{"id":5368,"date":"2010-11-13T19:40:09","date_gmt":"2010-11-13T19:40:09","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5368"},"modified":"2022-01-12T11:56:34","modified_gmt":"2022-01-12T11:56:34","slug":"rascunho-26","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5368","title":{"rendered":"Calcule a amplitude do \u00e2ngulo formado pela diagonal do cubo com qualquer das suas arestas"},"content":{"rendered":"<p><ul id='GTTabs_ul_5368' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5368' class='GTTabs_curr'><a  id=\"5368_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5368' ><a  id=\"5368_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_5368'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Seja $(O,\\vec{i},\\vec{j},\\vec{k})$ um referencial o. n. do espa\u00e7o.<\/p>\n<ol>\n<li>Calcule a amplitude do \u00e2ngulo formado pela diagonal de um cubo com qualquer das suas arestas.<\/li>\n<li>O vetor ${\\vec{u}}$ \u00a0\u00e9 tal que $\\vec{u}=2\\vec{i}+2\\vec{j}+2\\vec{k}$.<br \/>\nIndique, em radianos, uma medida de $(\\vec{u}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\vec{i})$, de\u00a0$(\\vec{u}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\vec{j})$ e de $(\\vec{u}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\vec{k})$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5368' onClick='GTTabs_show(1,5368)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5368'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><span class=\"alignright\"><script 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 0,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script><\/span>Consideremos um cubo com aresta de comprimento <strong>a<\/strong>, associado a um referencial o. n., como indicado na figura.\n<p>As coordenadas dos vetores assinalados s\u00e3o: $\\overrightarrow{OA}=(a,0,0)$, $\\overrightarrow{OC}=(0,a,0)$, $\\overrightarrow{OD}=(0,0,a)$ e $\\overrightarrow{OF}=(a,a,a)$.<\/p>\n<p>Ora,<\/p>\n<p>$\\begin{array}{*{35}{l}}<br \/>\n\\cos (\\overrightarrow{OA}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{OF}) &amp; = &amp; \\frac{(a,0,0).(a,a,a)}{\\sqrt{{{a}^{2}}}\\times \\sqrt{{{a}^{2}}+{{a}^{2}}+{{a}^{2}}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{{{a}^{2}}}{a\\times a\\sqrt{3}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\sqrt{3}}{3}\u00a0 \\\\<br \/>\n\\end{array}$<\/p>\n<p>$\\begin{array}{*{35}{l}}<br \/>\n\\cos (\\overrightarrow{OC}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{OF}) &amp; = &amp; \\frac{(0,a,0).(a,a,a)}{\\sqrt{{{a}^{2}}}\\times \\sqrt{{{a}^{2}}+{{a}^{2}}+{{a}^{2}}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{{{a}^{2}}}{a\\times a\\sqrt{3}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\sqrt{3}}{3}\u00a0 \\\\<br \/>\n\\end{array}$<\/p>\n<p>$\\begin{array}{*{35}{l}}<br \/>\n\\cos (\\overrightarrow{OD}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{OF}) &amp; = &amp; \\frac{(0,0,a).(a,a,a)}{\\sqrt{{{a}^{2}}}\\times \\sqrt{{{a}^{2}}+{{a}^{2}}+{{a}^{2}}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{{{a}^{2}}}{a\\times a\\sqrt{3}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\sqrt{3}}{3}\u00a0 \\\\<br \/>\n\\end{array}$<\/p>\n<p>Logo, $(\\overrightarrow{OA}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{OF})=(\\overrightarrow{OC}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{OF})=(\\overrightarrow{OD}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{OF})={{\\cos }^{-1}}(\\frac{\\sqrt{3}}{3})\\simeq 54,74{}^\\text{o}$.<\/p>\n<p>Isto \u00e9, \u00e9 invari\u00e1vel a amplitude do \u00e2ngulo formado pela diagonal de um cubo com qualquer das suas arestas, sendo essa amplitude de 55\u00ba, aproximadamente.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Dada a semelhan\u00e7a com a quest\u00e3o anterior,\u00a0temos: \\[\\begin{array}{*{35}{l}}<br \/>\n\\cos (\\overrightarrow{u}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{i})=\\cos (\\overrightarrow{u}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{j})=\\cos (\\overrightarrow{u}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{k}) &amp; = &amp; \\frac{(1,0,0).(2,2,2)}{\\sqrt{{{1}^{2}}}\\times \\sqrt{{{2}^{2}}+{{2}^{2}}+{{2}^{2}}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{2}{1\\times 2\\sqrt{3}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\sqrt{3}}{3}\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Logo, $(\\overrightarrow{u}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{i})=(\\overrightarrow{u}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{j})=(\\overrightarrow{u}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\overrightarrow{k})={{\\cos }^{-1}}(\\frac{\\sqrt{3}}{3})\\simeq 0,96\\,rad$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5368' onClick='GTTabs_show(0,5368)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Seja $(O,\\vec{i},\\vec{j},\\vec{k})$ um referencial o. n. do espa\u00e7o. Calcule a amplitude do \u00e2ngulo formado pela diagonal de um cubo com qualquer das suas arestas. O vetor ${\\vec{u}}$ \u00a0\u00e9 tal que $\\vec{u}=2\\vec{i}+2\\vec{j}+2\\vec{k}$.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19387,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,111],"series":[],"class_list":["post-5368","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","tag-vectores"],"views":4930,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Cubo_11.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5368","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=5368"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5368\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19387"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5368"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5368"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5368"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5368"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}