{"id":6394,"date":"2010-12-19T19:22:51","date_gmt":"2010-12-19T19:22:51","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6394"},"modified":"2022-01-21T23:16:43","modified_gmt":"2022-01-21T23:16:43","slug":"na-figura-estao-representados-tres-pontos","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6394","title":{"rendered":"Na figura est\u00e3o representados tr\u00eas pontos"},"content":{"rendered":"<p><ul id='GTTabs_ul_6394' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6394' class='GTTabs_curr'><a  id=\"6394_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6394' ><a  id=\"6394_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_6394'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6395\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6395\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59.jpg\" data-orig-size=\"342,230\" 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=\"Tr\u00eas pontos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59.jpg\" class=\"alignright wp-image-6395\" title=\"Tr\u00eas pontos\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59-300x201.jpg\" alt=\"\" width=\"270\" height=\"182\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59-300x201.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59-150x100.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59.jpg 342w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/a>Na figura est\u00e3o representados tr\u00eas pontos, num referencial o.n. Oxyz.<\/p>\n<p>Sabe-se que:<\/p>\n<ul>\n<li>ponto A tem coordenadas $(0,5,2)$;<\/li>\n<li>ponto B pertence ao plano xOz;<\/li>\n<li>ponto C pertence ao plano xOy;<\/li>\n<li>$(x,y,z)=(5,4,-1)+k(1,2,-1)\\,,\\,\\,k\\in \\mathbb{R}$ \u00e9 uma equa\u00e7\u00e3o vetorial da reta BC.<\/li>\n<\/ul>\n<ol>\n<li>Mostre que o ponto B tem coordenadas $(3,0,1)$ e que o ponto C tem coordenadas $(4,2,0)$.<\/li>\n<li>Mostre que o tri\u00e2ngulo [ABC] \u00e9 ret\u00e2ngulo em C.<\/li>\n<li>Considere a superf\u00edcie esf\u00e9rica de centro A, cuja intersec\u00e7\u00e3o com o plano xOy \u00e9 uma circunfer\u00eancia de raio 3.<br \/>\nEscreva uma equa\u00e7\u00e3o dessa superf\u00edcie esf\u00e9rica.<\/li>\n<li>De um plano $\\alpha $ sabe-se que passa em A e \u00e9 perpendicular a BC.<br \/>\nDetermine uma equa\u00e7\u00e3o de $\\alpha $.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6394' onClick='GTTabs_show(1,6394)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6394'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6395\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6395\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59.jpg\" data-orig-size=\"342,230\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;HP pstc4380&quot;,&quot;caption&quot;:&quot;&quot;,&quot;created_timestamp&quot;:&quot;0&quot;,&quot;copyright&quot;:&quot;&quot;,&quot;focal_length&quot;:&quot;0&quot;,&quot;iso&quot;:&quot;0&quot;,&quot;shutter_speed&quot;:&quot;0&quot;,&quot;title&quot;:&quot;&quot;}\" data-image-title=\"Tr\u00eas pontos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59.jpg\" class=\"alignright wp-image-6395\" title=\"Tr\u00eas pontos\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59-300x201.jpg\" alt=\"\" width=\"270\" height=\"182\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59-300x201.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59-150x100.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-59.jpg 342w\" sizes=\"auto, (max-width: 270px) 100vw, 270px\" \/><\/a>O ponto B pertence \u00e0 reta BC e ao plano xOz ($y=0$), logo as suas coordenadas t\u00eam de verificar ambas as condi\u00e7\u00f5es.\n<p>O \u00fanico ponto da reta BC com ordenada nula obt\u00e9m-se para $0=4+2k\\Leftrightarrow k=-2$.<\/p>\n<p>Logo, $B\\,(5-2\\times 1,0,-1-2\\times (-1))=(3,0,1)$.<\/p>\n<p>O ponto C pertence \u00e0 recta BC e ao plano xOy ($z=0$), logo as suas coordenadas t\u00eam de verificar ambas as condi\u00e7\u00f5es.<\/p>\n<p>O \u00fanico ponto da reta BC com cota nula obt\u00e9m-se para $0=-1-k\\Leftrightarrow k=-1$.<\/p>\n<p>Logo, $C\\,(5-1\\times (-1),4-1\\times 2,0)=(4,2,0)$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>Como $A\\,(0,5,2)$, $B\\,(3,0,1)$ e $C\\,(4,2,0)$, ent\u00e3o $\\overrightarrow{CA}=(-4,3,2)$ e $\\overrightarrow{CB}=(-1,-2,1)$.<\/p>\n<p>Ora, $\\overrightarrow{CA}\\,.\\,\\overrightarrow{CB}=-4\\times (-1)+3\\times (-2)+2\\times 1=4-6+2=0$.<br \/>\nLogo, os vetores s\u00e3o perpendiculares.<br \/>\nConsequentemente, o ret\u00e2ngulo [ABC] \u00e9 ret\u00e2ngulo em C.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag188-59.png\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6396\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6396\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag188-59.png\" data-orig-size=\"319,319\" data-comments-opened=\"1\" data-image-meta=\"{&quot;aperture&quot;:&quot;0&quot;,&quot;credit&quot;:&quot;&quot;,&quot;camera&quot;:&quot;&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=\"Circunfer\u00eancia\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag188-59.png\" class=\"alignright wp-image-6396 size-medium\" title=\"Circunfer\u00eancia\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag188-59-300x300.png\" alt=\"\" width=\"300\" height=\"300\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag188-59-300x300.png 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag188-59-150x150.png 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag188-59.png 319w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Na figura ao lado est\u00e1 representada a circunfer\u00eancia da intersec\u00e7\u00e3o da superf\u00edcie esf\u00e9rica com o plano xOy.\u00a0O raio da superf\u00edcie esf\u00e9rica \u00e9 $r=\\overline{AD}=\\sqrt{{{\\overline{AA&#8217;}}^{2}}+{{\\overline{A&#8217;D}}^{2}}}=\\sqrt{{{2}^{2}}+{{3}^{2}}}=\\sqrt{13}$.<\/p>\n<p>Logo, ${{x}^{2}}+{{(y-5)}^{2}}+{{(z-2)}^{2}}=13$ \u00e9 uma equa\u00e7\u00e3o da superf\u00edcie esf\u00e9rica considerada.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>O vetor $\\vec{r}=(1,2,-1)$ , diretor da reta BC, \u00e9 normal ao plano $\\alpha $.<\/p>\n<p>Assim, a equa\u00e7\u00e3o do plano $\\alpha $ \u00e9 da forma $x+2y-z+d=0$.<\/p>\n<p>Como A \u00e9 um ponto desse plano, ent\u00e3o as suas coordenadas t\u00eam de verificar a equa\u00e7\u00e3o anterior:\u00a0$0+2\\times 5-2+d=0\\Leftrightarrow d=-8$.<\/p>\n<p>Logo, $x+2y-z-8=0$ \u00e9 uma equa\u00e7\u00e3o do plano $\\alpha $.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6394' onClick='GTTabs_show(0,6394)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e3o representados tr\u00eas pontos, num referencial o.n. Oxyz. Sabe-se que: ponto A tem coordenadas $(0,5,2)$; ponto B pertence ao plano xOz; ponto C pertence ao plano xOy; $(x,y,z)=(5,4,-1)+k(1,2,-1)\\,,\\,\\,k\\in \\mathbb{R}$&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20844,"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-6394","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":3038,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag188-59_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6394","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=6394"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6394\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20844"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6394"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6394"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6394"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6394"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}