{"id":6411,"date":"2010-12-21T01:36:40","date_gmt":"2010-12-21T01:36:40","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6411"},"modified":"2022-01-21T23:55:04","modified_gmt":"2022-01-21T23:55:04","slug":"considere-num-referencial-ortonormado","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6411","title":{"rendered":"Considere num referencial ortonormado"},"content":{"rendered":"<p><ul id='GTTabs_ul_6411' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6411' class='GTTabs_curr'><a  id=\"6411_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6411' ><a  id=\"6411_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_6411'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6412\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6412\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67.jpg\" data-orig-size=\"616,357\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"pag-190-67\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67.jpg\" class=\"alignright wp-image-6412\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67-300x173.jpg\" alt=\"\" width=\"340\" height=\"197\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67-300x173.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67-150x86.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67-400x231.jpg 400w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67.jpg 616w\" sizes=\"auto, (max-width: 340px) 100vw, 340px\" \/><\/a>Considere num referencial ortonormado Oxyz:<\/p>\n<ul>\n<li>o ponto $A\\,(10,0,0)$<\/li>\n<li>o ponto $B\\,(0,2,1)$<\/li>\n<li>o ponto $C\\,(0,5,0)$<\/li>\n<li>a recta AB<\/li>\n<li>a recta BC<\/li>\n<\/ul>\n<ol>\n<li>Justifique que as retas AB e BC s\u00e3o complanares e mostre que o plano $\\alpha $ por elas definido admite como equa\u00e7\u00e3o $x+2y+6z=10$.<\/li>\n<li>Determine uma equa\u00e7\u00e3o vetorial da recta de intersec\u00e7\u00e3o do plano $\\alpha $ com o plano xOz.<\/li>\n<li>Calcule o volume da pir\u00e2mide [OBCA].<\/li>\n<\/ol>\n<p><span style=\"color: #0000ff;\">Exerc\u00edcio extra\u00eddo da <a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/ProvaModelo1999-135.pdf\" target=\"_blank\" rel=\"noopener\"><span style=\"color: #0000ff;\">Prova Modelo 1999<\/span><\/a> (quest\u00e3o 4 da 2.\u00aa Parte)<br \/>\nEXAME NACIONAL DO ENSINO SECUND\u00c1RIO, 12.\u00ba Ano de Escolaridade (Decreto-Lei n.\u00ba 286\/89, de 29 de Agosto), Cursos de Car\u00e1cter Geral e Cursos Tecnol\u00f3gicos<\/span><\/p>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6411' onClick='GTTabs_show(1,6411)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6411'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6412\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6412\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67.jpg\" data-orig-size=\"616,357\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"pag-190-67\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67.jpg\" class=\"alignright wp-image-6412\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67-300x173.jpg\" alt=\"\" width=\"340\" height=\"197\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67-300x173.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67-150x86.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67-400x231.jpg 400w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-67.jpg 616w\" sizes=\"auto, (max-width: 340px) 100vw, 340px\" \/><\/a>As retas AB e BC s\u00e3o concorrentes em B. Logo, s\u00e3o complanares.\n<p>As coordenadas dos pontos A, B e C verificam a equa\u00e7\u00e3o $x+2y+6z=10$, pois $10+2\\times 0+6\\times 0=10$, $0+2\\times 2+6\\times 1=10$ e $0+2\\times 5+6\\times 0=10$ s\u00e3o proposi\u00e7\u00f5es verdadeiras.<\/p>\n<p>Portanto, as retas concorrentes AB e BC definem um plano que admite por equa\u00e7\u00e3o a equa\u00e7\u00e3o dada.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>O plano xOz pode ser definido pela condi\u00e7\u00e3o $y=0$.\n<p>Ora,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx+2y+6z=10\u00a0 \\\\<br \/>\ny=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx+6z=10\u00a0 \\\\<br \/>\ny=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\frac{x-10}{-6}=\\frac{z}{1}\u00a0 \\\\<br \/>\ny=0\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto, uma equa\u00e7\u00e3o vetorial da recta pedida \u00e9 $(x,y,z)=(10,0,0)+k(-6,0,1)\\,,\\,\\,k\\in \\mathbb{R}$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>O volume da pir\u00e2mide [OBCA] \u00e9:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\nV &amp; = &amp; \\frac{1}{3}\\times {{A}_{[OBC]}}\\times \\overline{AO}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{1}{3}\\times \\frac{\\overline{OC}\\times \\overline{BB&#8217;}}{2}\\times \\overline{AO}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{1}{3}\\times \\frac{5\\times 1}{2}\\times 10\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{25}{3}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nunidades de volume.<br \/>\n(B&#8217; \u00e9 a proje\u00e7\u00e3o ortogonal do ponto B sobre o eixo Oy.)<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6411' onClick='GTTabs_show(0,6411)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n<\/p>\n<\/p>\n<h3>Adenda:<\/h3>\n<p>No sentido de facilitar a interpreta\u00e7\u00e3o das quest\u00f5es colocadas, pode ser \u00fatil explorar a anima\u00e7\u00e3o abaixo, onde j\u00e1 est\u00e3o representadas as retas AB e BC, bem como a pir\u00e2mide considerada.<\/p>\n<p>Pode acrescentar, ainda, os planos ABC (Plane3D <strong>e<\/strong>) e xOz (Plane3D <strong>d<\/strong>), bem como a sua intersec\u00e7\u00e3o \u2014 a reta AD (Line3D <strong>f<\/strong>).<\/p>\n<p>Entre outras possibilidades, sugere-se a utiliza\u00e7\u00e3o da ferramenta &#8220;Rodar a vista 3D&#8221; (2.\u00ba comando a contar da direita) para facilitar a interpreta\u00e7\u00e3o da posi\u00e7\u00e3o relativa dos v\u00e1rios elementos geom\u00e9tricos considerados.<\/p>\n<\/p>\n<p style=\"text-align: center;\"><iframe loading=\"lazy\" scrolling=\"no\" src=\"https:\/\/www.geogebratube.org\/material\/iframe\/id\/68228\/width\/900\/height\/618\/border\/888888\/rc\/false\/ai\/false\/sdz\/true\/smb\/false\/stb\/true\/stbh\/true\/ld\/false\/sri\/true\/at\/preferhtml5\" width=\"900px\" height=\"618px\" style=\"border:0px;\"> <\/iframe><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere num referencial ortonormado Oxyz: o ponto $A\\,(10,0,0)$ o ponto $B\\,(0,2,1)$ o ponto $C\\,(0,5,0)$ a recta AB a recta BC Justifique que as retas AB e BC s\u00e3o complanares e mostre&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20851,"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-6411","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":6167,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag190-67_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6411","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=6411"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6411\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20851"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6411"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6411"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6411"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6411"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}