{"id":5907,"date":"2010-11-21T03:06:04","date_gmt":"2010-11-21T03:06:04","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5907"},"modified":"2022-01-12T12:53:46","modified_gmt":"2022-01-12T12:53:46","slug":"escreva-uma-equacao-cartesiana-do-plano","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5907","title":{"rendered":"Escreva uma equa\u00e7\u00e3o cartesiana do plano"},"content":{"rendered":"<p><ul id='GTTabs_ul_5907' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5907' class='GTTabs_curr'><a  id=\"5907_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5907' ><a  id=\"5907_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_5907'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Seja $(O,\\vec{i},\\vec{j},\\vec{k})$\u00a0um referencial ortonormado.<\/p>\n<p>Escreva uma equa\u00e7\u00e3o cartesiana do plano:<\/p>\n<ol>\n<li>que passa pelo ponto\u00a0$A(3,1,2)$ e \u00e9 perpendicular a $\\vec{u}(3,41)$ ;<\/li>\n<li>que cont\u00e9m os pontos $A(3,0,0)$, $B(0,5,0)$ e $C(0,0,4)$;<\/li>\n<li>que passa por $A(2,1,5)$ e \u00e9 paralelo aos vetores $\\vec{u}(1,0,4)$\u00a0 e $\\vec{v}(2,-1,3)$ .<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5907' onClick='GTTabs_show(1,5907)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5907'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li><script src=\"https:\/\/cdn.geogebra.org\/apps\/deployggb.js\"><\/script>\r\n<div id=\"ggbApplet\" style=\"float: right;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet\",\r\n\"width\":258,\r\n\"height\":162,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49  50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet')};\r\n<\/script>Seja $P(x,y,z)$ em ponto gen\u00e9rico do plano $\\alpha $.\n<p>Como o vetor $\\vec{u}(3,4,1)$\u00a0 \u00e9 normal ao plano\u00a0\u00a0$\\alpha $, ent\u00e3o $\\overrightarrow{AP}\\bot \\vec{u}$ .<\/p>\n<p>Assim, temos:<\/p>\n<p>$$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{AP}\\bot \\vec{u}=0 &amp; \\Leftrightarrow\u00a0 &amp; (x-3,y-1,z-2).(3,4,1)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 3x-9+4y-4+z-2=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 3x+4y+z-15=0\u00a0 \\\\<br \/>\n\\end{array}$$<\/p>\n<p>Logo, $3x+4y+z-15=0$ \u00e9 uma equa\u00e7\u00e3o do plano pedido.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p><div id=\"ggbApplet2\" style=\"float: right;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet2\",\r\n\"width\":258,\r\n\"height\":162,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49  50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ \"material_id\":12345,\r\n\"ggbBase64\":\"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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet2 = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet'); applet2.inject('ggbApplet2')};\r\n<\/script>Comecemos por considerar um vetor $\\vec{n}\\bot \\alpha $ .<\/p>\n<p>Como os pontos A, B e C pertencem ao plano $\\alpha $, ent\u00e3o ter\u00e1 de ser $\\vec{n}\\bot \\overrightarrow{AB}\\wedge \\vec{n}\\bot \\overrightarrow{AC}$\u00a0 (por exemplo).<\/p>\n<p>Logo, designando $\\vec{n}(a,b,c)$ , temos:<\/p>\n<p>$$\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\vec{n}.\\overrightarrow{AB}=0\u00a0 \\\\<br \/>\n\\vec{n}.\\overrightarrow{AC}=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n(a,b,c).(-3,5,0)=0\u00a0 \\\\<br \/>\n(a,b,c).(-3,0,4)=0\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n-3a+5b=0\u00a0 \\\\<br \/>\n-3a+4c=0\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na=\\frac{5}{3}b\u00a0 \\\\<br \/>\n-5b+4c=0\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na=\\frac{5}{3}b\u00a0 \\\\<br \/>\nb=\\frac{4}{5}c\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n\\end{array}$$<\/p>\n<p>Portanto, $\\overrightarrow{{{n}_{1}}}=(20,12,15)$ (obtido para $c=15$) \u00e9 um vetor normal ao plano $\\alpha $.<\/p>\n<p>Dado que $\\overrightarrow{{{n}_{1}}}\\bot \\overrightarrow{AP}$, sendo $P(x,y,z)$ um ponto gen\u00e9rico de $\\alpha $, vem:<\/p>\n<p>$$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{{{n}_{1}}}\\bot \\overrightarrow{AP}=0 &amp; \\Leftrightarrow\u00a0 &amp; (20,12,15).(x-3,y,z)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 20x-60+12y+15z=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 20x+12y+15z-60=0\u00a0 \\\\<br \/>\n\\end{array}$$<\/p>\n<p>Logo, $20x+12y+15z-60=0$ \u00e9 uma equa\u00e7\u00e3o do plano pedido.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p><div id=\"ggbApplet3\" style=\"float: right;\"><\/div>\r\n<script>\r\nvar parameters = {\r\n\"id\": \"ggbApplet3\",\r\n\"width\":258,\r\n\"height\":162,\r\n\"showMenuBar\":false,\r\n\"showAlgebraInput\":false,\r\n\"showToolBar\":false,\r\n\"customToolBar\":\"0 39 | 1 501 67 , 5 19 , 72 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24  20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49  50 , 71 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 61 | 40 41 42 , 27 28 35 , 6\",\r\n\"showToolBarHelp\":false,\r\n\"showResetIcon\":true,\r\n\"enableLabelDrags\":false,\r\n\"enableShiftDragZoom\":false,\r\n\"enableRightClick\":false,\r\n\"errorDialogsActive\":false,\r\n\"useBrowserForJS\":false,\r\n\"preventFocus\":false,\r\n\"showFullscreenButton\":true,\r\n\"language\":\"pt\",\r\n\/\/ use this instead of ggbBase64 to load a material from GeoGebraTube\r\n\/\/ 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is3D=is 3D applet using 3D view, AV=Algebra View, SV=Spreadsheet View, CV=CAS View, EV2=Graphics View 2, CP=Construction Protocol, PC=Probability Calculator, DA=Data Analysis, FI=Function Inspector, PV=Python, macro=Macro View\r\nvar views = {'is3D': 0,'AV': 0,'SV': 0,'CV': 0,'EV2': 0,'CP': 1,'PC': 0,'DA': 0,'FI': 0,'PV': 0,'macro': 0};\r\nvar applet3 = new GGBApplet(parameters, '5.0', views);\r\nwindow.onload = function() {applet.inject('ggbApplet'); applet2.inject('ggbApplet2'); applet3.inject('ggbApplet3')};\r\n<\/script>Como os vetores livres ${\\vec{u}}$ \u00a0e ${\\vec{v}}$\u00a0 s\u00e3o paralelos ao plano pedido ($\\alpha $), aplicando-os no ponto $A\\in \\alpha $ esses vetores pertencer\u00e3o tamb\u00e9m ao plano pedido.<\/p>\n<p>Considerando um vetor $\\vec{n}\\bot \\alpha $ , ser\u00e1 $\\vec{n}\\bot \\vec{u}\\wedge \\vec{n}\\bot \\vec{v}$ .<\/p>\n<p>Logo, designando $\\vec{n}(a,b,c)$ , temos:<\/p>\n<p>$$\\begin{array}{*{35}{l}}<br \/>\n\\left\\{ \\begin{array}{*{35}{l}}<br \/>\n\\vec{n}.\\vec{u}=0\u00a0 \\\\<br \/>\n\\vec{n}.\\vec{v}=0\u00a0 \\\\<br \/>\n\\end{array} \\right. &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\n(a,b,c).(1,0,4)=0\u00a0 \\\\<br \/>\n(a,b,c).(2,-1,3)=0\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na+4c=0\u00a0 \\\\<br \/>\n2a-b+3c=0\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na=-4c\u00a0 \\\\<br \/>\n-8c-b+3c=0\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; \\left\\{ \\begin{array}{*{35}{l}}<br \/>\na=-4c\u00a0 \\\\<br \/>\nb=-5c\u00a0 \\\\<br \/>\n\\end{array} \\right.\u00a0 \\\\<br \/>\n\\end{array}$$<\/p>\n<p>Portanto, $\\overrightarrow{{{n}_{1}}}=(-4,-5,1)$ (obtido para $c=1$) \u00e9 um vetor normal ao plano $\\alpha $.<\/p>\n<p>Dado que $\\overrightarrow{{{n}_{1}}}\\bot \\overrightarrow{AP}$, sendo $P(x,y,z)$ um ponto gen\u00e9rico de $\\alpha $, vem:<\/p>\n<p>$$\\begin{array}{*{35}{l}}<br \/>\n\\overrightarrow{{{n}_{1}}}\\bot \\overrightarrow{AP}=0 &amp; \\Leftrightarrow\u00a0 &amp; (-4,-5,1).(x-2,y-1,z-5)=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; -4x+8-5y+5+z-5=0\u00a0 \\\\<br \/>\n{} &amp; \\Leftrightarrow\u00a0 &amp; 4x+5y-z-8=0\u00a0 \\\\<br \/>\n\\end{array}$$<\/p>\n<p>Logo,\u00a0$4x+5y-z-8=0$ \u00e9 uma equa\u00e7\u00e3o do plano pedido.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5907' onClick='GTTabs_show(0,5907)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Seja $(O,\\vec{i},\\vec{j},\\vec{k})$\u00a0um referencial ortonormado. Escreva uma equa\u00e7\u00e3o cartesiana do plano: que passa pelo ponto\u00a0$A(3,1,2)$ e \u00e9 perpendicular a $\\vec{u}(3,41)$ ; que cont\u00e9m os pontos $A(3,0,0)$, $B(0,5,0)$ e $C(0,0,4)$; que passa por&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":14083,"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-5907","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":14862,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2018\/03\/Mat28.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5907","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=5907"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5907\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5907"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5907"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5907"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5907"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}