{"id":6384,"date":"2010-12-14T16:46:58","date_gmt":"2010-12-14T16:46:58","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6384"},"modified":"2022-01-21T23:05:05","modified_gmt":"2022-01-21T23:05:05","slug":"um-paralelepipedo-abcodefg","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6384","title":{"rendered":"Um paralelep\u00edpedo [ABCODEFG]"},"content":{"rendered":"<p><ul id='GTTabs_ul_6384' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6384' class='GTTabs_curr'><a  id=\"6384_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6384' ><a  id=\"6384_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_6384'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6385\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6385\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54.jpg\" data-orig-size=\"394,324\" 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=\"Paralelep\u00edpedo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54.jpg\" class=\"alignright wp-image-6385\" title=\"Paralelep\u00edpedo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54-300x246.jpg\" alt=\"\" width=\"270\" height=\"222\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54-300x246.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54-150x123.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54.jpg 394w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/a>No referencial o.n. $(O,\\vec{i},\\vec{j},\\vec{k})$ da figura, est\u00e1 representado o paralelep\u00edpedo [ABCODEFG].<\/p>\n<ul>\n<li>$C\\in \\dot{O}y$ e $A\\in \\dot{O}x$<\/li>\n<li>$\\overrightarrow{OE}=2\\vec{i}+5\\vec{j}+3\\vec{k}$<\/li>\n<\/ul>\n<ol>\n<li>Indique as coordenadas dos v\u00e9rtices E, A e F.<\/li>\n<li>Defina, por uma condi\u00e7\u00e3o, o plano perpendicular ao vetor dado que passa pelo ponto A.<\/li>\n<li>Determine, com aproxima\u00e7\u00e3o \u00e0s cent\u00e9simas, a amplitude do \u00e2ngulo que o vetor $\\overrightarrow{OE}$ forma com o eixo Ox.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6384' onClick='GTTabs_show(1,6384)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6384'>\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\/pag187-54.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6385\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6385\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54.jpg\" data-orig-size=\"394,324\" 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=\"Paralelep\u00edpedo\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54.jpg\" class=\"alignright wp-image-6385\" title=\"Paralelep\u00edpedo\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54-300x246.jpg\" alt=\"\" width=\"270\" height=\"222\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54-300x246.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54-150x123.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag187-54.jpg 394w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/a>As coordenadas desses pontos s\u00e3o: $E\\,(2,5,3)$, $A\\,(2,0,0)$ e $F\\,(0,5,3)$.<br \/>\n\u00ad<\/li>\n<li>A equa\u00e7\u00e3o desse plano \u00e9 da forma $2x+5y+3z+d=0$.\n<p>Como A pertence a esse plano, as suas coordenadas t\u00eam de verificar a equa\u00e7\u00e3o anterior. Logo, vem: $2\\times 2+5\\times 0+3\\times 0+d=0\\Leftrightarrow d=-4$.<\/p>\n<p>Portanto, $2x+5y+3z-4=0$ \u00e9 uma equa\u00e7\u00e3o do plano considerado.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Ora, \\[\\cos (\\overrightarrow{OE}\\,\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\vec{i})=\\frac{(2,5,3).(1,0,0)}{\\sqrt{{{2}^{2}}+{{5}^{2}}+{{3}^{2}}}\\times 1}=\\frac{2}{\\sqrt{38}}=\\frac{2\\sqrt{38}}{38}=\\frac{\\sqrt{38}}{19}\\]<br \/>\nLogo, $\\overrightarrow{OE}\\,\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\vec{i}={{\\cos }^{-1}}(\\frac{\\sqrt{38}}{19})\\simeq 71,07{}^\\text{o}$ \u00e9 a amplitude pedida.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6384' onClick='GTTabs_show(0,6384)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado No referencial o.n. $(O,\\vec{i},\\vec{j},\\vec{k})$ da figura, est\u00e1 representado o paralelep\u00edpedo [ABCODEFG]. $C\\in \\dot{O}y$ e $A\\in \\dot{O}x$ $\\overrightarrow{OE}=2\\vec{i}+5\\vec{j}+3\\vec{k}$ Indique as coordenadas dos v\u00e9rtices E, A e F. Defina, por uma condi\u00e7\u00e3o, o&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20841,"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-6384","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":2469,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag187-54_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6384","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=6384"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6384\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20841"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6384"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6384"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6384"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6384"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}