{"id":5175,"date":"2010-11-08T23:03:09","date_gmt":"2010-11-08T23:03:09","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5175"},"modified":"2022-01-12T01:14:07","modified_gmt":"2022-01-12T01:14:07","slug":"dados-os-ponto-a-b-e-c","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5175","title":{"rendered":"Dados os ponto A, B e C"},"content":{"rendered":"<p><ul id='GTTabs_ul_5175' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5175' class='GTTabs_curr'><a  id=\"5175_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5175' ><a  id=\"5175_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_5175'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Considere um referencial ortonormado $(O,\\vec{i},\\vec{j},\\vec{k})$.<\/p>\n<p>Dados os pontos $A\\,(-2,1,5)$, $B\\,(1,-3,0)$ e $C\\,(2,2,-1)$<\/p>\n<ol>\n<li>Determine as coordenadas do ponto M, tal que $\\overrightarrow{BM}=3\\,\\overrightarrow{AB}+2\\,\\overrightarrow{AC}$.<\/li>\n<li>Determine as coordenadas do ponto N, tal que $2\\,\\overrightarrow{NA}=3\\,\\overrightarrow{NB}$.<\/li>\n<li>Calcule as coordenadas do ponto m\u00e9dio I de [MN].<\/li>\n<li>Calcule as coordenadas do sim\u00e9trico de C em rela\u00e7\u00e3o a I.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5175' onClick='GTTabs_show(1,5175)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5175'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Ora,<br \/>\n$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{BM} &amp; = &amp; 3\\,\\overrightarrow{AB}+2\\,\\overrightarrow{AC}\u00a0 \\\\<br \/>\n{} &amp; = &amp; 3(1+2,-3-1,0-5)+2(2+2,2-1,-1-5)\u00a0 \\\\<br \/>\n{} &amp; = &amp; (9,-12,-15)+(8,2,-12)\u00a0 \\\\<br \/>\n{} &amp; = &amp; (17,-10,-27)\u00a0 \\\\<br \/>\n\\end{array}$<br \/>\nlogo:<br \/>\n$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{BM}=(17,-10,-27) &amp; \\Leftrightarrow\u00a0 &amp; M-B=(17,-10,-27)\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; M=B+(17,-10,-27)\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; M=(1,-3,0)+(17,-10,-27)\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; M=(18,-13,-27)\u00a0 \\\\<br \/>\n\\end{array}$<br \/>\n\u00ad<\/li>\n<li>Considerando $N\\,(x,y,z)$, vem:<br \/>\n$\\begin{array}{*{35}{l}}<br \/>\n2\\,\\overrightarrow{NA}=3\\,\\overrightarrow{NB} &amp; \\Leftrightarrow\u00a0 &amp; 2(-2-x,1-y,5-z)=3(1-x,-3-y,0-z)\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; (-4-2x,2-2y,10-2z)=(3-3x,-9-3y,-3z)\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n-4-2x=3-3x\u00a0 \\\\<br \/>\n2-2y=-9-3y\u00a0 \\\\<br \/>\n10-2z=-3z\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=7\u00a0 \\\\<br \/>\ny=-11\u00a0 \\\\<br \/>\nz=-10\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n\\end{array}$<br \/>\nLogo, $N\\,(7,-11,-10)$.<\/p>\n<p><strong>ALTERNATIVA<\/strong>:<\/p>\n<p>$\\begin{array}{*{35}{l}}<br \/>\n2\\,\\overrightarrow{NA}=3\\,\\overrightarrow{NB} &amp; \\Leftrightarrow\u00a0 &amp; 2(A-N)=3(B-N)\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 3N-2N=3B-2A\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; N=3(1,-3,0)-2(-2,1,5)\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; N=(7,-11,-10)\u00a0 \\\\<br \/>\n\\end{array}$<br \/>\n\u00ad<\/p>\n<\/li>\n<li>As coordenadas de I s\u00e3o $(\\frac{18+7}{2},\\frac{-13+(-11)}{2},\\frac{-27+(-10)}{2})=(\\frac{25}{2},-12,-\\frac{37}{2})$.<br \/>\n\u00ad<\/li>\n<li>Seja C&#8217; o sim\u00e9trico de C em rela\u00e7\u00e3o a I.<br \/>\nOra, C&#8217; \u00e9 tal que $C&#8217;=I+\\overrightarrow{CI}$. (Verifique graficamente)<br \/>\nLogo,<br \/>\n$\\begin{array}{*{35}{l}}<br \/>\nC&#8217; &amp; = &amp; (\\frac{25}{2},-12,-\\frac{37}{2})+(\\frac{25}{2}-2,-12-2,-\\frac{37}{2}+1)\u00a0 \\\\<br \/>\n{} &amp; = &amp; (23,-26,-36)\u00a0 \\\\<br \/>\n\\end{array}$<\/p>\n<p><strong>ALTERNATIVA<\/strong>:<\/p>\n<p>O ponto m\u00e9dio do segmento [CC&#8217;] ser\u00e1 o ponto I (Verifique graficamente). Logo, sendo C&#8217; (<em>x<\/em>,<em>y<\/em>.<em>z<\/em>), vem:<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\n(\\frac{x+2}{2},\\frac{y+2}{2},\\frac{z-1}{2})=(\\frac{25}{2},-12,-\\frac{37}{2}) &amp; \\Leftrightarrow\u00a0 &amp; (x+2,y+2,z-1)=(25,-24,-37)\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; (x,y,z)=(23,-26,-36)\u00a0 \\\\<br \/>\n\\end{array}\\]<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5175' onClick='GTTabs_show(0,5175)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Considere um referencial ortonormado $(O,\\vec{i},\\vec{j},\\vec{k})$. Dados os pontos $A\\,(-2,1,5)$, $B\\,(1,-3,0)$ e $C\\,(2,2,-1)$ Determine as coordenadas do ponto M, tal que $\\overrightarrow{BM}=3\\,\\overrightarrow{AB}+2\\,\\overrightarrow{AC}$. Determine as coordenadas do ponto N, tal que $2\\,\\overrightarrow{NA}=3\\,\\overrightarrow{NB}$. Calcule&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19374,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[98,97,110],"tags":[422,67,111],"series":[],"class_list":["post-5175","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-vectores"],"views":2581,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Dados_os_pontos_A_B_C.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5175","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=5175"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5175\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19374"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5175"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5175"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5175"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5175"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}