{"id":6290,"date":"2010-11-30T00:00:42","date_gmt":"2010-11-30T00:00:42","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6290"},"modified":"2022-01-12T14:02:53","modified_gmt":"2022-01-12T14:02:53","slug":"rascunho-38","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6290","title":{"rendered":"Determine a intersec\u00e7\u00e3o dos planos \u03b1, \u03b2 e \u03b3"},"content":{"rendered":"<p><ul id='GTTabs_ul_6290' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6290' class='GTTabs_curr'><a  id=\"6290_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6290' ><a  id=\"6290_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_6290'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span> Determine a intersec\u00e7\u00e3o dos planos \u03b1, \u03b2 e \u03b3, tais que:<\/p>\n<ol>\n<li>\u03b1: $2x-y+z-1=0$, \u03b2: $5x-3y+2z=5$ e\u00a0\u03b3: $4x-3y+7z=7$<\/li>\n<li>\u03b1: $x+y-z=0$, \u03b2:\u00a0$x-y+z=0$ e\u00a0\u03b3: $3x+y-z=0$<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6290' onClick='GTTabs_show(1,6290)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6290'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Resolvendo o sistema, vem:<br \/>\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 2x-y+z-1=0\u00a0 \\\\ \u00a0\u00a0 5x-3y+2z=5\u00a0 \\\\ \u00a0\u00a0 4x-3y+7z=7\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}} \u00a0\u00a0 -3\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 2x-y+z=1\u00a0 \\\\ \u00a0\u00a0 5x-3y+2z=5\u00a0 \\\\ \u00a0\u00a0 4x-3y+7z=7\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\begin{array}{*{35}{r}} \u00a0\u00a0 {}\u00a0 \\\\ \u00a0\u00a0 -2\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 2x-y+z=1\u00a0 \\\\ \u00a0\u00a0 -x-z=2\u00a0 \\\\ \u00a0\u00a0 -2x+4z=4\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\ \u00a0\u00a0 {} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 2x-y+z=1\u00a0 \\\\ \u00a0\u00a0 -x-z=2\u00a0 \\\\ \u00a0\u00a0 6z=0\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 y=-5\u00a0 \\\\ \u00a0\u00a0 x=-2\u00a0 \\\\ \u00a0\u00a0 z=0\u00a0 \\\\ \\end{array} \\right. &amp; {}\u00a0 \\\\ \\end{array}\\]<br \/>\nA intersec\u00e7\u00e3o dos tr\u00eas planos \u00e9 o ponto de coordenadas $(-2,-5,0)$.\u00ad<br \/>\n\u00ad<\/li>\n<li>Resolvendo o sistema, vem:<br \/>\n\\[\\begin{array}{*{35}{l}} \u00a0\u00a0 \\begin{array}{*{35}{r}} \u00a0\u00a0 +\u00a0 \\\\ \u00a0\u00a0 1\\times\u00a0\u00a0 \\\\ \u00a0\u00a0 +\u00a0 \\\\ \\end{array}\\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+y-z=0\u00a0 \\\\ \u00a0\u00a0 x-y+z=0\u00a0 \\\\ \u00a0\u00a0 3x+y-z=0\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x+y-z=0\u00a0 \\\\ \u00a0\u00a0 2x=0\u00a0 \\\\ \u00a0\u00a0 4x=0\u00a0 \\\\ \\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}} \u00a0\u00a0 x=0\u00a0 \\\\ \u00a0\u00a0 y=z\u00a0 \\\\ \\end{array} \\right.\u00a0 \\\\ \\end{array}\\]<br \/>\nA intersec\u00e7\u00e3o dos tr\u00eas planos \u00e9 a recta definida por: $x=0\\wedge y=z$.<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6290' onClick='GTTabs_show(0,6290)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Determine a intersec\u00e7\u00e3o dos planos \u03b1, \u03b2 e \u03b3, tais que: \u03b1: $2x-y+z-1=0$, \u03b2: $5x-3y+2z=5$ e\u00a0\u03b3: $4x-3y+7z=7$ \u03b1: $x+y-z=0$, \u03b2:\u00a0$x-y+z=0$ e\u00a0\u03b3: $3x+y-z=0$ Resolu\u00e7\u00e3o &gt;&gt; Resolu\u00e7\u00e3o &lt;&lt; Enunciado<\/p>\n","protected":false},"author":1,"featured_media":19407,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,119],"series":[],"class_list":["post-6290","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":2398,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Intersecao_de_tres_planos.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6290","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=6290"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6290\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19407"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6290"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6290"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6290"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6290"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}