{"id":6375,"date":"2010-12-13T19:53:28","date_gmt":"2010-12-13T19:53:28","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6375"},"modified":"2022-01-21T22:17:32","modified_gmt":"2022-01-21T22:17:32","slug":"o-cubo-abcdefgh","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6375","title":{"rendered":"O cubo [ABCDEFGH]"},"content":{"rendered":"<p><ul id='GTTabs_ul_6375' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6375' class='GTTabs_curr'><a  id=\"6375_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6375' ><a  id=\"6375_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_6375'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-49.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6376\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6376\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-49.jpg\" data-orig-size=\"279,316\" 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\/pag186-49.jpg\" class=\"alignright wp-image-6376\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-49.jpg\" alt=\"\" width=\"240\" height=\"272\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-49.jpg 279w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-49-264x300.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-49-132x150.jpg 132w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>No referencial o.n. $(O,\\vec{i},\\vec{j},\\vec{k})$ est\u00e1 representado o cubo [ABCDEFGH].<\/p>\n<p>$BF\\parallel OZ$ e as coordenadas dos v\u00e9rtices opostos B e H s\u00e3o, respetivamente, $(0,-3,3)$ e $(4,1,-1)$.<\/p>\n<ol>\n<li>Quais s\u00e3o as coordenadas dos outros v\u00e9rtices do cubo?<\/li>\n<li>Escreva uma equa\u00e7\u00e3o da reta DG.<\/li>\n<li>Determine uma equa\u00e7\u00e3o da reta que passa por H e \u00e9 paralela a GB.<\/li>\n<li>Escreva uma equa\u00e7\u00e3o do plano que cont\u00e9m a face [ABCD].<\/li>\n<li>Considere o lugar geom\u00e9trico definido pelo cubo e escreva uma condi\u00e7\u00e3o que o caracterize.<\/li>\n<li>Determine uma equa\u00e7\u00e3o da superf\u00edcie esf\u00e9rica circunscrita ao cubo.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6375' onClick='GTTabs_show(1,6375)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6375'>\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\/pag186-49.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6376\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6376\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-49.jpg\" data-orig-size=\"279,316\" 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\/pag186-49.jpg\" class=\"alignright wp-image-6376\" title=\"Cubo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-49.jpg\" alt=\"\" width=\"240\" height=\"272\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-49.jpg 279w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-49-264x300.jpg 264w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag186-49-132x150.jpg 132w\" sizes=\"auto, (max-width: 240px) 100vw, 240px\" \/><\/a>As coordenadas dos outros v\u00e9rtices do cubo s\u00e3o: $A\\,(4,-3,3)$, $C\\,(0,1,3)$, $D\\,(4,1,3)$, $E\\,(4,-3,-1)$, $F\\,(0,-3,-1)$ e $G\\,(0,1,-1)$.<br \/>\n\u00ad<\/li>\n<li>Como $\\overrightarrow{GD}=(4,0,4)$ \u00e9 um vetor diretor da reta DG, ent\u00e3o $(x,y,z)=(4,1,3)+k(4,0,4)\\,,\\,\\,k\\in \\mathbb{R}$ \u00e9 uma equa\u00e7\u00e3o vetorial dessa reta.<br \/>\n\u00ad<\/li>\n<li>Como $\\overrightarrow{GB}=(0,-4,4)$, ent\u00e3o $(x,y,z)=(4,1,-1)+k(0,-4,4)\\,,\\,\\,k\\in \\mathbb{R}$ \u00e9 uma equa\u00e7\u00e3o vetorial da reta pedida.<br \/>\n\u00ad<\/li>\n<li>O plano que cont\u00e9m a face [ABCD] pode ser definido pela equa\u00e7\u00e3o $z=3$.<br \/>\n\u00ad<\/li>\n<li>O cubo pode ser caracterizado pela condi\u00e7\u00e3o $\\begin{matrix}<br \/>\n0\\le x\\le 4 &amp; \\wedge\u00a0 &amp; -3\\le y\\le 1 &amp; \\wedge\u00a0 &amp; -1\\le z\\le 3\u00a0 \\\\<br \/>\n\\end{matrix}$.<br \/>\n\u00ad<\/li>\n<li>O centro dessa superf\u00edcie esf\u00e9rica \u00e9 o centro do cubo,\u00a0o ponto de coordenadas $(2,-1,1)$, e o raio \u00e9 metade da sua diagonal espacial: $r=\\frac{\\overline{AG}}{2}=\\frac{4\\sqrt{3}}{2}=2\\sqrt{3}$.\n<p>Logo, essa superf\u00edcie esf\u00e9rica pode ser definida pela equa\u00e7\u00e3o ${{(x-2)}^{2}}+{{(y+1)}^{2}}+{{(z-1)}^{2}}=12$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6375' onClick='GTTabs_show(0,6375)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado No referencial o.n. $(O,\\vec{i},\\vec{j},\\vec{k})$ est\u00e1 representado o cubo [ABCDEFGH]. $BF\\parallel OZ$ e as coordenadas dos v\u00e9rtices opostos B e H s\u00e3o, respetivamente, $(0,-3,3)$ e $(4,1,-1)$. Quais s\u00e3o as coordenadas dos outros&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20836,"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-6375","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":2716,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag186-49_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6375","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=6375"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6375\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20836"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6375"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6375"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6375"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6375"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}