{"id":6366,"date":"2010-12-12T18:13:54","date_gmt":"2010-12-12T18:13:54","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6366"},"modified":"2022-01-12T14:53:21","modified_gmt":"2022-01-12T14:53:21","slug":"posicao-de-uma-recta-em-relacao-a-um-plano","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6366","title":{"rendered":"Posi\u00e7\u00e3o de uma reta em rela\u00e7\u00e3o a um plano"},"content":{"rendered":"<p><ul id='GTTabs_ul_6366' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6366' class='GTTabs_curr'><a  id=\"6366_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6366' ><a  id=\"6366_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_6366'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere o plano de equa\u00e7\u00e3o $x-y+z-3=0$ e a reta que passa por $A\\,(1,1,1)$ e tem a dire\u00e7\u00e3o do vetor $\\vec{u}\\,(1,-1,1)$ .<\/p>\n<ol>\n<li>Qual a posi\u00e7\u00e3o relativa da reta em rela\u00e7\u00e3o ao plano? Justifique.<\/li>\n<li>Determine o ponto de intersec\u00e7\u00e3o da reta com o plano.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6366' onClick='GTTabs_show(1,6366)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6366'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Considere o plano de equa\u00e7\u00e3o $x-y+z-3=0$ e a reta que passa por $A\\,(1,1,1)$ e tem a dire\u00e7\u00e3o do vetor $\\vec{u}\\,(1,-1,1)$ .<\/p>\n<\/blockquote>\n<ol>\n<li>\n<blockquote>\n<p>Qual a posi\u00e7\u00e3o relativa da reta em rela\u00e7\u00e3o ao plano? Justifique.<\/p>\n<\/blockquote>\n<\/li>\n<li>\n<blockquote>\n<p>Determine o ponto de intersec\u00e7\u00e3o da reta com o plano.<\/p>\n<\/blockquote>\n<p>\u00ad<\/p>\n<\/li>\n<\/ol>\n<ol>\n<li>Um vetor normal ao plano \u00e9 $\\vec{n}\\,(1,-1,1)$.\u00a0 (As coordenadas deste vetor normal ao plano s\u00e3o os coeficientes de x, y e z, respetivamente, da equa\u00e7\u00e3o do plano.)\n<p>Ora, os vetores ${\\vec{u}}$ \u00a0e ${\\vec{n}}$ \u00a0s\u00e3o colineares. Logo, a reta \u00e9 perpendicular ao plano.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Uma equa\u00e7\u00e3o cartesiana da reta \u00e9 $\\frac{x-1}{1}=\\frac{y-1}{-1}=\\frac{z-1}{1}$.<br \/>\nResolvendo o sistema, temos:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx-y+z-3=0\u00a0 \\\\<br \/>\n\\frac{x-1}{1}=\\frac{y-1}{-1}=\\frac{z-1}{1}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx-y+z=3\u00a0 \\\\<br \/>\n-x+1=y-1\u00a0 \\\\<br \/>\nx-1=z-1\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}}<br \/>\n(+) &amp; (+)\u00a0 \\\\<br \/>\n{} &amp; (1\\times )\u00a0 \\\\<br \/>\n(-1\\times ) &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx-y+z=3\u00a0 \\\\<br \/>\n-x-y=-2\u00a0 \\\\<br \/>\nx-z=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}}<br \/>\n{}\u00a0 \\\\<br \/>\n(+)\u00a0 \\\\<br \/>\n(-2\\times )\u00a0 \\\\<br \/>\n\\end{array}\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx-y+z=3\u00a0 \\\\<br \/>\n-2y+z=1\u00a0 \\\\<br \/>\n-y+2z=3\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx-y+z=3\u00a0 \\\\<br \/>\n-2y+z=1\u00a0 \\\\<br \/>\n-3z=-5\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=y-\\frac{5}{3}+3\u00a0 \\\\<br \/>\ny=\\frac{1-\\frac{5}{3}}{-2}\u00a0 \\\\<br \/>\nz=\\frac{5}{3}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=\\frac{5}{3}\u00a0 \\\\<br \/>\ny=\\frac{1}{3}\u00a0 \\\\<br \/>\nz=\\frac{5}{3}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nO ponto de intersec\u00e7\u00e3o da reta com o plano tem coordenadas \\((\\frac{5}{3},\\frac{1}{3},\\frac{5}{3})\\).<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6366' onClick='GTTabs_show(0,6366)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere o plano de equa\u00e7\u00e3o $x-y+z-3=0$ e a reta que passa por $A\\,(1,1,1)$ e tem a dire\u00e7\u00e3o do vetor $\\vec{u}\\,(1,-1,1)$ . Qual a posi\u00e7\u00e3o relativa da reta em rela\u00e7\u00e3o ao plano?&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19408,"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],"series":[],"class_list":["post-6366","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"],"views":2104,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/Posicao_relativa_de_uma_reta_e_um_plano.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6366","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=6366"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6366\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19408"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6366"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6366"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6366"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6366"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}