{"id":6409,"date":"2010-12-21T00:45:42","date_gmt":"2010-12-21T00:45:42","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6409"},"modified":"2022-01-21T23:50:30","modified_gmt":"2022-01-21T23:50:30","slug":"considere-os-pontos-a-e-b","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6409","title":{"rendered":"Considere os pontos A e B"},"content":{"rendered":"<p><ul id='GTTabs_ul_6409' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6409' class='GTTabs_curr'><a  id=\"6409_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6409' ><a  id=\"6409_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_6409'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6410\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6410\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66.jpg\" data-orig-size=\"334,203\" 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=\"Dois pontos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66.jpg\" class=\"alignright wp-image-6410 size-medium\" title=\"Dois pontos\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66-300x182.jpg\" alt=\"\" width=\"300\" height=\"182\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66-300x182.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66-150x91.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66.jpg 334w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>Considere, num referencial o.n. Oxyz, os pontos $A\\,(5,0,0)$ e $B\\,(0,3,1)$.<\/p>\n<ol>\n<li>Mostre que a reta AB est\u00e1 contida no plano de equa\u00e7\u00e3o $x+2y-z=5$.<\/li>\n<li>Determine as coordenadas de um ponto C, pertencente ao eixo Oz e de cota positiva, de tal modo que o tri\u00e2ngulo [ABC] seja ret\u00e2ngulo em C.<\/li>\n<li>Determine o volume do cone que resulta da rota\u00e7\u00e3o do tri\u00e2ngulo [AOB] em torno do eixo Ox.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6409' onClick='GTTabs_show(1,6409)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6409'>\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-190-66.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6410\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6410\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66.jpg\" data-orig-size=\"334,203\" 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=\"Dois pontos\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66.jpg\" class=\"alignright wp-image-6410 size-medium\" title=\"Dois pontos\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66-300x182.jpg\" alt=\"\" width=\"300\" height=\"182\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66-300x182.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66-150x91.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-190-66.jpg 334w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a>As coordenadas dos pontos A e B verificam a equa\u00e7\u00e3o do plano considerado: $5+2\\times 0-0=5$ e $0+2\\times 3-1=5$ s\u00e3o ambas proposi\u00e7\u00f5es verdadeiras.\n<p>Assim, os pontos A e B pertencem ao plano considerado. Consequentemente, a recta AB est\u00e1 contida nesse mesmo plano.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>As coordenadas do ponto C s\u00e3o do tipo $(0,0,c)$, com $c&gt;0$.\n<p>Para que o tri\u00e2ngulo seja ret\u00e2ngulo em C tem de se verificar: $\\overrightarrow{AC}\\,.\\,\\overrightarrow{BC}=0$.<\/p>\n<p>Como $\\overrightarrow{AC}=(-5,0,c)$ e $\\overrightarrow{BC}=(0,-3,c-1)$, vem:<\/p>\n<p>\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n(-5,0,c).(0,-3,c-1)=0\u00a0 \\\\<br \/>\nc&gt;0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nc(c-1)=0\u00a0 \\\\<br \/>\nc&gt;0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; c=1\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<p>Portanto, $C\\,(0,0,1)$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>O cone obtido tem altura [AO] e a base \u00e9 o c\u00edrculo, contido no plano yOz,\u00a0com centro em O e raio [OB].\n<p>Assim, o volume desse cone \u00e9:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\nV &amp; = &amp; \\frac{1}{3}\\times \\pi \\times {{\\left( \\sqrt{0+{{3}^{2}}+{{1}^{2}}} \\right)}^{2}}\\times 5\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{50\\pi }{3}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nunidades de volume.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6409' onClick='GTTabs_show(0,6409)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere, num referencial o.n. Oxyz, os pontos $A\\,(5,0,0)$ e $B\\,(0,3,1)$. Mostre que a reta AB est\u00e1 contida no plano de equa\u00e7\u00e3o $x+2y-z=5$. Determine as coordenadas de um ponto C, pertencente ao&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20850,"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-6409","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":1506,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag190-66_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6409","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=6409"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6409\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20850"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6409"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6409"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6409"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6409"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}