{"id":6379,"date":"2010-12-13T23:38:37","date_gmt":"2010-12-13T23:38:37","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6379"},"modified":"2022-01-12T21:01:03","modified_gmt":"2022-01-12T21:01:03","slug":"tres-pontos-a-b-e-c","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6379","title":{"rendered":"Tr\u00eas pontos: A, B e C"},"content":{"rendered":"<p><ul id='GTTabs_ul_6379' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6379' class='GTTabs_curr'><a  id=\"6379_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6379' ><a  id=\"6379_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_6379'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere, num referencial o.n. Oxyz, os pontos $A\\,(-6,6,0)$, $B\\,(-2,10,0)$ e $C\\,(0,0,8)$.<\/p>\n<ol>\n<li>Determine uma equa\u00e7\u00e3o cartesiana do plano $\\alpha $ definido por A, B e C.<\/li>\n<li>Escreva equa\u00e7\u00f5es cartesianas da reta de intersec\u00e7\u00e3o do plano $\\alpha $ com o plano coordenado xOz.<\/li>\n<li>Prove que $\\overrightarrow{OA}$ \u00e9 perpendicular a $\\overrightarrow{AB}$ e determine as coordenadas do ponto D de modo que [OABD] seja um ret\u00e2ngulo.<\/li>\n<li>[OABD] \u00e9 a base inferior e C um v\u00e9rtice da base superior do paralelep\u00edpedo ret\u00e2ngulo [OABDCEFG].<br \/>\nDetermine as coordenadas dos outros v\u00e9rtices do s\u00f3lido referido.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6379' onClick='GTTabs_show(1,6379)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6379'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Comecemos por determinar um vetor normal ao plano definido pelos pontos A, B e C, isto \u00e9, um vetor $\\vec{n}=(a,b,c)$ perpendicular aos vetores $\\overrightarrow{AB}=(4,4,0)$ e $\\overrightarrow{AC}=(6,-6,8)$.<br \/>\nOra,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\vec{n}.\\overrightarrow{AB}=0\u00a0 \\\\<br \/>\n\\vec{n}.\\overrightarrow{AC}=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n(a,b,c).(4,4,0)=0\u00a0 \\\\<br \/>\n(a,b,c).(6,-6,8)=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n4a+4b=0\u00a0 \\\\<br \/>\n6a-6b+8c=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nb=-a\u00a0 \\\\<br \/>\n6a+6a+8c=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nb=-a\u00a0 \\\\<br \/>\nc=-\\frac{3}{2}a\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; {}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nLogo, um vector normal a $\\alpha $ \u00e9, por exemplo, ${{{\\vec{n}}}_{1}}=(2,-2,-3)$.<\/p>\n<p>Assim, a equa\u00e7\u00e3o pedida \u00e9 da forma $2x-2y-3z+d=0$. Como o ponto C pertence a esse plano, vem $2\\times 0-2\\times 0-3\\times 8+d=0\\Leftrightarrow d=24$.<\/p>\n<p>Logo, $2x-2y-3z+24=0$ \u00e9 uma equa\u00e7\u00e3o do plano $\\alpha $.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>O plano xOz pode ser definido por $y=0$.\n<p>Logo, temos $2x-2y-3z+24=0\\wedge y=0\\Leftrightarrow 2x-3z+24=0\\wedge y=0$.<\/p>\n<p>Pelo que $2x-3z+24=0\\wedge y=0$ podem constituir as equa\u00e7\u00f5es pedidas.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Como $\\overrightarrow{OA}.\\overrightarrow{AB}=(-6,6,0).(4,4,0)=-24+24+0=0$, ent\u00e3o os vetores considerados s\u00e3o perpendiculares.\n<p>O ponto D tem de satisfazer a condi\u00e7\u00e3o $D=O+\\overrightarrow{AB}$.<br \/>\nLogo, $D=(0,0,0)+(4,4,0)=(4,4,0)$.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Os pontos O, A, B e D pertencem ao plano xOy.\n<p>Logo, o ponto $C\\,(0,0,8)$ apenas pode ser extremo da aresta lateral [OC].<\/p>\n<p>Assim, vem: $E\\,(-6,6,8)$, $F\\,(-2,10,8)$ e $G\\,(4,4,8)$.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6379' onClick='GTTabs_show(0,6379)'>&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 $A\\,(-6,6,0)$, $B\\,(-2,10,0)$ e $C\\,(0,0,8)$. Determine uma equa\u00e7\u00e3o cartesiana do plano $\\alpha $ definido por A, B e C. Escreva equa\u00e7\u00f5es cartesianas da reta de&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14114,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,119],"series":[],"class_list":["post-6379","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-interseccao-de-planos"],"views":2060,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat56.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6379","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=6379"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6379\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14114"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6379"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6379"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6379"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6379"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}