{"id":5356,"date":"2010-11-13T19:04:48","date_gmt":"2010-11-13T19:04:48","guid":{"rendered":"https:\/\/www.acasinhadamatematica.pt\/?p=5356"},"modified":"2022-01-12T01:41:47","modified_gmt":"2022-01-12T01:41:47","slug":"determine-o-angulo-que-a-recta-r-faz-com-a-recta-s","status":"publish","type":"post","link":"https:\/\/www.acasinhadamatematica.pt\/?p=5356","title":{"rendered":"Determine o \u00e2ngulo que a recta $r$ faz com a recta $s$"},"content":{"rendered":"<p><ul id='GTTabs_ul_5356' class='GTTabs' style='display:none'>\n<li id='GTTabs_li_0_5356' class='GTTabs_curr'><a  id=\"5356_0\" onMouseOver=\"GTTabsShowLinks('Enunciado'); return true;\"  onMouseOut=\"GTTabsShowLinks();\"  class='GTTabsLinks'>Enunciado<\/a><\/li>\n<li id='GTTabs_li_1_5356' ><a  id=\"5356_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_5356'>\n<span class='GTTabs_titles'><b>Enunciado<\/b><\/span><\/p>\n<p>Seja $(O,\\vec{i},\\vec{j})$ um referencial o. n. do plano.<\/p>\n<p>Determine o \u00e2ngulo que a reta r faz com a reta s:<\/p>\n<ol>\n<li>r: $(x,y)=(1,3)+k.(-2,-2)\\,,\\,\\,k\\in \\mathbb{R}$ e s: $3y-x-2=0$;<\/li>\n<li>r: $x+2y+5=0$ e s: $y=\\frac{3}{4}x-3$.<\/li>\n<\/ol>\n<p><div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_next'><a href='#GTTabs_ul_5356' onClick='GTTabs_show(1,5356)'>Resolu\u00e7\u00e3o &gt;&gt;<\/a><\/span><\/div><\/div>\n\n<div class='GTTabs_divs' id='GTTabs_1_5356'>\n<span class='GTTabs_titles'><b>Resolu\u00e7\u00e3o<\/b><\/span><!--more--><\/p>\n<ol>\n<li>A equa\u00e7\u00e3o reduzida da reta s \u00e9 $y=\\frac{1}{3}x+\\frac{2}{3}$ , pelo que o seu declive \u00e9 ${{m}_{s}}=\\frac{1}{3}$.<br \/>\nEnt\u00e3o, os vetores $\\vec{r}(-2,-2)$ \u00a0e $\\vec{s}(3,1)$\u00a0 (Porqu\u00ea?) s\u00e3o diretores das retas r e s, respetivamente.<\/p>\n<p>Como \\[\\begin{array}{*{35}{l}}<br \/>\n\\cos (\\widehat{\\vec{r}\\,\\vec{s}}) &amp; = &amp; \\frac{(-2,-2).(3,1)}{\\sqrt{{{(-2)}^{2}}+{{(-2)}^{2}}}\\times \\sqrt{{{3}^{2}}+{{1}^{2}}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{-2\\times 3-2\\times 1}{2\\sqrt{2}\\times \\sqrt{10}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; -\\frac{4}{2\\sqrt{5}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; -\\frac{2\\sqrt{5}}{5}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nent\u00e3o $\\widehat{\\vec{r}\\,\\vec{s}}={{\\cos }^{-1}}(-\\frac{2\\sqrt{5}}{5})\\simeq 153,4{}^\\text{o}$\u00a0 e, portanto, o \u00e2ngulo entre as retas r e s \u00e9 de, aproximadamente, 26,6\u00ba.<br \/>\n\u00ad<\/p>\n<\/li>\n<li>\n<p>A equa\u00e7\u00e3o reduzida da reta r \u00e9 $y=-\\frac{1}{2}x-\\frac{5}{2}$, pelo que o seu declive \u00e9 ${{m}_{r}}=-\\frac{1}{2}$.<br \/>\nEnt\u00e3o, os vetores $\\vec{r}(2,-1)$\u00a0 e $\\vec{s}(4,3)$\u00a0 (Porqu\u00ea?) s\u00e3o diretores das retas r e s, respetivamente.<\/p>\n<p>Como \\[\\begin{array}{*{35}{l}}<br \/>\n\\cos (\\widehat{\\vec{r}\\,\\vec{s}}) &amp; = &amp; \\frac{(2,-1).(4,3)}{\\sqrt{{{2}^{2}}+{{(-1)}^{2}}}\\times \\sqrt{{{4}^{2}}+{{3}^{2}}}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{2\\times 4-1\\times 3}{\\sqrt{5}\\times 5}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{5}{5\\sqrt{5}}\u00a0 \\\\<br \/>\n{} &amp; = &amp; \\frac{\\sqrt{5}}{5}\u00a0 \\\\<br \/>\n\\end{array}\\]<br \/>\nent\u00e3o $\\widehat{\\vec{r}\\,\\vec{s}}={{\\cos }^{-1}}(\\frac{\\sqrt{5}}{5})\\simeq 63,4{}^\\text{o}$ , que \u00e9 tamb\u00e9m a amplitude do \u00e2ngulo entre as retas r e s.<\/p>\n<\/li>\n<\/ol>\n<div class='GTTabsNavigation' style='display:none'><span class='GTTabs_nav_prev'><a href='#GTTabs_ul_5356' onClick='GTTabs_show(0,5356)'>&lt;&lt; Enunciado<\/a><\/span><\/div><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Enunciado Resolu\u00e7\u00e3o Enunciado Seja $(O,\\vec{i},\\vec{j})$ um referencial o. n. do plano. Determine o \u00e2ngulo que a reta r faz com a reta s: r: $(x,y)=(1,3)+k.(-2,-2)\\,,\\,\\,k\\in \\mathbb{R}$ e s: $3y-x-2=0$; r: $x+2y+5=0$ e s: $y=\\frac{3}{4}x-3$.&#46;&#46;&#46;<\/p>\n","protected":false},"author":1,"featured_media":19385,"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-5356","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":1538,"jetpack_featured_media_url":"https:\/\/www.acasinhadamatematica.pt\/wp-content\/uploads\/2010\/11\/Angulo_de_duas_retas.png","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5356","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=5356"}],"version-history":[{"count":0,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/posts\/5356\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=\/wp\/v2\/media\/19385"}],"wp:attachment":[{"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5356"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5356"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5356"},{"taxonomy":"series","embeddable":true,"href":"https:\/\/www.acasinhadamatematica.pt\/index.php?rest_route=%2Fwp%2Fv2%2Fseries&post=5356"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}