{"id":6403,"date":"2010-12-20T01:24:05","date_gmt":"2010-12-20T01:24:05","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6403"},"modified":"2022-01-12T22:41:10","modified_gmt":"2022-01-12T22:41:10","slug":"uma-recta-e-a-sua-interseccao-com-dois-planos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6403","title":{"rendered":"Uma reta e a sua intersec\u00e7\u00e3o com dois planos"},"content":{"rendered":"<p><ul id='GTTabs_ul_6403' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6403' class='GTTabs_curr'><a  id=\"6403_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6403' ><a  id=\"6403_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_6403'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Suponha que a reta r: $x=y=z$ intersecta o plano $\\alpha $: $x-2y-z=2$ no ponto P e o plano $\\beta $: $x-2y-z=4$, no ponto Q.<\/p>\n<p>Qual \u00e9, ent\u00e3o, na unidade considerada, a norma do vector $PQ$?<\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6403' onClick='GTTabs_show(1,6403)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6403'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<blockquote>\n<p>Suponha que a reta r: $x=y=z$ intersecta o plano $\\alpha $: $x-2y-z=2$ no ponto P e o plano $\\beta $: $x-2y-z=4$, no ponto Q.<\/p>\n<p>Qual \u00e9, ent\u00e3o, na unidade considerada, a norma do vector $PQ$?<\/p>\n<\/blockquote>\n<p>Comecemos por determinar as coordenadas dos pontos P e Q:<\/p>\n<p>\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx-2y-z=2\u00a0 \\\\<br \/>\nx=y=z\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx-2y-z=2\u00a0 \\\\<br \/>\nx-y=0\u00a0 \\\\<br \/>\nx-z=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx-2y-z=2\u00a0 \\\\<br \/>\nx-y=0\u00a0 \\\\<br \/>\n-2y=2\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=-1\u00a0 \\\\<br \/>\ny=-1\u00a0 \\\\<br \/>\nz=-1\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx-2y-z=4\u00a0 \\\\<br \/>\nx=y=z\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx-2y-z=4\u00a0 \\\\<br \/>\nx-y=0\u00a0 \\\\<br \/>\nx-z=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx-2y-z=4\u00a0 \\\\<br \/>\nx-y=0\u00a0 \\\\<br \/>\n-2y=4\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=-2\u00a0 \\\\<br \/>\ny=-2\u00a0 \\\\<br \/>\nz=-2\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Logo, $P\\,(-1,-1,-1)$ e $Q\\,(-2,-2,-2)$.<\/p>\n<p>Ent\u00e3o, na unidade considerada, a norma do vector $PQ$ \u00e9:<br \/>\n$$\\left\\| \\overrightarrow{PQ} \\right\\|=\\sqrt{{{(-2+1)}^{2}}+{{(-2+1)}^{2}}+{{(-2+1)}^{2}}}=\\sqrt{3}$$<\/p><\/p>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6403' onClick='GTTabs_show(0,6403)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Suponha que a reta r: $x=y=z$ intersecta o plano $\\alpha $: $x-2y-z=2$ no ponto P e o plano $\\beta $: $x-2y-z=4$, no ponto Q. Qual \u00e9, ent\u00e3o, na unidade considerada, a&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19170,"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-6403","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":2882,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat61.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6403","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=6403"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6403\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19170"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6403"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6403"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6403"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6403"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}