{"id":5879,"date":"2010-11-21T01:04:17","date_gmt":"2010-11-21T01:04:17","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5879"},"modified":"2022-01-12T12:45:00","modified_gmt":"2022-01-12T12:45:00","slug":"um-vector-perpendicular-a-outros-dois","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5879","title":{"rendered":"Um vetor perpendicular a outros dois"},"content":{"rendered":"<p><ul id='GTTabs_ul_5879' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5879' class='GTTabs_curr'><a  id=\"5879_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5879' ><a  id=\"5879_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_5879'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Num referencial ortonormado do espa\u00e7o, indique um vetor que seja perpendicular a $\\vec{u}(1,4,7)$ \u00a0e a $\\vec{v}(2,-1,5)$ .<br \/>\nObserve que qualquer outro vetor nas mesmas condi\u00e7\u00f5es \u00e9 colinear com ele.<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5879' onClick='GTTabs_show(1,5879)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5879'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<p><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"float: right;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":256,\r\n\"height\":192,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/p>\n<p>Apliquemos os vetores livres $\\vec{u}(1,4,7)$ e $\\vec{v}(2,-1,5)$ num ponto A.<\/p>\n<p>Como os vetores n\u00e3o s\u00e3o colineares, ent\u00e3o definem um plano $\\alpha $.<\/p>\n<p>O vetor pretendido n\u00e3o \u00e9 mais do que um vetor normal ao plano $\\alpha $.<\/p>\n<\/p>\n<p>Seja $\\vec{w}(a,b,c)$ .<\/p>\n<p>Como $\\vec{w}\\bot \\vec{u}\\wedge \\vec{w}\\bot \\vec{v}$ , vem:\u00a0\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\vec{w}.\\vec{u}=0\u00a0 \\\\<br \/>\n\\vec{w}.\\vec{v}=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n(a,b,c)(1,4,7)=0\u00a0 \\\\<br \/>\n(a,b,c)(2,-1,5)=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na+4b+7c=0\u00a0 \\\\<br \/>\n2a-b+5c=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nb=2a+5c\u00a0 \\\\<br \/>\na+8a+20c+7c=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nb=2a+5c\u00a0 \\\\<br \/>\na=-3c\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nb=-c\u00a0 \\\\<br \/>\na=-3c\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; {} &amp; {} &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto, $\\vec{w}=(-3c,-c,c)\\,,\\,\\,c\\in \\mathbb{R}$\u00a0 traduz a fam\u00edlia de vetores perpendiculares aos vetores dados.<\/p>\n<p>Logo, um vector perpendicular aos vetores dados \u00e9, por exemplo, $\\overrightarrow{{{w}_{1}}}=(-3,-1,1)$.<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5879' onClick='GTTabs_show(0,5879)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Num referencial ortonormado do espa\u00e7o, indique um vetor que seja perpendicular a $\\vec{u}(1,4,7)$ \u00a0e a $\\vec{v}(2,-1,5)$ . Observe que qualquer outro vetor nas mesmas condi\u00e7\u00f5es \u00e9 colinear com ele. Resolu\u00e7\u00e3o &gt;&gt;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19405,"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-5879","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":12851,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Um_vetor_perpendicular_a_outros_dois.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5879","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=5879"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5879\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19405"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5879"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5879"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5879"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5879"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}