{"id":6386,"date":"2010-12-14T17:16:18","date_gmt":"2010-12-14T17:16:18","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6386"},"modified":"2022-01-12T21:27:57","modified_gmt":"2022-01-12T21:27:57","slug":"considere-o-plano-%ce%b1","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6386","title":{"rendered":"Considere o plano \u03b1"},"content":{"rendered":"<p><ul id='GTTabs_ul_6386' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6386' class='GTTabs_curr'><a  id=\"6386_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6386' ><a  id=\"6386_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_6386'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere o plano $\\alpha $: $x+2y-2z+7=0$.<\/p>\n<ol>\n<li>Determine os pontos de intersec\u00e7\u00e3o de $\\alpha $ com os tr\u00eas eixos coordenados.<\/li>\n<li>Calcule a amplitude do \u00e2ngulo formado por Oz com qualquer reta normal ao plano dado.<br \/>\n(Aproxime o resultado \u00e0 d\u00e9cima do grau.)<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6386' onClick='GTTabs_show(1,6386)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6386'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>O eixo Ox pode ser definido pela condi\u00e7\u00e3o: $y=0\\wedge z=0$\n<p>O eixo Oy pode ser definido pela condi\u00e7\u00e3o: $x=0\\wedge z=0$<\/p>\n<p>O eixo Oz pode ser definido pela condi\u00e7\u00e3o: $x=0\\wedge y=0$<\/p>\n<p><span style=\"text-decoration: underline;\">Ponto de intersec\u00e7\u00e3o com o eixo Ox<\/span>: $A\\,(-7,0,0)$<\/p>\n<p>\\[x+2y-2z+7=0\\wedge y=0\\wedge z=0\\Leftrightarrow x=-7\\wedge y=0\\wedge z=0\\]<\/p>\n<p><span style=\"text-decoration: underline;\">Ponto de intersec\u00e7\u00e3o com o eixo Oy<\/span>: $B\\,(0,-\\frac{7}{2},0)$<\/p>\n<p>\\[x+2y-2z+7=0\\wedge x=0\\wedge z=0\\Leftrightarrow x=0\\wedge y=-\\frac{7}{2}\\wedge z=0\\]<\/p>\n<p><span style=\"text-decoration: underline;\">Ponto de intersec\u00e7\u00e3o com o eixo Oz<\/span>: $C\\,(0,0,\\frac{7}{2})$<\/p>\n<p>\\[x+2y-2z+7=0\\wedge x=0\\wedge y=0\\Leftrightarrow x=0\\wedge y=0\\wedge z=\\frac{7}{2}\\]<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Um vetor normal ao plano $\\alpha $ \u00e9 $\\vec{n}=(1,2,-2)$ .<\/p>\n<p>Ora,<br \/>\n\\[\\cos (\\vec{n}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\vec{k})=\\frac{(1,2,-2).(0,0,1)}{\\sqrt{{{1}^{2}}+{{2}^{2}}+{{2}^{2}}}\\times 1}=\\frac{-2}{3}=-\\frac{2}{3}\\]<br \/>\nLogo, $\\vec{n}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\vec{k}={{\\cos }^{-1}}(-\\frac{2}{3})\\simeq 131,8{}^\\text{o}$, pelo que a amplitude do \u00e2ngulo pedido \u00e9 $\\beta =180{}^\\text{o}-(\\vec{n}\\overset{\\hat{\\ }}{\\mathop{{}}}\\,\\vec{k})\\simeq 48,2{}^\\text{o}$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6386' onClick='GTTabs_show(0,6386)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere o plano $\\alpha $: $x+2y-2z+7=0$. Determine os pontos de intersec\u00e7\u00e3o de $\\alpha $ com os tr\u00eas eixos coordenados. Calcule a amplitude do \u00e2ngulo formado por Oz com qualquer reta normal&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19180,"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-6386","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":1781,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2021\/12\/Mat71.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6386","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=6386"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6386\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6386"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6386"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6386"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6386"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}