{"id":6360,"date":"2010-12-08T23:19:57","date_gmt":"2010-12-08T23:19:57","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=6360"},"modified":"2022-01-12T14:30:24","modified_gmt":"2022-01-12T14:30:24","slug":"equacoes-cartesianas-de-duas-rectas","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=6360","title":{"rendered":"Equa\u00e7\u00f5es cartesianas de duas rectas"},"content":{"rendered":"<p><ul id='GTTabs_ul_6360' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_6360' class='GTTabs_curr'><a  id=\"6360_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_6360' ><a  id=\"6360_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_6360'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Seja um referencial ortonormado $(O,\\vec{i},\\vec{j},\\vec{k})$.<\/p>\n<p>Dados os pontos $A\\,(2,3,-1)$ e $B\\,(2,-1,4)$ e o vetor $\\vec{u}\\,(1,4,-2)$ , determine:<\/p>\n<ol>\n<li>uma equa\u00e7\u00e3o vetorial da reta que passa em A e \u00e9 paralela a ${\\vec{u}}$ ;<\/li>\n<li>equa\u00e7\u00f5es cartesianas da reta que passa em A e tem a dire\u00e7\u00e3o de ${\\vec{u}}$ ;<\/li>\n<li>equa\u00e7\u00f5es cartesianas da reta AB.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_6360' onClick='GTTabs_show(1,6360)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_6360'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>Como $\\vec{u}\\,(1,4,-2)$\u00a0 \u00e9 um vetor diretor da reta pedida, ent\u00e3o $(x,y,z)=(2,3,-1)+k(1,4,-2)\\,,\\,\\,k\\in \\mathbb{R}$ \u00e9 uma sua equa\u00e7\u00e3o vetorial.<br \/>\n\u00ad<\/li>\n<li>Da equa\u00e7\u00e3o vetorial da reta da al\u00ednea anterior, vem:<br \/>\n\\[\\begin{matrix}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=2+k\u00a0 \\\\<br \/>\ny=3+4k\u00a0 \\\\<br \/>\nz=-1-2k\u00a0 \\\\<br \/>\n\\end{array},\\,k\\in \\mathbb{R} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nk=x-2\u00a0 \\\\<br \/>\nk=\\frac{y-3}{4}\u00a0 \\\\<br \/>\nk=\\frac{z+1}{-2}\u00a0 \\\\<br \/>\n\\end{array},\\,k\\in \\mathbb{R} \\right. &amp; \\Leftrightarrow\u00a0 &amp; x-2=\\frac{y-3}{4}=\\frac{z+1}{-2}\u00a0 \\\\<br \/>\n\\end{matrix}\\]<\/p>\n<p>Logo, \\[x-2=\\frac{y-3}{4}=\\frac{z+1}{-2}\\]<br \/>\nquer \\[4x-y=5\\wedge 2x+z=3\\]<br \/>\ns\u00e3o equa\u00e7\u00f5es cartesianas da reta pedida.<\/p>\n<p>Note que:<br \/>\n\\[x-2=\\frac{y-3}{4}=\\frac{z+1}{-2}\\Leftrightarrow 4(x-2)=y-3\\wedge -2(x-2)=z+1\\Leftrightarrow 4x-y=5\\wedge 2x+z=3\\]<br \/>\n\u00ad<\/p>\n<\/li>\n<li>Como $\\overrightarrow{AB}=(2,-1,4)-(2,3,-1)=(0,-4,5)$, ent\u00e3o $(x,y,z)=(2,3,-1)+k(0,-4,5)\\,,\\,\\,k\\in \\mathbb{R}$ \u00e9 uma equa\u00e7\u00e3o vetorial da recta AB.\n<p>De forma an\u00e1loga, temos:<br \/>\n\\[\\begin{matrix}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=2+0k\u00a0 \\\\<br \/>\ny=3-4k\u00a0 \\\\<br \/>\nz=-1+5k\u00a0 \\\\<br \/>\n\\end{array},\\,k\\in \\mathbb{R} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\nx=2\u00a0 \\\\<br \/>\nk=\\frac{y-3}{-4}\u00a0 \\\\<br \/>\nk=\\frac{z+1}{5}\u00a0 \\\\<br \/>\n\\end{array},\\,k\\in \\mathbb{R} \\right. &amp; \\Leftrightarrow\u00a0 &amp; x=2\\wedge \\frac{y-3}{-4}=\\frac{z+1}{5}\u00a0 \\\\<br \/>\n\\end{matrix}\\]<br \/>\nLogo, \\[x=2\\wedge \\frac{y-3}{-4}=\\frac{z+1}{5}\\] s\u00e3o equa\u00e7\u00f5es cartesianas da reta AB.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_6360' onClick='GTTabs_show(0,6360)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Seja um referencial ortonormado $(O,\\vec{i},\\vec{j},\\vec{k})$. Dados os pontos $A\\,(2,3,-1)$ e $B\\,(2,-1,4)$ e o vetor $\\vec{u}\\,(1,4,-2)$ , determine: uma equa\u00e7\u00e3o vetorial da reta que passa em A e \u00e9 paralela a ${\\vec{u}}$&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14113,"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-6360","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":2342,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat55.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6360","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=6360"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/6360\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/14113"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=6360"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=6360"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=6360"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=6360"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}