{"id":6370,"date":"2010-12-12T23:14:42","date_gmt":"2010-12-12T23:14:42","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6370"},"modified":"2022-01-12T15:48:10","modified_gmt":"2022-01-12T15:48:10","slug":"um-ponto-e-tres-planos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6370","title":{"rendered":"Um ponto e tr\u00eas planos"},"content":{"rendered":"<p><ul id='GTTabs_ul_6370' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6370' class='GTTabs_curr'><a  id=\"6370_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6370' ><a  id=\"6370_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_6370'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere o ponto $A\\,(2,3,1)$ e os planos de equa\u00e7\u00f5es:<\/p>\n<ul>\n<li>$\\alpha $: $x+y+z=2$<\/li>\n<li>$\\beta $: $x-2y+2z+3=0$<\/li>\n<li>$\\gamma $: $2x-3y-4z=0$<\/li>\n<\/ul>\n<ol>\n<li>Verifique se $A\\in \\beta $.<\/li>\n<li>Determine uma equa\u00e7\u00e3o da reta perpendicular a $\\alpha $ e que passa por A.<\/li>\n<li>Determine a intersec\u00e7\u00e3o dos tr\u00eas planos.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6370' onClick='GTTabs_show(1,6370)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6370'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>As coordenadas de A n\u00e3o verificam a equa\u00e7\u00e3o do plano $\\beta $:<br \/>\n$$2-2\\times 3+2\\times 1+3=0\\Leftrightarrow 2-6+2+3=0\\Leftrightarrow 1=0$$<\/p>\n<p>Logo, $A\\notin \\beta $.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Um vector normal ao plano $\\alpha $ \u00e9 $\\overrightarrow{{{n}_{\\alpha }}}(1,1,1)$, que ser\u00e1, ent\u00e3o, diretor da reta pedida.\n<p>Logo, uma equa\u00e7\u00e3o vetorial da reta pedida \u00e9:<br \/>\n$$(x,y,z)=(2,3,1)+k(1,1,1)\\,,\\,\\,k\\in \\mathbb{R}$$<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Resolvendo o sistema de equa\u00e7\u00f5es dos planos dados, temos:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\begin{array}{*{35}{r}}<br \/>\n{} &amp; (-1\\times )\u00a0 \\\\<br \/>\n(-2\\times ) &amp; (+)\u00a0 \\\\<br \/>\n(+) &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx+y+z=2\u00a0 \\\\<br \/>\nx-2y+2z=-3\u00a0 \\\\<br \/>\n2x-3y-4z=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}}<br \/>\n{}\u00a0 \\\\<br \/>\n+\u00a0 \\\\<br \/>\n(3\\times )\u00a0 \\\\<br \/>\n\\end{array}\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx+y+z=2\u00a0 \\\\<br \/>\n-3y+z=-5\u00a0 \\\\<br \/>\ny-8z=6\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx+y+z=2\u00a0 \\\\<br \/>\n-3y+z=-5\u00a0 \\\\<br \/>\n-23z=13\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{13}{23}+2\u00a0 \\\\<br \/>\ny=\\frac{z+5}{3}\u00a0 \\\\<br \/>\nz=-\\frac{13}{23}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=\\frac{25}{23}\u00a0 \\\\<br \/>\ny=\\frac{34}{23}\u00a0 \\\\<br \/>\nz=-\\frac{13}{23}\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nA intersec\u00e7\u00e3o dos tr\u00eas planos \u00e9 o ponto de coordenadas $(\\frac{25}{23},\\frac{34}{23},-\\frac{13}{23})$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6370' onClick='GTTabs_show(0,6370)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere o ponto $A\\,(2,3,1)$ e os planos de equa\u00e7\u00f5es: $\\alpha $: $x+y+z=2$ $\\beta $: $x-2y+2z+3=0$ $\\gamma $: $2x-3y-4z=0$ Verifique se $A\\in \\beta $. Determine uma equa\u00e7\u00e3o da reta perpendicular a $\\alpha&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19177,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[98,97,110],"tags":[422,67,119],"series":[],"class_list":["post-6370","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-interseccao-de-planos"],"views":1723,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat68.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6370","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=6370"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6370\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6370"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6370"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6370"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6370"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}