{"id":436,"date":"2026-10-26T08:15:19","date_gmt":"2026-10-26T07:15:19","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=436"},"modified":"2026-10-25T17:45:54","modified_gmt":"2026-10-25T16:45:54","slug":"alg-el-espacio-afin-euclideo-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=436","title":{"rendered":"ALG: El espacio af\u00edn eucl\u00eddeo con maxima"},"content":{"rendered":"<p>Repasemos un poco lo visto:<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1nto suman los coeficientes que multiplican a \\(x\\) e \\(y\\) en la ecuaci\u00f3n impl\u00edcita de la recta del plano af\u00edn que pasa por los puntos P(4,1) y Q(7,-4)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2yu() {\n  var htmlShow2yu = document.getElementById(\"html-show2yu\");\n  if (htmlShow2yu.style.display === \"none\") {\n    htmlShow2yu.style.display = \"block\";\n  } else {\n    htmlShow2yu.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2yu()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2yu\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGrect_afin02.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<blockquote><p><strong>Ejemplo:<\/strong> Determinar si los puntos P(4,1), Q(3,-4) y R(2,-1) son colineales.<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1yu() {\n  var htmlShow1yu = document.getElementById(\"html-show1yu\");\n  if (htmlShow1yu.style.display === \"none\") {\n    htmlShow1yu.style.display = \"block\";\n  } else {\n    htmlShow1yu.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1yu()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1yu\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGrect_afin03.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Determinar las ecuaciones param\u00e9tricas de la recta que pasa por los puntos P(4,1,2) y Q(3,-4,0)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv3f() {\n  var htmlShow3f = document.getElementById(\"html-show3f\");\n  if (htmlShow3f.style.display === \"none\") {\n    htmlShow3f.style.display = \"block\";\n  } else {\n    htmlShow3f.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3f()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3f\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGrect_afin04.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Determinar las ecuaciones impl\u00edcitas de la recta que pasa por los puntos P(4,1,2) y Q(3,-4,0)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv3w() {\n  var htmlShow3w = document.getElementById(\"html-show3w\");\n  if (htmlShow3w.style.display === \"none\") {\n    htmlShow3w.style.display = \"block\";\n  } else {\n    htmlShow3w.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv3w()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3w\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"input\"><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">4<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">2<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">4<\/span>,<span class=\"code_number\">0<\/span>]<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Dados estos dos puntos la recta vendr\u00e1 dada por uno de ellos, por ejemplo P, y el vector director obtenido de la resta de ambos, \\(\\overrightarrow{PQ}\\)<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td><span class=\"input\"><span class=\"code_variable\">r<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>As\u00ed, las ecuaciones impl\u00edcitas las determinaran las ecuaciones que impliquen rango igual a 1 en la matriz:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">transpose<\/span>(<span class=\"code_function\">matrix<\/span>(<span class=\"code_variable\">r<\/span>,[<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">y<\/span>,<span class=\"code_variable\">z<\/span>]<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span>))<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"center\"><mtd><mtext>(A)<\/mtext><\/mtd><mtd><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mrow><mi>\u2212<\/mi><mn>1<\/mn><\/mrow><\/mtd><mtd><mrow><mi>x<\/mi><mi>\u2212<\/mi><mn>4<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mrow><mi>\u2212<\/mi><mn>5<\/mn><\/mrow><\/mtd><mtd><mrow><mi>y<\/mi><mi>\u2212<\/mi><mn>1<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mrow><mi>\u2212<\/mi><mn>2<\/mn><\/mrow><\/mtd><mtd><mrow><mi>z<\/mi><mi>\u2212<\/mi><mn>2<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Para conseguir estas ecuaciones, consideremas un menor de orden 1 distinto de cero y consigamos escalonar la matriz en base a dicho menor:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"input\"><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span>(<span class=\"code_variable\">A<\/span>,<span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_variable\">A<\/span>[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">\/<\/span><span class=\"code_variable\">A<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">A<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rowop<\/span>(<span class=\"code_variable\">A<\/span>,<span class=\"code_number\">3<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_variable\">A<\/span>[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">1<\/span>]<span class=\"code_operator\">\/<\/span><span class=\"code_variable\">A<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>])<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"center\"><mtd><mtext\/><\/mtd><mtd><mo>0 errores, 0 advertencias<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"center\"><mtd><mtext>(A)<\/mtext><\/mtd><mtd><mrow><mo>(<\/mo><mrow><mtable><mtr><mtd><mrow><mi>\u2212<\/mi><mn>1<\/mn><\/mrow><\/mtd><mtd><mrow><mi>x<\/mi><mi>\u2212<\/mi><mn>4<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mrow><mi>y<\/mi><mi>\u2212<\/mi><mn>5<\/mn><mo>\u2062<\/mo><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mi>\u2212<\/mi><mn>4<\/mn><\/mrow><mo>)<\/mo><\/mrow><mi>\u2212<\/mi><mn>1<\/mn><\/mrow><\/mtd><\/mtr><mtr><mtd><mn>0<\/mn><\/mtd><mtd><mrow><mi>z<\/mi><mi>\u2212<\/mi><mn>2<\/mn><mo>\u2062<\/mo><mrow><mo>(<\/mo><mrow><mi>x<\/mi><mi>\u2212<\/mi><mn>4<\/mn><\/mrow><mo>)<\/mo><\/mrow><mi>\u2212<\/mi><mn>2<\/mn><\/mrow><\/mtd><\/mtr><\/mtable><\/mrow><mo>)<\/mo><\/mrow><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Las ecuaciones que buscamos se corresponde con las ecuaciones que hacen cero las dos lineas por debajo del menor escogido:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i8)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"input\"><span class=\"code_function\">rat<\/span>(<span class=\"code_variable\">A<\/span>[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">2<\/span>])<span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">rat<\/span>(<span class=\"code_variable\">A<\/span>[<span class=\"code_number\">3<\/span>,<span class=\"code_number\">2<\/span>])<span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"center\"><mtd><mtext>(%o7)\/R\/ <\/mtext><\/mtd><mtd><mi>y<\/mi><mi>\u2212<\/mi><mn>5<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mo>+<\/mo><mn>19<\/mn><mi>=<\/mi><mn>0<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"center\"><mtd><mtext>(%o8)\/R\/ <\/mtext><\/mtd><mtd><mi>z<\/mi><mi>\u2212<\/mi><mn>2<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mo>+<\/mo><mn>6<\/mn><mi>=<\/mi><mn>0<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong> Determinar la ecuaci\u00f3n impl\u00edcita del plano que pasa por el punto P(4,1,2) y tiene por vectores directores \\(\\vec{v}(1,5,3)\\) y \\(\\vec{u}(3,-2,1)\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4() {\n  var htmlShow4 = document.getElementById(\"html-show4\");\n  if (htmlShow4.style.display === \"none\") {\n    htmlShow4.style.display = \"block\";\n  } else {\n    htmlShow4.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGrect_afin05.html\" width=\"650\" height=\"210\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea S(\\(x\\),4,-3), \u00bfexiste alg\u00fan valor de \\(x\\) que hace a S coplanario con P(0,-2,1), Q(1,-2,-3) y R(1,-3,1)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv4a() {\n  var htmlShow4a = document.getElementById(\"html-show4a\");\n  if (htmlShow4a.style.display === \"none\") {\n    htmlShow4a.style.display = \"block\";\n  } else {\n    htmlShow4a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4a\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i9)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">S<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_variable\">x<\/span>,<span class=\"code_number\">4<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>]<span class=\"code_endofline\">$<\/span><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">0<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">$<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>]<span class=\"code_endofline\">$<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">3<\/span>,<span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">pq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">pr<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">ps<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">S<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span>(<span class=\"code_function\">determinant<\/span>(<span class=\"code_function\">matrix<\/span>(<span class=\"code_variable\">pq<\/span>,<span class=\"code_variable\">pr<\/span>,<span class=\"code_variable\">ps<\/span>)))<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_function\">solve<\/span>(<span class=\"code_variable\">eq<\/span>,<span class=\"code_variable\">x<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(eq)<\/mtext><\/mtd><mtd><mi>\u2212<\/mi><mn>4<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mi>\u2212<\/mi><mn>20<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o9) <\/mtext><\/mtd><mtd><mo>[<\/mo><mi>x<\/mi><mi>=<\/mi><mi>\u2212<\/mi><mn>5<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<h2>El espacio eucl\u00eddeo<\/h2>\n<p>En este tema nos proponemos proveer de una m\u00e9trica a los espacios afines de \\(\\mathbb{R}^2\\) y \\(\\mathbb{R}^3\\). Esta m\u00e9trica nos permitir\u00e1 definir distancias, el \u00e1ngulo entre dos vectores y el concepto de perpendicularidad.<\/p>\n<blockquote><p>Definimos el producto escalar de dos vectores \\(\\mathbf{v},\\mathbf{u}\\in\\mathbb{R}^n\\), siendo \\(\\mathbf{v}=[v_1,\\ldots,v_n]\\) y \\(\\mathbf{u}=[u_1,\\ldots,u_n]\\) como \\[\\mathbf{v}\\bullet \\mathbf{u}=\\sum{i=1}^nv_iu_i\\in\\mathbb{R}\\]\n<\/p><\/blockquote>\n<blockquote><p><strong>Propiedades:<\/strong> El producto escalar verifica que para todo vector, \\(\\mathbf{v},\\mathbf{u},\\mathbf{w}\\in\\mathbb{R}^n\\) es <\/p>\n<ul>\n<li>\\(\\mathbf{v}\\bullet \\mathbf{u}=\\mathbf{a}\\bullet \\mathbf{v}\\)<\/li>\n<li>\\(\\mathbf{v}\\bullet (\\lambda\\mathbf{u})=\\lambda(\\mathbf{v}\\bullet \\mathbf{u}) \\forall \\lambda\\in\\mathbb{R}\\)<\/li>\n<li>\\(\\mathbf{v}\\bullet (\\lambda\\mathbf{u}+\\mu\\mathbf{w})=\\lambda\\mathbf{v}\\bullet\\mathbf{u}+\\mu\\mathbf{v}\\bullet\\mathbf{w} \\forall \\lambda,\\mu\\in\\mathbb{R}\\)<\/li>\n<\/ul>\n<\/blockquote>\n<p>El producto escalar nos da pie a definir la norma de un vector como la ra\u00edz cuadrada del producto escalar de un vector por si mismo: \\[||\\vec{v}||=\\sqrt{\\vec{v}\\bullet\\vec{v}}\\]<\/p>\n<p>En el caso de \\(\\mathbb{R}^n\\): \\[||(v_1,v_2,\\ldots,v_n)||=\\sqrt{v_1^2 +v_2^2+\\ldots + v_n^2}\\]<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Determina un vector en la direcci\u00f3n del subespacio \\(U=\\mbox{Gen}\\{(2,0,1)\\}\\) de longitud 6.<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv52() {\n  var htmlShow52 = document.getElementById(\"html-show52\");\n  if (htmlShow52.style.display === \"none\") {\n    htmlShow52.style.display = \"block\";\n  } else {\n    htmlShow52.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv52()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show52\" style=\"display: none;\">\n<!-- Text cell --> <\/p>\n<div class=\"comment\">Que un vector, \\(\\vec{v}\\), est\u00e9 en la direcci\u00f3n del subespacio significa que \\(\\vec{v}\\in\\mbox{Gen}\\{(2,0,1)\\}\\); es decir, existe un n\u00famero real, \\(\\lambda\\), tal que \\(\\vec{v}=\\lambda(2,0,1)\\). Que su longitud sea 6, nos dice que \\(\\parallel\\vec{v}\\parallel=\\sqrt{v_1^2+v_2^2+v_3^2}=6\\). Luego \\<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i2)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">\u03bb<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\sqrt{5} \\left| \\lambda \\right| \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Y como tiene que ser 6:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">solve<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">\u03bb<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ \\left| \\lambda \\right| =\\frac{6}{\\sqrt{5}}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">El vector que buscamos es<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_operator\">(<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">5<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ \\frac{12}{\\sqrt{5}}{,}0{,}\\frac{6}{\\sqrt{5}}\\right] \\]<\/p>\n<\/div>\n<hr \/>\n<p>Adem\u00e1s, definimos el producto vectorial de dos vectores no nulos de \\(\\mathbb{R}^3\\), estudiando propiedades que m\u00e1s tarde utilizaremos. Por \u00faltimo, hemos definido el producto mixto de tres vectores de \\(\\mathbb{R}^3\\).<\/p>\n<p>Adem\u00e1s, hemos aprendido a expresar de una nueva forma un plano af\u00edn en \\(\\mathbb{R}^3\\). Si \\(\\pi:\\{P+\\lambda\\vec{v}+\\mu\\vec{u}|P\\in\\mathbb{R}^3, \\vec{v},\\vec{u}\\in\\mathbb{R}^3,\\lambda,\\mu\\in\\mathbb{R}\\}\\), llamamos forma general a \\[(x-p_1,y-p_2,z-p_3)\\cdot(\\vec{v}\\times\\vec{u})=0.\\]<\/p>\n<p>El s\u00edmbolo \\(\\times\\) hace referencia al producto vectorial, que calculamos mediante:<br \/>\n\\[\\vec{v}\\times\\vec{u}=\\begin{vmatrix}\\vec{i} &#038; \\vec{j} &#038;\\vec{k}\\\\ v_1 &#038; v_2 &#038; v_3 \\\\ u_1 &#038; u_2 &#038; u_3\\end{vmatrix}\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> Sean \\(\\vec{v}=(1,-2,3)\\) y \\(\\vec{u}=(2,1,-1)\\), calcular \\(\\vec{v}\\times\\vec{u}\\).<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1rt() {\n  var htmlShow1rt = document.getElementById(\"html-show1rt\");\n  if (htmlShow1rt.style.display === \"none\") {\n    htmlShow1rt.style.display = \"block\";\n  } else {\n    htmlShow1rt.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1rt()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1rt\" style=\"display: none;\">\n <!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[5 k+7 j-i\\]<\/p>\n<\/div>\n<hr \/>\n<p>Este resultado junto con el anterior nos permite deducir la forma general del plano \\((x-p_1,y-p_2,z-p_3)\\cdot(\\vec{v}\\times\\vec{u})=0,\\) como<br \/>\n\\[\\begin{vmatrix}x-p_1 &#038; y-p_2 &#038;z-p_3\\\\ v_1 &#038; v_2 &#038; v_3 \\\\ u_1 &#038; u_2 &#038; u_3\\end{vmatrix}=0\\]<\/p>\n<p>El producto vectorial solo hace referencia a vectores en el espacio vectorial \\(\\mathbb{R}^3\\), y cumple:<\/p>\n<ul>\n<li>\\( (\\vec{v} \\times \\vec{u} )\\times \\vec {w} \\neq \\vec {v} \\times (\\vec {u} \\times \\vec {w} )\\); el producto vectorial no es asociativo<\/li>\n<li>\\({\\vec{v}}\\times {\\vec{u}}=-({\\vec{u}}\\times {\\vec{v}})\\); anticonmutatividad<\/li>\n<li>\\( \\vec{v}\\bullet (\\vec{v}\\times \\vec{u})=0\\); cancelaci\u00f3n por ortogonalidad.<\/li>\n<li>Si \\({\\vec{v}}\\times {\\vec{u}}={\\vec{0}}\\) con \\(\\vec{v}\\neq \\vec{0}\\) y \\( \\vec{u}\\neq \\vec{0}\\), \\(\\Rightarrow \\vec{v}\\|\\vec{u}\\); la anulaci\u00f3n del producto vectorial proporciona la condici\u00f3n de paralelismo entre dos direcciones.<\/li>\n<li>\\(({\\vec{v}}+{\\vec{u}})\\times {\\vec{w}}={\\vec{v}}\\times {\\vec{w}}+{\\vec{u}}\\times {\\vec{w}}\\); distributividad por derecha e izquierda<\/li>\n<li>\\({\\vec{v}}\\times ({\\vec{u}}\\times {\\vec{w}})={\\vec{u}}({\\vec{v}}\\bullet {\\vec{w}})-{\\vec{w}}({\\vec{v}}\\bullet {\\vec{u}})\\); conocida como regla de la expulsi\u00f3n.<\/li>\n<li>\\( \\left\\|\\vec{v}\\times \\vec{u}\\right\\|=\\left\\|\\vec{v}\\right\\|\\left\\|\\vec{u}\\right\\|\\left|\\sin \\theta \\right|\\). En la expresi\u00f3n del t\u00e9rmino de la derecha, ser\u00eda el m\u00f3dulo de los vectores \\( \\vec{v}\\) y \\( \\vec{u}\\), siendo \\(\\theta\\) el \u00e1ngulo menor entre los vectores \\( \\vec{v}\\) y \\( \\vec{u}\\); esta expresi\u00f3n relaciona al producto vectorial con el \u00e1rea del paralelogramo que definen ambos vectores.<\/li>\n<li>\\(\\lambda (\\vec{v}\\times \\vec{u})=(\\lambda \\vec{v})\\times \\vec{u}=\\vec{v}\\times (\\lambda \\vec{u})\\); el producto vectorial es bihomog\u00e9neo<\/li>\n<li>El m\u00f3dulo o norma del producto vectorial puede calcularse f\u00e1cilmente sin hacer el producto vectorial: \\(\\|\\vec{v}\\times \\vec{u}\\|=\\left(\\|\\vec{v}\\|^{2}\\|\\vec{u}\\|^{2}-(\\vec{v}\\bullet \\vec{u})^{2}\\right)^{1\/2}\\)<\/li>\n<li>El vector unitario \\( {\\hat {\\mathbf {n} }}=\\frac{\\vec{v}\\times \\vec{u}}{\\|\\vec{v}\\times \\vec{u}\\|}\\) es normal al plano que contiene a los vectores \\( \\vec{v}\\) y \\( \\vec{u}\\).<\/li>\n<\/ul>\n<\/ul>\n<blockquote><p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1l es la norma del vector perpendicular al subespacio generado por \\(\\vec{v}:(1,-1,5)\\) y \\(\\vec{u}:(2,3,-1)\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv1g2() {\n  var htmlShow1g2 = document.getElementById(\"html-show1g2\");\n  if (htmlShow1g2.style.display === \"none\") {\n    htmlShow1g2.style.display = \"block\";\n  } else {\n    htmlShow1g2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1g2()\">Soluci\u00f3n:<\/button><\/p>\n<p><div id=\"html-show1g2\" style=\"display: none;\">\nLo que buscamos es \\(\\|\\vec{v}\\times\\vec{u}\\|\\).<br \/>\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">fpprintprec<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">5<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">3<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">vu<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span>(<span class=\"code_function\">determinant<\/span>(<span class=\"code_function\">matrix<\/span>([<span class=\"code_variable\">i<\/span>,<span class=\"code_variable\">j<\/span>,<span class=\"code_variable\">k<\/span>],<span class=\"code_variable\">v<\/span>,<span class=\"code_variable\">u<\/span>)),<span class=\"code_variable\">k<\/span>,<span class=\"code_variable\">j<\/span>,<span class=\"code_variable\">i<\/span>)<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_variable\">vu<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_function\">coeff<\/span>(<span class=\"code_variable\">vu<\/span>,<span class=\"code_variable\">i<\/span>),<span class=\"code_function\">coeff<\/span>(<span class=\"code_variable\">vu<\/span>,<span class=\"code_variable\">j<\/span>),<span class=\"code_function\">coeff<\/span>(<span class=\"code_variable\">vu<\/span>,<span class=\"code_variable\">k<\/span>)]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00abLa norma de \u00ab<\/span>,<span class=\"code_variable\">vu<\/span>,<span class=\"code_string\">\u00abes\u00bb<\/span>,<span class=\"code_function\">float<\/span>(<span class=\"code_function\">sqrt<\/span>(<span class=\"code_variable\">vu<\/span>.<span class=\"code_variable\">vu<\/span>)))<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(vu)<\/mtext><\/mtd><mtd><mi>\u2212<\/mi><mn>14<\/mn><mo>\u2062<\/mo><mi>i<\/mi><mo>+<\/mo><mn>11<\/mn><mo>\u2062<\/mo><mi>j<\/mi><mo>+<\/mo><mn>5<\/mn><mo>\u2062<\/mo><mi>k<\/mi><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>La norma de <\/mo><mo\/><mo>[<\/mo><mi>\u2212<\/mi><mn>14<\/mn><mo>,<\/mo><mn>11<\/mn><mo>,<\/mo><mn>5<\/mn><mo>]<\/mo><mo\/><mo>es<\/mo><mo\/><mn>18.4932<\/mn><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> \u00bfCu\u00e1l es, en valor absoluto, el producto escalar del vector \\([1,2,4]\\) por el vector normal unitario de la recta definida por las ecuaciones \\(\\pi_1:-6z+9y+x-1=0\\) y \\(\\pi_2:15z-18y-4x-5=0\\)?  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv81() {\n  var htmlShow81 = document.getElementById(\"html-show81\");\n  if (htmlShow81.style.display === \"none\") {\n    htmlShow81.style.display = \"block\";\n  } else {\n    htmlShow81.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv81()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show81\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">Vector normal de la recta definida por las ecuaciones \\(\\pi_1:-6z+9y+x-1=0\\) y \\(\\pi_2:15z-18y-4x-5=0\\) vendr\u00e1 dado por el producto vectorial de los vectores normales de cada plano:<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i3)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">9<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">6<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">18<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">15<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[18 k+9 j+27 i\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Luego el vector normal es [27,9,18]. Ahora lo normalizamos (hacerlo unitario):<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">n<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">27<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">9<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">18<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">(<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\u00b7<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ \\frac{3}{\\sqrt{14}}{,}\\frac{1}{\\sqrt{14}}{,}\\frac{2}{\\sqrt{14}}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Por \u00faltimo,<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">4<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\frac{13}{\\sqrt{14}}\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[3.474396144861517\\]<\/p>\n<\/div>\n<hr \/>\n<p>Utilizando el producto escalar, podemos definir el coseno de dos vectores: \\(\\vec{v},\\vec{u}\\in\\mathbb{R}^n\\), donde \\(n\\in\\{2,3\\}\\), como \\[\\textbf{cos}(\\vec{v},\\vec{u})=\\frac{\\vec{v}\\bullet\\vec{u}}{||\\vec{v}||\\cdot ||\\vec{u}||}\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong>  Sean \\(\\vec{v}=(1,-2,3)\\) y  \\(\\vec{u}=(2,1,-1)\\), \u00bfcu\u00e1l es la segunda cifra decimal del \\(\\cos(\\vec{v},\\vec{u})\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2g4() {\n  var htmlShow2g4 = document.getElementById(\"html-show2g4\");\n  if (htmlShow2g4.style.display === \"none\") {\n    htmlShow2g4.style.display = \"block\";\n  } else {\n    htmlShow2g4.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2g4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2g4\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">fpprintprec<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">3<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">cos<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">v<\/span>.<span class=\"code_variable\">u<\/span><span class=\"code_operator\">\/<\/span>(<span class=\"code_function\">sqrt<\/span>(<span class=\"code_variable\">v<\/span>.<span class=\"code_variable\">v<\/span>)<span class=\"code_operator\">*<\/span><span class=\"code_function\">sqrt<\/span>(<span class=\"code_variable\">u<\/span>.<span class=\"code_variable\">u<\/span>))<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00abEl coseno es \u00ab<\/span>,<span class=\"code_function\">float<\/span>(<span class=\"code_variable\">cos<\/span>))<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>El coseno es <\/mo><mo\/><mi>\u2212<\/mi><mn>0.327327<\/mn><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<p>Otro vector que podemos definir es la proyecci\u00f3n de un vector sobre otro, como \\[\\textbf{proy}_\\vec{v}(\\vec{u})=\\frac{\\vec{v}\\bullet\\vec{u}}{||\\vec{v}||^2}\\vec{v}\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1l es el producto escalar del vector [1,-1,1] por la proyecci\u00f3n de \\(\\vec{u}=(2,1,-1)\\) sobre \\(\\vec{v}=(1,-2,3)\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2a5() {\n  var htmlShow2a5 = document.getElementById(\"html-show2a5\");\n  if (htmlShow2a5.style.display === \"none\") {\n    htmlShow2a5.style.display = \"block\";\n  } else {\n    htmlShow2a5.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2a5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2a5\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">fpprintprec<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">3<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">proy<\/span><span class=\"code_operator\">:<\/span>(<span class=\"code_variable\">v<\/span>.<span class=\"code_variable\">u<\/span><span class=\"code_operator\">\/<\/span>(<span class=\"code_variable\">v<\/span>.<span class=\"code_variable\">v<\/span>))<span class=\"code_operator\">*<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">$<\/span><br \/>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>].<span class=\"code_variable\">proy<\/span>,<span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o5) <\/mtext><\/mtd><mtd><mi>\u2212<\/mi><mn>1.28571<\/mn><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<p>La componente de \\(\\mathbf{u}\\) en la direcci\u00f3n de \\(\\mathbf{v}\\), vendr\u00e1 dada por \\[\\textbf{comp}_\\mathbf{v}(\\mathbf{u})=\\frac{\\mathbf{v}\\bullet\\mathbf{u}}{||\\mathbf{v}||}\\]<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Determinar la componente del vector \\([3,2,1]\\) en la direcci\u00f3n del vector \\([2,-1,0]\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv31c() {\n  var htmlShow31c = document.getElementById(\"html-show31c\");\n  if (htmlShow31c.style.display === \"none\") {\n    htmlShow31c.style.display = \"block\";\n  } else {\n    htmlShow31c.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv31c()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show31c\" style=\"display: none;\">\n<strong>Soluci\u00f3n<\/strong>:<br \/>\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">u<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\frac{4}{\\sqrt{5}}\\]<\/p>\n<p>\\[1.788854381999831\\]<\/p>\n<\/div>\n<hr \/>\n<p>Con la norma podemos definir la distancia entre dos puntos \\(P\\) y \\(Q\\in \\mathbb{R}^3\\) como: \\[d(P,Q)=||\\vec{QP}||=\\sqrt{(q_1-p_1)^2 +(q_2-p_2)^2 + (q_3-p_3)^2}\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1l es la distancia entre los puntos \\(P(1,-1,2)\\) y \\(Q(0,2,-2)\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2b() {\n  var htmlShow2b = document.getElementById(\"html-show2b\");\n  if (htmlShow2b.style.display === \"none\") {\n    htmlShow2b.style.display = \"block\";\n  } else {\n    htmlShow2b.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2b()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2b\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">Recordemos que \\(d(P,Q)=||\\vec{QP}||\\). Luego<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">PQ<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">PQ<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">PQ<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\sqrt{26}\\]<\/p>\n<p>\\[5.099019513592784\\]<\/p>\n<\/div>\n<hr \/>\n<p>Del mismo modo definimos la distancia de una recta \\(r=\\{P+\\mathbf{Gen}\\{\\vec{v}\\}\\}\\in \\mathbb{R}^3\\) a un punto \\(Q\\) como:\\[d(Q,r)=\\frac{||\\vec{PQ}\\times\\vec{v}||}{||\\vec{v}||}\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1l es la distancia del punto \\(Q(0,2,-2)\\) a la recta \\(r:\\{(1,-1,2)+\\mathbf{Gen}\\{(3,2,-1)\\}\\}\\in \\mathbb{R}^3\\) ?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2c() {\n  var htmlShow2c = document.getElementById(\"html-show2c\");\n  if (htmlShow2c.style.display === \"none\") {\n    htmlShow2c.style.display = \"block\";\n  } else {\n    htmlShow2c.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2c()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2c\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">PQ<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">i<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">j<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">k<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">PQ<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[-11 k-13 j+5 i\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i8)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">PQxv<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">13<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">11<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">PQxv<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">PQxv<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\frac{3 \\sqrt{35}}{\\sqrt{14}}\\]<\/p>\n<p>\\[4.743416490252569\\]<\/p>\n<\/div>\n<hr \/>\n<p>Sin embargo, si queremos calcular la distancia entre un punto \\(P\\) y el plano \\(\\pi:ax+by+cz+d=0\\), que no lo contiene, lo haremos mediante: \\[d(P,\\pi)=\\frac{|ap_1+bp_2+cp_3+d|}{\\sqrt{a^2+b^2+c^2}}\\]<\/p>\n<blockquote><p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1l es la distancia del punto \\(Q(0,2,-2)\\) al plano \\(\\pi:\\{(1,-1,2)+\\mathbf{Gen}\\{(3,2,-1),(0,1,1)\\}\\}\\in \\mathbb{R}^3\\) ?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2d() {\n  var htmlShow2d = document.getElementById(\"html-show2d\");\n  if (htmlShow2d.style.display === \"none\") {\n    htmlShow2d.style.display = \"block\";\n  } else {\n    htmlShow2d.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2d\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">PQ<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[3 \\left( z-2\\right) -3 \\left( y+1\\right) +3 \\left( x-1\\right) \\]<\/p>\n<p>\\[3 z-3 y+3 x-12\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">n<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">abs<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">12<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\frac{8}{\\sqrt{3}}\\]<\/p>\n<p>\\[4.618802153517007\\]<\/p>\n<\/div>\n<hr \/>\n<h2>Ecuaci\u00f3n normalizada<\/h2>\n<p>Para algunos ejercicios utilizaremos la siguiente definici\u00f3n:<\/p>\n<blockquote><p><strong>Definici\u00f3n:<\/strong> Sea \\(ax+by+cz+d=0\\) una ecuaci\u00f3n de un plano en el espacio af\u00edn. Llamaremos <strong>ecuaci\u00f3n normalizada<\/strong> a la ecuaci\u00f3n resultado de<br \/>\n\\[\\frac{1}{\\sqrt{a^2+b^2+c^2+d^2}}(ax+by+cz+d=0)\\]<\/p><\/blockquote>\n<blockquote><p><strong>Ejemplo:<\/strong> Sea \\(ax+by+cz+d=0\\) la ecuaci\u00f3n del plano af\u00edn, normalizada, que pasa por el punto P(0,1,-3) y tiene por vectores directores \\(\\textbf{v}=[2,-1,1]\\) y \\(\\textbf{u}=[1, 2, 3]\\)  . \u00bfCu\u00e1l es el resultado, en valor absoluto, de evaluar \\([x,y,z]\\) por [1,1,-1]?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2dx4d() {\n  var htmlShow2dx4d = document.getElementById(\"html-show2dx4d\");\n  if (htmlShow2dx4d.style.display === \"none\") {\n    htmlShow2dx4d.style.display = \"block\";\n  } else {\n    htmlShow2dx4d.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2dx4d()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2dx4d\" style=\"display: none;\">\nRecordad que para calcular la ecuaci\u00f3n impl\u00edcita del plano utilizamos el determinante \\[\\begin{vmatrix} [x,y,z]-P\\\\ v\\\\ u\\end{vmatrix}=0\\]<br \/>\nEs importante considerarlo en esta disposici\u00f3n; en otro caso, al sustituir, puede dar un resultado diferente.<br \/>\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">0<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">v<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">P<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">v<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">u<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">rat<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">0<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\begin{bmatrix}x &amp; y-1 &amp; z+3\\\\2 &amp; -1 &amp; 1\\\\1 &amp; 2 &amp; 3\\end{bmatrix}\\]<\/p>\n<p>\\[5 z-5 y-5 x+20=0\\]<\/p>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i7)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">n<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">20<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">:<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_variable\">n<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\left[ -\\frac{1}{\\sqrt{19}}\\operatorname{,}-\\frac{1}{\\sqrt{19}}\\operatorname{,}\\frac{1}{\\sqrt{19}}\\operatorname{,}\\frac{4}{\\sqrt{19}}\\right] \\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">la ecuaci\u00f3n normalizada ser\u00eda<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i8)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">n<\/span><span class=\"code_endofline\">.<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\frac{z}{\\sqrt{19}}-\\frac{y}{\\sqrt{19}}-\\frac{x}{\\sqrt{19}}+\\frac{4}{\\sqrt{19}}\\]<\/p>\n<p><!-- Text cell --><\/p>\n<div class=\"comment\">Nuestra pregunta era: \u00bfCu\u00e1l es el resultado, en valor absoluto, de evaluar [x,y,z]<br \/> por [1,1,-1]?<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i10) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_function\">ev<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">[<\/span><span class=\"code_variable\">x<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">y<\/span><span class=\"code_operator\">=<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">z<\/span><span class=\"code_operator\">=<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">%<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">numer<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[\\frac{1}{\\sqrt{19}}\\]<\/p>\n<p>\\[0.2294157338705617\\]<\/p>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong>  Cu\u00e1l es el producto escalar del vector [1,2,3] por el vector normal unitario del plano que pasa por los puntos \\(P(4,-2,3)\\), \\(Q(2,-1,1)\\) y \\(R(0,2,5)\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2b8() {\n  var htmlShow2b8 = document.getElementById(\"html-show2b8\");\n  if (htmlShow2b8.style.display === \"none\") {\n    htmlShow2b8.style.display = \"block\";\n  } else {\n    htmlShow2b8.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2b8()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2b8\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i9)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"input\"><span class=\"code_variable\">P<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">4<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">2<\/span>,<span class=\"code_number\">3<\/span>]<span class=\"code_endofline\">$<\/span><span class=\"code_variable\">Q<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">2<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>]<span class=\"code_endofline\">$<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">0<\/span>,<span class=\"code_number\">2<\/span>,<span class=\"code_number\">5<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">rat<\/span>(<span class=\"code_function\">determinant<\/span>(<span class=\"code_function\">matrix<\/span>([<span class=\"code_variable\">x<\/span>,<span class=\"code_variable\">y<\/span>,<span class=\"code_variable\">z<\/span>]<span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span>,<span class=\"code_variable\">Q<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span>,<span class=\"code_variable\">R<\/span><span class=\"code_operator\">&#8211;<\/span><span class=\"code_variable\">P<\/span>)))<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00abEcuaci\u00f3n del plano 0=\u00bb<\/span>,<span class=\"code_variable\">eq<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">n<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_function\">coeff<\/span>(<span class=\"code_variable\">eq<\/span>,<span class=\"code_variable\">x<\/span>),<span class=\"code_function\">coeff<\/span>(<span class=\"code_variable\">eq<\/span>,<span class=\"code_variable\">y<\/span>),<span class=\"code_function\">coeff<\/span>(<span class=\"code_variable\">eq<\/span>,<span class=\"code_variable\">x<\/span>)]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00abVector normal del plano: \u00ab<\/span>,<span class=\"code_variable\">n<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">print<\/span>(<span class=\"code_string\">\u00abProducto escalar\u00bb<\/span>)<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_function\">print<\/span>([<span class=\"code_number\">1<\/span>,<span class=\"code_number\">2<\/span>,<span class=\"code_number\">3<\/span>],<span class=\"code_string\">\u00ab.\u00bb<\/span>,<span class=\"code_variable\">n<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span>(<span class=\"code_variable\">n<\/span>.<span class=\"code_variable\">n<\/span>),<span class=\"code_string\">\u00ab=\u00bb<\/span>,[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">2<\/span>,<span class=\"code_number\">3<\/span>].<span class=\"code_variable\">n<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span>(<span class=\"code_variable\">n<\/span>.<span class=\"code_variable\">n<\/span>))<span class=\"code_endofline\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>Ecuaci\u00f3n del plano 0=<\/mo><mo\/><mi>\u2212<\/mi><mn>4<\/mn><mo>\u2062<\/mo><mi>z<\/mi><mo>+<\/mo><mn>12<\/mn><mo>\u2062<\/mo><mi>y<\/mi><mo>+<\/mo><mn>10<\/mn><mo>\u2062<\/mo><mi>x<\/mi><mi>\u2212<\/mi><mn>4<\/mn><mo\/><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>Vector normal del plano: <\/mo><mo\/><mo>[<\/mo><mn>10<\/mn><mo>,<\/mo><mn>12<\/mn><mo>,<\/mo><mn>-4<\/mn><mo>]<\/mo><mo\/><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>Producto escalar<\/mo><mo\/><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>[<\/mo><mn>1<\/mn><mo>,<\/mo><mn>2<\/mn><mo>,<\/mo><mn>3<\/mn><mo>]<\/mo><mo\/><mo>.<\/mo><mo\/><mo>[<\/mo><mfrac><mn>5<\/mn><msqrt><mn>65<\/mn><\/msqrt><\/mfrac><mo>,<\/mo><mfrac><mn>6<\/mn><msqrt><mn>65<\/mn><\/msqrt><\/mfrac><mo>,<\/mo><mfrac><mn>-2<\/mn><msqrt><mn>65<\/mn><\/msqrt><\/mfrac><mo>]<\/mo><mo\/><mo>=<\/mo><mo\/><mfrac><mn>11<\/mn><msqrt><mn>65<\/mn><\/msqrt><\/mfrac><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> \u00bfCu\u00e1l es, en valor absoluto, el producto escalar del vector \\([1,-1,1]\\) por el vector normal unitario de la recta definida por las ecuaciones \\(\\pi_1:-6z+9y+x-1=0\\) y \\(\\pi_2:15z-18y-4x-5=0\\)?  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv81s() {\n  var htmlShow81s = document.getElementById(\"html-show81s\");\n  if (htmlShow81s.style.display === \"none\") {\n    htmlShow81s.style.display = \"block\";\n  } else {\n    htmlShow81s.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv81s()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show81s\" style=\"display: none;\">\n<!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>El vector normal de la recta definida por las ecuaciones \\(\\pi_1:-6z+9y+x-1=0\\) y \\(\\pi_2:15z-18y-4x-5=0\\) vendr\u00e1 dado por el producto vectorial de los vectores normales de cada plano:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i4) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"input\"><span class=\"code_variable\">p1<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">1<\/span>,<span class=\"code_number\">9<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">6<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">p2<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">4<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">18<\/span>,<span class=\"code_number\">15<\/span>]<span class=\"code_endofline\">$<\/span><br \/><span class=\"code_variable\">eq<\/span><span class=\"code_operator\">:<\/span><span class=\"code_function\">determinant<\/span>(<span class=\"code_function\">matrix<\/span>([<span class=\"code_variable\">i<\/span>,<span class=\"code_variable\">j<\/span>,<span class=\"code_variable\">k<\/span>],<span class=\"code_variable\">p1<\/span>,<span class=\"code_variable\">p2<\/span>))<span class=\"code_endofline\">;<\/span><br \/><span class=\"code_variable\">vn<\/span><span class=\"code_operator\">:<\/span>[<span class=\"code_function\">coeff<\/span>(<span class=\"code_variable\">eq<\/span>,<span class=\"code_variable\">i<\/span>),<span class=\"code_function\">coeff<\/span>(<span class=\"code_variable\">eq<\/span>,<span class=\"code_variable\">j<\/span>),<span class=\"code_function\">coeff<\/span>(<span class=\"code_variable\">eq<\/span>,<span class=\"code_variable\">k<\/span>)]<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(eq)<\/mtext><\/mtd><mtd><mn>18<\/mn><mo>\u2062<\/mo><mi>k<\/mi><mo>+<\/mo><mn>9<\/mn><mo>\u2062<\/mo><mi>j<\/mi><mo>+<\/mo><mn>27<\/mn><mo>\u2062<\/mo><mi>i<\/mi><\/mtd><\/mlabeledtr><\/mtable><\/math><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(vn)<\/mtext><\/mtd><mtd><mo>[<\/mo><mn>27<\/mn><mo>,<\/mo><mn>9<\/mn><mo>,<\/mo><mn>18<\/mn><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Ahora normalizamos el vector:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"input\">(<span class=\"code_number\">1<\/span><span class=\"code_operator\">\/<\/span><span class=\"code_function\">sqrt<\/span>(<span class=\"code_variable\">vn<\/span>.<span class=\"code_variable\">vn<\/span>)).<span class=\"code_variable\">vn<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o5) <\/mtext><\/mtd><mtd><mo>[<\/mo><mfrac><mn>3<\/mn><msqrt><mn>14<\/mn><\/msqrt><\/mfrac><mo>,<\/mo><mfrac><mn>1<\/mn><msqrt><mn>14<\/mn><\/msqrt><\/mfrac><mo>,<\/mo><mfrac><mn>2<\/mn><msqrt><mn>14<\/mn><\/msqrt><\/mfrac><mo>]<\/mo><\/mtd><\/mlabeledtr><\/mtable><\/math><!-- Text cell --><\/p>\n<div class=\"comment\">\n<p>Por \u00faltimo, multiplicamos seg\u00fan el enunciado:<\/p>\n<\/div>\n<p><!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i6) <\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"> <span class=\"input\"><span class=\"input\"><span class=\"code_function\">abs<\/span>([<span class=\"code_number\">1<\/span>,<span class=\"code_number\">&#8211;<\/span><span class=\"code_number\">1<\/span>,<span class=\"code_number\">1<\/span>].<span class=\"code_variable\">%<\/span>)<span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(%o6) <\/mtext><\/mtd><mtd><mfrac><mn>4<\/mn><msqrt><mn>14<\/mn><\/msqrt><\/mfrac><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<h2>Sistemas de ecuaciones <\/h2>\n<p>Hoy abordamos la soluci\u00f3n de sistemas, que es el paso de las ecuaciones impl\u00edcitas a ecuaciones param\u00e9tricas. Ve\u00e1moslo con los siguientes ejemplos.<\/p>\n<ul>\n<li><strong>linsolve<\/strong>(\\([eq_1, &#8230;, eq_m], [x_1, &#8230;, x_n]\\)): Solves the list of simultaneous linear equations for the list of variables. The expressions must each be polynomials in the variables and may be equations. <\/li>\n<\/ul>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Resolver el sistema de ecuaciones \\[\\begin{matrix}2x+y-z=1 \\\\x-3y+2z=1 \\\\ -x+2y-4z=2\\end{matrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv16y() {\n  var htmlShow16y = document.getElementById(\"html-show16y\");\n  if (htmlShow16y.style.display === \"none\") {\n    htmlShow16y.style.display = \"block\";\n  } else {\n    htmlShow16y.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv16y()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show16y\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_sist01.html\" width=\"650\" height=\"150\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<p>El sistema puede tener infinitas soluciones, en cuyo caso estas se dan en forma param\u00e9trica:<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Cu\u00e1nto suman, en valor absoluto, las coordenadas del vector director unitario de la recta af\u00edn definida por las ecuaciones impl\u00edcitas\\[\\begin{matrix}3x-y-z=2 \\\\ x+y-2z=1\\end{matrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2a45() {\n  var htmlShow2a45 = document.getElementById(\"html-show2a45\");\n  if (htmlShow2a45.style.display === \"none\") {\n    htmlShow2a45.style.display = \"block\";\n  } else {\n    htmlShow2a45.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv2a45()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2a45\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGsistR3_01.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea  \\(\\pi:2x-3y+z=5\\in\\mathbb{R}^3\\) y \\((a,3,1)\\) perteneciente al subespacio director de \\(\\pi\\), \u00bfcu\u00e1l es el valor de \\(a\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3a() {\n  var htmlShow3a = document.getElementById(\"html-show3a\");\n  if (htmlShow3a.style.display === \"none\") {\n    htmlShow3a.style.display = \"block\";\n  } else {\n    htmlShow3a.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv3a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3a\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGsistR3_02.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<p>Si el sistema no tiene soluci\u00f3n <strong>linsolve<\/strong> devuelve una lista vac\u00eda. Veamos m\u00e1s ejemplos de su utilizaci\u00f3n.<\/p>\n<blockquote><p>Ejercicio:<\/strong> Determinar la suma de \\(a+b\\) para que la soluci\u00f3n sistema \\[\\begin{array}{l}8x-2y-z=1 \\\\ 9x-4y+bz=-2 \\\\ 2x+y+az=2\\end{array}\\] sea (1\/3,1,-1\/3)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv8() {\n  var htmlShow8 = document.getElementById(\"html-show8\");\n  if (htmlShow8.style.display === \"none\") {\n    htmlShow8.style.display = \"block\";\n  } else {\n    htmlShow8.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv8()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show8\" style=\"display: none;\">\n<p><iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGsistR3_04.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> \u00bfCu\u00e1les son las ecuaciones param\u00e9tricas de la variedad \\(\\{(x,y,z,t,u)\\in\\mathbb{R}^5;x-2=y+3=z-1=t+2u\\}\\)?<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv18() {\n  var htmlShow18 = document.getElementById(\"html-show18\");\n  if (htmlShow18.style.display === \"none\") {\n    htmlShow18.style.display = \"block\";\n  } else {\n    htmlShow18.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv18()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show18\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGsistema02.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<p>El sistema puede tener infinitas soluciones, en cuyo caso estas se dan en forma param\u00e9trica:<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Resolver el sistema de ecuaciones \\[\\begin{matrix}3x-y-z+t=2 \\\\ x+y-2z-5t=1 \\end{matrix}\\]<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv2() {\n  var htmlShow2 = document.getElementById(\"html-show2\");\n  if (htmlShow2.style.display === \"none\") {\n    htmlShow2.style.display = \"block\";\n  } else {\n    htmlShow2.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/Ejer_sist02.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Sea  \\(\\pi:3x-y-z+t=2\\in\\mathbb{R}^4\\) y \\((a,3,2,1)\\) perteneciente al subespacio director de \\(\\pi\\), \u00bfcu\u00e1l es el valor de \\(a\\)? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv3() {\n  var htmlShow3 = document.getElementById(\"html-show3\");\n  if (htmlShow3.style.display === \"none\") {\n    htmlShow3.style.display = \"block\";\n  } else {\n    htmlShow3.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show3\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGsistema03.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<p>Si el sistema no tiene soluci\u00f3n <strong>linsolve<\/strong> devuelve una lista vac\u00eda.<\/p>\n<blockquote>\n<p><strong>Ejemplo:<\/strong> Determinar el n\u00facleo de la aplicaci\u00f3n \\(f:\\mathbb{R}^3\\to\\mathcal{M}_2(\\mathbb{R})\\), dada por \\[f(a,b,c)=\\begin{bmatrix}-2c+b+a&amp; -c-b\\\\ b+c &amp; c-2b-a\\end{bmatrix}\\]  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv4er() {\n  var htmlShow4er = document.getElementById(\"html-show4er\");\n  if (htmlShow4er.style.display === \"none\") {\n    htmlShow4er.style.display = \"block\";\n  } else {\n    htmlShow4er.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv4er()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show4er\" style=\"display: none;\">\n<iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGnucleo01b.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n<hr \/>\n<h2>\u00c1rea de un paralelogramo<\/h2>\n<p>Veamos una aplicaci\u00f3n del determinante en el caso de un paralelogramo en el plano. Consideremos el paralelogramo dado por los puntos \\(P\\), \\(Q\\) y \\(R\\), del plano af\u00edn:<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/uploads.jesussoto.es\/paralelogramo.svg\" alt=\"Paralelogramo PQR con vectores u y v\"><\/p>\n<p>Observemos que si \\(P(1,3)\\), \\(Q(5,3)\\) y \\(R(0,5)\\), los vectores \\(\\vec{u} = \\overrightarrow{PQ} = (4,0)\\), y<br \/>\n\\(\\vec{v} = \\overrightarrow{PR} = (-1,2)\\). Entonces el \u00e1rea del paralelogramo \\(PQR\\) vendr\u00e1 dada por<br \/>\n\\[<br \/>\n\\begin{vmatrix}<br \/>\n\\vec{u}&#038; \\vec{v}<br \/>\n\\end{vmatrix}=<br \/>\n\\begin{vmatrix}<br \/>\n4&#038; -1 \\\\ 0 &#038; 2<br \/>\n\\end{vmatrix}=8.<br \/>\n\\]<\/p>\n<p>Recordad que, siempre, en valor absoluto.<\/p>\n<p>Ve\u00e1moslo dicho de otra forma:<\/p>\n<blockquote><p>\n<strong>Proposici\u00f3n:<\/strong> El \u00e1rea del paralelogramo dado por los puntos \\(P(p_1,p_2)\\), \\(Q(q_1,q_2)\\) y \\(R(r_1,r_2)\\) viene dada por<br \/>\n\\[<br \/>\n\\mathbf{abs}\\left(\\begin{vmatrix}<br \/>\np_1 &#038; p_2 &#038; 1 \\\\<br \/>\nq_1 &#038; q_2 &#038; 1 \\\\<br \/>\nr_1 &#038; r_2 &#038; 1<br \/>\n\\end{vmatrix}\\right)<br \/>\n\\]\n<\/p><\/blockquote>\n<blockquote><p>\n<strong>Corolario:<\/strong> El \u00e1rea del tri\u00e1ngulo dado por los puntos \\(P(p_1,p_2)\\), \\(Q(q_1,q_2)\\) y \\(R(r_1,r_2)\\) viene dada por<br \/>\n\\[<br \/>\n\\frac{1}{2}\\mathbf{abs}\\left(\\begin{vmatrix}<br \/>\np_1 &#038; p_2 &#038; 1 \\\\<br \/>\nq_1 &#038; q_2 &#038; 1 \\\\<br \/>\nr_1 &#038; r_2 &#038; 1<br \/>\n\\end{vmatrix}\\right)<br \/>\n\\]\n<\/p><\/blockquote>\n<h2>Producto mixto<\/h2>\n<blockquote><p><strong>Definici\u00f3n:<\/strong> El producto mixto de los vectores \\({\\displaystyle {\\vec {u}},{\\vec {v}},{\\vec {w}}} \\), denotado por<br \/>\n\\({\\displaystyle [{\\vec {u}},{\\vec {v}},{\\vec {w}}]}\\), est\u00e1 definido como \\({\\displaystyle [{\\vec {u}},{\\vec {v}},{\\vec {w}}]={\\vec {u}}\\bullet ({\\vec {v}}\\times {\\vec {w}})}\\) <\/p><\/blockquote>\n<blockquote><p><strong>Teorema:<\/strong> \\[{\\displaystyle [{\\vec {u}},{\\vec {v}},{\\vec {w}}]={\\begin{vmatrix}u_{1}&#038;u_{2}&#038;u_{3}\\\\v_{1}&#038;v_{2}&#038;v_{3}\\\\w_{1}&#038;w_{2}&#038;w_{3}\\end{vmatrix}}}\\]<\/p><\/blockquote>\n<blockquote><p><strong>Interpretaci\u00f3n geom\u00e9trica:<\/strong> Si \\({\\displaystyle {\\vec {u}},{\\vec {v}},{\\vec {w}}}\\) son vectores tridimensionales, entonces \\({\\displaystyle |{\\vec {u}}\\bullet ({\\vec {v}}\\times {\\vec {w}})|}\\) es igual al volumen del paralelep\u00edpedo definido por \\({\\displaystyle {\\vec {u}},{\\vec {v}},{\\vec {w}}.}\\)<\/p><\/blockquote>\n<blockquote><p><strong>Ejemplo:<\/strong> Sean los puntos de coordenadas \\(O\\)(-1,1,3), \\(R\\)(2,-1,1), \\(S\\)(5,2,-3) y \\(T\\)(4, -1,2). \u00bfCu\u00e1l es el volumen del paralelep\u00edpedo definido por los vectores \\(\\vec{OR}\\), \\(\\vec{OS}\\) y \\(\\vec{OT}\\)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2dx3() {\n  var htmlShow2dx3 = document.getElementById(\"html-show2dx3\");\n  if (htmlShow2dx3.style.display === \"none\") {\n    htmlShow2dx3.style.display = \"block\";\n  } else {\n    htmlShow2dx3.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2dx3()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2dx3\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">O<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">1<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">S<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">5<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">3<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_variable\">T<\/span><span class=\"code_operator\">:<\/span><span class=\"code_operator\">[<\/span><span class=\"code_number\">4<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_number\">1<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_number\">2<\/span><span class=\"code_operator\">]<\/span><span class=\"code_endofline\">$<\/span><span class=\"code_endofline\"><br \/><\/span><span class=\"code_function\">determinant<\/span><span class=\"code_operator\">(<\/span><span class=\"code_function\">matrix<\/span><span class=\"code_operator\">(<\/span><span class=\"code_variable\">R<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">O<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">S<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">O<\/span><span class=\"code_endofline\">,<\/span><span class=\"code_variable\">T<\/span><span class=\"code_operator\">\u2212<\/span><span class=\"code_variable\">O<\/span><span class=\"code_operator\">)<\/span><span class=\"code_operator\">)<\/span><span class=\"code_endofline\">;<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p>\\[43\\]<\/p>\n<\/div>\n<hr \/>\n<table id=\"yzpi\" border=\"0\" width=\"100%\" cellspacing=\"0\" cellpadding=\"3\" bgcolor=\"#999999\">\n<tbody>\n<tr>\n<td width=\"100%\"><strong>Ejercicio:<\/strong> Sea \\(f:\\mathbb{R}^4\\to\\mathcal{M}_2(\\mathbb{R})\\), dada por \\[f(a,b,c,d)=\\begin{bmatrix}a+b-2d&amp; a-3d\\\\ b+2c-2d &amp; -d+2c-b-a\\end{bmatrix}\\] y \\(\\vec{v}\\in\\mathbf{Ker}\\,f\\), entonces la suma, en valor absoluto, de las coordenadas de dicho vector normalizado es<\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>1.24<\/li>\n<li>2.51<\/li>\n<li>3.64<\/li>\n<\/ul>\n<\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><script>\nfunction showHtmlDiv(id) {\n  var htmlShow = document.getElementById(id);\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script><br \/>\n<p><em>La soluci\u00f3n se mostrar\u00e1 m\u00e1s adelante.<\/em><\/p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Repasemos un poco lo visto: Ejemplo: \u00bfCu\u00e1nto suman los coeficientes que multiplican a \\(x\\) e \\(y\\) en la ecuaci\u00f3n impl\u00edcita de la recta del plano af\u00edn que pasa por los puntos P(4,1)&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[20],"class_list":["post-436","post","type-post","status-publish","format-standard","hentry","category-algebra","tag-practicas-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/436","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=436"}],"version-history":[{"count":12,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/436\/revisions"}],"predecessor-version":[{"id":451,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/436\/revisions\/451"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=436"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=436"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=436"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}