \/>\n\\end{matrix}$<\/li>\n<li>$\\begin{matrix}<br \/>\n\\overrightarrow{v}.\\overrightarrow{i}=-4 &amp; \\wedge\u00a0 &amp; \\overrightarrow{v}.\\overrightarrow{j}=-5\u00a0 \\\\<br \/>\n\\end{matrix}$<\/li>\n<\/ul>\n<ol>\n<li>Quais s\u00e3o as coordenadas de $\\overrightarrow{u}$ e de\u00a0$\\overrightarrow{v}$?<\/li>\n<li>Calcule $\\overrightarrow{u}.\\overrightarrow{v}$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5229' onClick='GTTabs_show(1,5229)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5229'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>$\\overrightarrow{u}=(2,-3)$ e $\\overrightarrow{v}=(-4,-5)$.<br \/>\nCom efeito, para $\\overrightarrow{u}=({{u}_{1}},{{u}_{2}})$, obt\u00e9m-se:<\/p>\n<p>$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{u}.\\overrightarrow{i}=2 &amp; \\Leftrightarrow\u00a0 &amp; ({{u}_{1}}\\overrightarrow{i}+{{u}_{2}}\\overrightarrow{j})\\overrightarrow{i}=2\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{u}_{1}}\\overrightarrow{i}.\\overrightarrow{i}+{{u}_{2}}\\overrightarrow{j}.\\overrightarrow{i}=2\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{u}_{1}}\\times 1+{{u}_{2}}\\times 0=2\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{u}_{1}}=2\u00a0 \\\\<br \/>\n\\end{array}$<br \/>\ne<\/p>\n<p>$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{u}.\\overrightarrow{j}=-3 &amp; \\Leftrightarrow\u00a0 &amp; ({{u}_{1}}\\overrightarrow{i}+{{u}_{2}}\\overrightarrow{j})\\overrightarrow{j}=-3\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{u}_{1}}\\overrightarrow{i}.\\overrightarrow{j}+{{u}_{2}}\\overrightarrow{j}.\\overrightarrow{j}=-3\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{u}_{1}}\\times 0+{{u}_{2}}\\times 1=-3\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; {{u}_{2}}=-3\u00a0 \\\\<br \/>\n\\end{array}$<br \/>\nLogo, $\\overrightarrow{u}=(2,-3)$ e, de forma an\u00e1loga, $\\overrightarrow{v}=(-4,-5)$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Como $\\overrightarrow{u}=(2,-3)$ e $\\overrightarrow{v}=(-4,-5)$, ent\u00e3o:<br \/>\n$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{u}.\\overrightarrow{v} &amp; = &amp; (2\\overrightarrow{i}-3\\overrightarrow{j}).(-4\\overrightarrow{i}-5\\overrightarrow{j})\u00a0 \\\\<br \/>\n{} &amp; = &amp; -8\\overrightarrow{i}.\\overrightarrow{i}-10\\overrightarrow{i}.\\overrightarrow{j}+12\\overrightarrow{i}.\\overrightarrow{j}+15\\overrightarrow{j}.\\overrightarrow{j}\u00a0 \\\\<br \/>\n{} &amp; = &amp; -8-0+0+15\u00a0 \\\\<br \/>\n{} &amp; = &amp; 7\u00a0 \\\\<br \/>\n\\end{array}$<\/p>\n<p>Ou, ainda: $\\overrightarrow{u}.\\overrightarrow{v}=(2,-3).(-4,-5)=2\\times (-4)+(-3)\\times (-5)=-8+15=7$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5229' onClick='GTTabs_show(0,5229)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Os vetores $\\overrightarrow{u}$ e $\\overrightarrow{v}$ verificam a condi\u00e7\u00f5es, em $(O,\\overrightarrow{i},\\overrightarrow{j})$: $\\begin{matrix} \\overrightarrow{u}.\\overrightarrow{i}=2 &amp; \\wedge\u00a0 &amp; \\overrightarrow{u}.\\overrightarrow{j}=-3\u00a0 \\\\ \\end{matrix}$ $\\begin{matrix} \\overrightarrow{v}.\\overrightarrow{i}=-4 &amp; \\wedge\u00a0 &amp; \\overrightarrow{v}.\\overrightarrow{j}=-5\u00a0 \\\\ \\end{matrix}$ Quais s\u00e3o as coordenadas de&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19379,"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-5229","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":2003,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Mais_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\/5229","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=5229"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5229\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19379"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5229"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5229"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5229"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5229"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}