{"id":6380,"date":"2010-12-14T00:51:45","date_gmt":"2010-12-14T00:51:45","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6380"},"modified":"2022-01-21T22:43:31","modified_gmt":"2022-01-21T22:43:31","slug":"mais-um-cubo","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6380","title":{"rendered":"Mais um cubo"},"content":{"rendered":"<p><ul id='GTTabs_ul_6380' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6380' class='GTTabs_curr'><a  id=\"6380_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6380' ><a  id=\"6380_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_6380'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6381\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6381\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52.jpg\" data-orig-size=\"283,304\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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=\"Cubo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52.jpg\" class=\"alignright wp-image-6381\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52-279x300.jpg\" alt=\"\" width=\"240\" height=\"258\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52-279x300.jpg 279w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52-139x150.jpg 139w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52.jpg 283w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Num referencial ortonormado do espa\u00e7o, considere o cubo [ABCDEFGH] com 6 unidades de aresta.<\/p>\n<p>A face [ABFE] \u00e9 paralela ao plano zOy, a face [ABCD] \u00e9 paralela ao plano xOy e $F\\,(2,1,4)$.<\/p>\n<ol>\n<li>Mostre que o tri\u00e2ngulo [BED] \u00e9 equil\u00e1tero.<\/li>\n<li>Determine uma equa\u00e7\u00e3o cartesiana do plano que o cont\u00e9m.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6380' onClick='GTTabs_show(1,6380)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6380'>\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\/pag186-52.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6381\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6381\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52.jpg\" data-orig-size=\"283,304\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&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=\"Cubo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52.jpg\" class=\"alignright wp-image-6381\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52-279x300.jpg\" alt=\"\" width=\"240\" height=\"258\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52-279x300.jpg 279w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52-139x150.jpg 139w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-52.jpg 283w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>Os lados do tri\u00e2ngulo [BED] s\u00e3o diagonais faciais do cubo, logo s\u00e3o geometricamente iguais. Por isso, o tri\u00e2ngulo \u00e9 equil\u00e1tero.\n<p>Mas, de qualquer forma&#8230;<\/p>\n<p>Sendo $B\\,(2,1,-2)$, $D\\,(-4,-5-2)$ e $E\\,(2,-5,4)$, temos:<\/p>\n<p>$\\overrightarrow{BE}=(0,-6,6)$, $\\overrightarrow{ED}=(-6,0,-6)$ e $\\overrightarrow{DB}=(6,6,0)$.<\/p>\n<p>Donde, $\\left\\| \\overrightarrow{BE} \\right\\|=\\left\\| \\overrightarrow{ED} \\right\\|=\\left\\| \\overrightarrow{DB} \\right\\|=\\sqrt{{{6}^{2}}+{{6}^{2}}}=6\\sqrt{2}$.<\/p>\n<p>Logo, o tri\u00e2ngulo \u00e9 equil\u00e1tero.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Comecemos por determinar um vetor normal ao plano definido pelos pontos B, E e D, isto \u00e9, um vetor $\\vec{n}=(a,b,c)$ perpendicular aos vetores $\\overrightarrow{BE}=(0,-6,6)$ e $\\overrightarrow{ED}=(-6,0,-6)$.<br \/>\nOra,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\vec{n}.\\overrightarrow{BE}=0\u00a0 \\\\<br \/>\n\\vec{n}.\\overrightarrow{ED}=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n(a,b,c).(0,-6,6)=0\u00a0 \\\\<br \/>\n(a,b,c).(-6,0,-6)=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n-6b+6c=0\u00a0 \\\\<br \/>\n-6a-6c=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nb=c\u00a0 \\\\<br \/>\na=-c\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nLogo, um vetor normal ao plano BED \u00e9, por exemplo, $\\overrightarrow{{{n}_{1}}}=(-1,1,1)$.<\/p>\n<p>Assim, a equa\u00e7\u00e3o pedida \u00e9 da forma $-x+y+z+d=0$.<br \/>\nComo o ponto B pertence a esse plano, vem $-2+1-2+d=0\\Leftrightarrow d=3$.<\/p>\n<p>Logo, $-x+y+z+3=0$ \u00e9 uma equa\u00e7\u00e3o do plano BED.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6380' onClick='GTTabs_show(0,6380)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Num referencial ortonormado do espa\u00e7o, considere o cubo [ABCDEFGH] com 6 unidades de aresta. A face [ABFE] \u00e9 paralela ao plano zOy, a face [ABCD] \u00e9 paralela ao plano xOy e&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20839,"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-6380","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":3269,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag186-52_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6380","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=6380"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6380\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20839"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6380"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6380"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6380"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6380"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}