{"id":5163,"date":"2010-11-05T00:25:26","date_gmt":"2010-11-05T00:25:26","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5163"},"modified":"2022-01-21T16:34:43","modified_gmt":"2022-01-21T16:34:43","slug":"rascunho-19","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5163","title":{"rendered":"Um ponto e um vetor"},"content":{"rendered":"<p><ul id='GTTabs_ul_5163' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5163' class='GTTabs_curr'><a  id=\"5163_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5163' ><a  id=\"5163_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_5163'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p><a href=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"5166\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=5166\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2.jpg\" data-orig-size=\"349,247\" 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=\"pag176-2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2.jpg\" class=\"alignright wp-image-5166 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2.jpg\" alt=\"\" width=\"349\" height=\"247\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2.jpg 349w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2-300x212.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2-150x106.jpg 150w\" sizes=\"auto, (max-width: 349px) 100vw, 349px\" \/><\/a>Na figura est\u00e1 representado um referencial o. n. $(O,\\vec{u},\\vec{v},\\vec{w})$, um ponto P e um vetor $\\vec{t}\\,(\\vec{t}=\\overrightarrow{OT})$.<\/p>\n<p>Sendo P&#8217; e T&#8217; as proje\u00e7\u00f5es ortogonais de P e T no plano xOy,<\/p>\n<ol>\n<li>Indique as coordenadas de P e do vetor ${\\vec{t}}$.<\/li>\n<li>Determine as coordenadas dos pontos Q, R e S, tais que: $Q=P+\\vec{t}$, $R=P-2\\vec{v}+\\vec{t}$ e $S=P-2\\vec{t}$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5163' onClick='GTTabs_show(1,5163)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5163'>\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\/11\/pag176-2.jpg\"><img loading=\"lazy\" decoding=\"async\" data-attachment-id=\"5166\" data-permalink=\"https:\/\/www.acasinhadamatematica.pt\/?attachment_id=5166\" data-orig-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2.jpg\" data-orig-size=\"349,247\" 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=\"pag176-2\" data-image-description=\"\" data-image-caption=\"\" data-large-file=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2.jpg\" class=\"alignright wp-image-5166 size-full\" src=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2.jpg\" alt=\"\" width=\"349\" height=\"247\" srcset=\"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2.jpg 349w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2-300x212.jpg 300w, https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/pag176-2-150x106.jpg 150w\" sizes=\"auto, (max-width: 349px) 100vw, 349px\" \/><\/a>$P\\,(1,2,\\frac{3}{2})$.<br \/>\nComo $T\\,(1,3,1)$, ent\u00e3o $\\vec{t}(1,3,1)$, pois $\\vec{t}=\\overrightarrow{OT}$.<br \/>\n\u00ad<\/li>\n<li>Ora,<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\nQ &amp; = &amp; P+\\vec{t}\u00a0 \\\\<br \/>\n{} &amp; = &amp; (1,2,\\frac{3}{2})+(1,3,1)\u00a0 \\\\<br \/>\n{} &amp; = &amp; (2,5,\\frac{5}{2})\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\nR &amp; = &amp; P-2\\vec{v}+\\vec{t}\u00a0 \\\\<br \/>\n{} &amp; = &amp; (1,2,\\frac{3}{2})-2(0,1,0)+(1,3,1)\u00a0 \\\\<br \/>\n{} &amp; = &amp; (1,2,\\frac{3}{2})-(0,2,0)+(1,3,1)\u00a0 \\\\<br \/>\n{} &amp; = &amp; (2,3,\\frac{5}{2})\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\n\\[\\begin{array}{*{35}{l}}<br \/>\nS &amp; = &amp; P-2\\vec{t}\u00a0 \\\\<br \/>\n{} &amp; = &amp; (1,2,\\frac{3}{2})-2(1,3,1)\u00a0 \\\\<br \/>\n{} &amp; = &amp; (1,2,\\frac{3}{2})-(2,6,2)\u00a0 \\\\<br \/>\n{} &amp; = &amp; (-1,-4,-\\frac{1}{2})\u00a0 \\\\<br \/>\n\\end{array}\\]<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5163' onClick='GTTabs_show(0,5163)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Na figura est\u00e1 representado um referencial o. n. $(O,\\vec{u},\\vec{v},\\vec{w})$, um ponto P e um vetor $\\vec{t}\\,(\\vec{t}=\\overrightarrow{OT})$. Sendo P&#8217; e T&#8217; as proje\u00e7\u00f5es ortogonais de P e T no plano xOy, Indique&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":20824,"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-5163","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":2312,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/11V1Pag176-2_520x245.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5163","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=5163"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5163\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/20824"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5163"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5163"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5163"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5163"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}