{"id":6392,"date":"2010-12-19T17:43:06","date_gmt":"2010-12-19T17:43:06","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6392"},"modified":"2022-01-21T23:12:55","modified_gmt":"2022-01-21T23:12:55","slug":"um-prisma-triangular-regular","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6392","title":{"rendered":"Um prisma triangular regular"},"content":{"rendered":"<p><ul id='GTTabs_ul_6392' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6392' class='GTTabs_curr'><a  id=\"6392_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6392' ><a  id=\"6392_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_6392'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6393\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6393\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58.jpg\" data-orig-size=\"353,318\" 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=\"Prisma\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58.jpg\" class=\"alignright wp-image-6393\" title=\"Prisma\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58-300x270.jpg\" alt=\"\" width=\"260\" height=\"234\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58-300x270.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58-150x135.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58.jpg 353w\" sizes=\"auto, (max-width: 260px) 100vw, 260px\" \/><\/a>Na figura est\u00e1 representado, em referencial o.n. Oxyz, um prisma triangular regular:<\/p>\n<p>Sabe-se que:<\/p>\n<ul>\n<li>o v\u00e9rtice O coincide com a origem do referencial;<\/li>\n<li>o v\u00e9rtice P pertence ao semieixo positivo Ox;<\/li>\n<li>o v\u00e9rtice R pertence ao semieixo positivo Oy;<\/li>\n<li>o segmento [QR] tem comprimento 6.<\/li>\n<\/ul>\n<ol>\n<li>Indique, justificando, o valor do produto escalar $\\overrightarrow{TS}\\,.\\,\\overrightarrow{TR}$.<\/li>\n<li>Determine uma equa\u00e7\u00e3o vetorial da reta de intersec\u00e7\u00e3o do plano PQS com o plano de equa\u00e7\u00e3o $x+y+z=5$.<\/li>\n<li>Sabendo que a \u00e1rea lateral do prisma \u00e9 72, determine as coordenadas do ponto S.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6392' onClick='GTTabs_show(1,6392)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6392'>\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-188-58.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"6393\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=6393\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58.jpg\" data-orig-size=\"353,318\" 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=\"Prisma\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58.jpg\" class=\"alignright wp-image-6393\" title=\"Prisma\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58-300x270.jpg\" alt=\"\" width=\"260\" height=\"234\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58-300x270.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58-150x135.jpg 150w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/pag-188-58.jpg 353w\" sizes=\"auto, (max-width: 260px) 100vw, 260px\" \/><\/a>O produto escalar $\\overrightarrow{TS}\\,.\\,\\overrightarrow{TR}$ \u00e9 nulo, pois os vetores s\u00e3o perpendiculares, visto o prisma ser regular (as arestas laterais s\u00e3o perpendiculares \u00e0s arestas da base).<br \/>\n\u00ad<\/li>\n<li>O plano PQS pode ser definido pela condi\u00e7\u00e3o $x=6$.\n<p>Ora, $\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx+y+z=5\u00a0 \\\\<br \/>\nx=6\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\ny+z=-1\u00a0 \\\\<br \/>\nx=6\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\frac{y}{-1}=\\frac{z+1}{1}\u00a0 \\\\<br \/>\nx=6\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n\\end{array}$<\/p>\n<p>Portanto, um ponto dessa reta \u00e9, por exemplo, o ponto de coordenadas $(6,0,-1)$; um vetor diretor da reta \u00e9, por exemplo, $\\vec{r}=(0,-1,1)$ .<\/p>\n<p>Logo, $(x,y,z)=(6,0,-1)+k(0,-1,1)\\,,\\,\\,k\\in \\mathbb{R}$ \u00e9 uma equa\u00e7\u00e3o vetorial da reta pedida.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Como o prisma \u00e9 regular, a \u00e1rea lateral ser\u00e1 dada por ${{A}_{L}}=3\\times \\overline{QR}\\times \\overline{PQ}$.\n<p>Assim, $72=3\\times 6\\times \\overline{PQ}\\Leftrightarrow \\overline{PQ}=4$.<\/p>\n<p>Logo, $Q\\,(6,4,0)$ e $S\\,(6,2,{{z}_{S}})$, com ${{z}_{S}}&gt;0$.<\/p>\n<p>Ora, \\[tg\\,(S\\hat{P}Q)=\\frac{{{z}_{S}}}{\\frac{\\overline{PQ}}{2}}\\]<\/p>\n<p>Logo, $tg\\,60{}^\\text{o}=\\frac{{{z}_{S}}}{2}\\Leftrightarrow {{z}_{S}}=2\\sqrt{3}$ e, portanto, $S\\,(6,2,2\\sqrt{3})$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6392' onClick='GTTabs_show(0,6392)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e1 representado, em referencial o.n. Oxyz, um prisma triangular regular: Sabe-se que: o v\u00e9rtice O coincide com a origem do referencial; o v\u00e9rtice P pertence ao semieixo positivo Ox;&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20843,"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-6392","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":5896,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/12\/11V1Pag188-58_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6392","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=6392"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6392\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20843"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6392"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6392"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6392"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6392"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}