{"id":424,"date":"2025-11-18T08:15:57","date_gmt":"2025-11-18T07:15:57","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=424"},"modified":"2025-12-16T10:51:28","modified_gmt":"2025-12-16T09:51:28","slug":"alg-el-espacio-afin-euclideo-mathbbr3-y-sistemas-de-ecuaciones-con-maxima","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=424","title":{"rendered":"ALG: El espacio af\u00edn eucl\u00eddeo \\(\\mathbb{R}^3\\) y sistemas de ecuaciones con maxima"},"content":{"rendered":"<h2>El espacio af\u00edn eucl\u00eddeo<\/h2>\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 showHtmlDiv1g() {\n  var htmlShow1g = document.getElementById(\"html-show1g\");\n  if (htmlShow1g.style.display === \"none\") {\n    htmlShow1g.style.display = \"block\";\n  } else {\n    htmlShow1g.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv1g()\">Soluci\u00f3n:<\/button><\/p>\n<p><div id=\"html-show1g\" style=\"display: none;\">\nLo que buscamos es \\(\\|\\vec{v}\\times\\vec{u}\\|\\).<\/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\">(%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\">$<\/span><\/span><\/td>\n<\/tr>\n<\/table>\n<p><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"block\"><mtable><mlabeledtr columnalign=\"left\"><mtd><mtext>(vu)<\/mtext><\/mtd><mtd><mi>\u2212<\/mi><mn>14<\/mn><mo>\u2062<\/mo><mi>i<\/mi><mo>+<\/mo><mn>11<\/mn><mo>\u2062<\/mo><mi>j<\/mi><mo>+<\/mo><mn>5<\/mn><mo>\u2062<\/mo><mi>k<\/mi><\/mtd><\/mlabeledtr><mlabeledtr columnalign=\"left\"><mtd><mtext\/><\/mtd><mtd><mo>La norma de <\/mo><mo\/><mo>[<\/mo><mi>\u2212<\/mi><mn>14<\/mn><mo>,<\/mo><mn>11<\/mn><mo>,<\/mo><mn>5<\/mn><mo>]<\/mo><mo\/><mo>es<\/mo><mo\/><mn>18.4932<\/mn><mo\/><\/mtd><\/mlabeledtr><\/mtable><\/math>\n<\/div>\n<hr \/>\n<blockquote><p><strong>Ejemplo:<\/strong>  Sean \\(\\vec{v}=(1,-2,3)\\) y  \\(\\vec{u}=(2,1,-1)\\), \u00bfcu\u00e1l es la segunda cifra decimal del \\(\\cos(\\vec{v},\\vec{u})\\)?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2g() {\n  var htmlShow2g = document.getElementById(\"html-show2g\");\n  if (htmlShow2g.style.display === \"none\") {\n    htmlShow2g.style.display = \"block\";\n  } else {\n    htmlShow2g.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2g()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2g\" style=\"display: none;\">\n<!-- Code cell --><\/p>\n<table>\n<tr style=\"border: 0px;\">\n<td style=\"width: 70px;vertical-align: top;padding: 1mm;\"><span class=\"prompt\">(%i5)<\/span><\/td>\n<td style=\"vertical-align: top;padding: 1mm;\"><span class=\"input\"><span class=\"code_variable\">fpprintprec<\/span><span class=\"code_operator\">:<\/span><span class=\"code_number\">6<\/span><span class=\"code_endofline\">$<\/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<blockquote><p><strong>Ejemplo:<\/strong>  Cu\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 showHtmlDiv2a() {\n  var htmlShow2a = document.getElementById(\"html-show2a\");\n  if (htmlShow2a.style.display === \"none\") {\n    htmlShow2a.style.display = \"block\";\n  } else {\n    htmlShow2a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2a\" 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<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 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<!-- 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><p><strong>Ejemplo:<\/strong> Cu\u00e1nto suman las coordenadas del punto que se encuentra a dos unidades del punto P(2,-5,3), sobre la recta que pasa por los puntos P y Q(1,3,4)<\/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<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Ecuaciones param\u00e9tricas de una recta en \u211d3. Ej.2 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/le4yHfy8eok?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\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 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\">\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 showHtmlDiv1() {\n  var htmlShow1 = document.getElementById(\"html-show1\");\n  if (htmlShow1.style.display === \"none\") {\n    htmlShow1.style.display = \"block\";\n  } else {\n    htmlShow1.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv1()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show1\" 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 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\/EjrALGnucleo01b.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\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() {\n  var htmlShow = document.getElementById(\"html-show\");\n  if (htmlShow.style.display === \"none\") {\n    htmlShow.style.display = \"block\";\n  } else {\n    htmlShow.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show\" style=\"display: none;\">\n<p><strong>A.)<\/strong><\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/uploads.jesussoto.es\/maxima\/EjrALGnucleo01.html\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>El espacio af\u00edn eucl\u00eddeo Ejemplo: \u00bfCu\u00e1l es la norma del vector perpendicular al subespacio generado por \\(\\vec{v}:(1,-1,5)\\) y \\(\\vec{u}:(2,3,-1)\\)? Soluci\u00f3n: Lo que buscamos es \\(\\|\\vec{v}\\times\\vec{u}\\|\\). (%i6) fpprintprec:6$v:[1,&#8211;1,5]$u:[2,3,&#8211;1]$vu:rat(determinant(matrix([i,j,k],v,u)),k,j,i);vu:[coeff(vu,i),coeff(vu,j),coeff(vu,k)]$print(\u00abLa norma de \u00ab,vu,\u00abes\u00bb,float(sqrt(vu.vu)))$ (vu)\u221214\u2062i+11\u2062j+5\u2062kLa norma&#8230;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[7],"class_list":["post-424","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\/424","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=424"}],"version-history":[{"count":6,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/424\/revisions"}],"predecessor-version":[{"id":429,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/424\/revisions\/429"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=424"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=424"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=424"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}