{"id":144,"date":"2025-10-02T09:15:54","date_gmt":"2025-10-02T07:15:54","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=144"},"modified":"2025-10-02T13:28:07","modified_gmt":"2025-10-02T11:28:07","slug":"mathbio-aplicacion-de-los-determinantes","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=144","title":{"rendered":"MathBio: Aplicaci\u00f3n de los determinantes"},"content":{"rendered":"<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz \\[A=\\begin{bmatrix}<br \/>\n2 &#038; 4 &#038; 1 &#038; 12 \\\\<br \/>\n-1 &#038; 1 &#038; 0 &#038; 3 \\\\<br \/>\n0 &#038; -1 &#038; 9 &#038; -3 \\\\<br \/>\n7 &#038; 3 &#038; 6 &#038; 9<br \/>\n\\end{bmatrix}\\in\\mathcal{M}_4(\\mathbb{R}),\\] \u00bfes regular? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv62() {\n  var htmlShow62 = document.getElementById(\"html-show62\");\n  if (htmlShow62.style.display === \"none\") {\n    htmlShow62.style.display = \"block\";\n  } else {\n    htmlShow62.style.display = \"none\";\n  }\n}\n<\/script> <\/p>\n<p><button onclick=\"showHtmlDiv62()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show62\" style=\"display: none;\">\nNo, es suficiente con comprobar que hay dos columnas linealmente dependientes.\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Dada la matriz \\[A=\\begin{bmatrix}<br \/>\n-\\alpha &#038; \\alpha-1 &#038; \\alpha+1 \\\\<br \/>\n 1 &#038; 2 &#038; 3 \\\\<br \/>\n2-\\alpha &#038; \\alpha+3 &#038; \\alpha+7<br \/>\n\\end{bmatrix}\\in\\mathcal{M}_3(\\mathbb{R}),\\] \u00bfpara que valores de \\(\\alpha\\) la matriz no es regular? <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv62q() {\n  var htmlShow62q = document.getElementById(\"html-show62q\");\n  if (htmlShow62q.style.display === \"none\") {\n    htmlShow62q.style.display = \"block\";\n  } else {\n    htmlShow62q.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv62q()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show62q\" style=\"display: none;\">\nObervar que<br \/>\n\\[\\begin{bmatrix}-\\alpha  &#038; \\alpha -1 &#038; \\alpha +1\\\\<br \/>\n1 &#038; 2 &#038; 3\\\\<br \/>\n2-\\alpha  &#038; \\alpha +3 &#038; \\alpha +7\\end{bmatrix}\\overset{f_3-f_1}{\\sim }<br \/>\n\\begin{bmatrix}-\\alpha  &#038; \\alpha -1 &#038; \\alpha +1\\\\<br \/>\n1 &#038; 2 &#038; 3\\\\<br \/>\n2 &#038; 4 &#038; 6\\end{bmatrix}\\]\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Cu\u00e1l es el valor de \\(\\alpha\\) para que el vector \\([\\alpha,2,-2]\\) pertenezca al subespacio \\(\\mbox{Gen}\\{[1,-2,-1],[3,2,-1]\\}\\)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv6() {\n  var htmlShow6 = document.getElementById(\"html-show6\");\n  if (htmlShow6.style.display === \"none\") {\n    htmlShow6.style.display = \"block\";\n  } else {\n    htmlShow6.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv6()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show6\" style=\"display: none;\">\nObservemos que \\[ u\\in U\\Rightarrow  \\exists \\lambda ,\\mu \\in\\mathbb{R};\\,  [\\alpha ,2,-2]=\\lambda [1,-2,-1]+\\mu [3,2,-1]<br \/>\n\\]<br \/>\nEs decir, el vector \\([\\alpha,2,-2]\\) es linealmente dependientes de los vectores que generan el subespacio. Luego \\[\\begin{vmatrix}<br \/>\n1 &#038; -2 &#038; -1 \\\\<br \/>\n3 &#038; 2 &#038; -1 \\\\<br \/>\n\\alpha  &#038; 2 &#038; -2 \\\\<br \/>\n\\end{vmatrix}=0\\]<br \/>\nResolvemos el determinate:<br \/>\n\\[\\begin{vmatrix}<br \/>\n1 &#038; -2 &#038; -1 \\\\<br \/>\n3 &#038; 2 &#038; -1 \\\\<br \/>\n\\alpha  &#038; 2 &#038; -2 \\\\<br \/>\n\\end{vmatrix}\\overset{\\underset{f_2+f_1}{f_3+f_1}}{\\sim } \\begin{vmatrix}<br \/>\n1 &#038; -2 &#038; -1 \\\\<br \/>\n4 &#038; 0 &#038; -2 \\\\<br \/>\n\\alpha+1  &#038; 0 &#038; -3 \\\\<br \/>\n\\end{vmatrix}=2\\begin{vmatrix}<br \/>\n4 &#038; -2 \\\\<br \/>\n\\alpha+1  &#038; -3 \\\\<br \/>\n\\end{vmatrix}\\Rightarrow 4\\alpha -20=0\\]<br \/>\nAs\u00ed pues, la soluci\u00f3n es \\(\\alpha=5\\).\n<\/div>\n<hr \/>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Justificar si el punto S(2,4,-13) es coplanario con los puntos P(1,-3,-1), Q(2,-2,1) y R(3,2,-4)<\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv6q() {\n  var htmlShow6q = document.getElementById(\"html-show6q\");\n  if (htmlShow6q.style.display === \"none\") {\n    htmlShow6q.style.display = \"block\";\n  } else {\n    htmlShow6q.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv6q()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show6q\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Puntos Coplanarios. Ej.2 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/9g6oZ805A5Y?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<h2>Ecuaciones impl\u00edcitas<\/h2>\n<p>Utilizando los determinantes podemos construir la ecuaci\u00f3n impl\u00edcita de la recta en el plano af\u00edn que pasa por los puntos \\(P(p_1,p_2)\\) y \\(Q(q_1,q_2)\\), esta vendr\u00e1 dada por:<br \/>\n\\[\\begin{vmatrix} x &#038; y &#038; 1\\\\ p_1 &#038; p_2 &#038; 1\\\\ q_1 &#038; q_2 &#038; 1 \\end{vmatrix}=0\\]<\/p>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> Si \\(Ax+By+D=0\\) es la ecuaci\u00f3n impl\u00edcita de la recta pasa por los puntos \\(P(1,3)\\), \\(Q(-1,2)\\), \u00bfcu\u00e1l es el valor del producto escalar del vector \\([1,1]\\) por el vector \\(\\frac{1}{\\parallel[A,B]\\parallel}[A,B]\\)?  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv72a() {\n  var htmlShow72a = document.getElementById(\"html-show72a\");\n  if (htmlShow72a.style.display === \"none\") {\n    htmlShow72a.style.display = \"block\";\n  } else {\n    htmlShow72a.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv72a()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show72a\" style=\"display: none;\">\nSi aplicamos lo anterior, la recta pasa por los puntos \\(P(1,3)\\), \\(Q(-1,2)\\) vendr\u00e1 dada por el determinante:<br \/>\n\\[\\begin{vmatrix}x &#038; y &#038; 1\\\\<br \/>\n1 &#038; 3 &#038; 1\\\\<br \/>\n-1 &#038; 2 &#038; 1\\end{vmatrix}=0\\Rightarrow -2y+x+5=0\\]<br \/>\nPor tanto, el vector unitario normal del plano es: \\[\\frac{1}{\\parallel[1,-2]\\parallel}[1,-2]=\\left[\\frac{1}{\\sqrt{5}},\\frac{-2}{\\sqrt{5}}\\right]\\]<br \/>\nEl producto escalar por el vector \\([1,1]\\) equivale a la suma de sus componentes:\\[\\frac{1}{\\sqrt{5}}-\\frac{2}{\\sqrt{5}}=-\\frac{1}{\\sqrt{5}}\\approx -0.44\\]\n<\/div>\n<hr \/>\n<p>En el caso del espacio af\u00edn, la recta \\(r:\\{P(p_1,p_2,p_3)+\\lambda(v_1,v_2,v_3)\\}\\), vendr\u00e1 dada por las ecuaciones que plantean dos menores de orden dos de la matriz \\[\\begin{bmatrix}v_1 &#038;x-p_1\\\\ v_2 &#038;y-p_2\\\\ v_3 &#038;z-p_3\\\\\\end{bmatrix}\\]<\/p>\n<blockquote>\n<p><strong>Proposici\u00f3n:<\/strong> Sea la recta \\(r:\\{P(p_1,p_2,p_3)+\\lambda(v_1,v_2,v_3)\\}\\) y \\(v_1\\neq 0\\) las ecuaciones impl\u00edcitas que definen a la recta estar\u00e1n dadas por las dos ecuaciones: \\[ \\begin{vmatrix}  v_1 &#038; x-p_1\\\\ v_2 &#038; y-p_2 \\end{vmatrix}=0\\, , \\begin{vmatrix}  v_1 &#038; x-p_1\\\\ v_3 &#038; z-p_3 \\end{vmatrix}=0\\]<\/p>\n<\/blockquote>\n<p>Si deseamos la ecuaci\u00f3n impl\u00edcita del plano en el espacio af\u00edn que pasa por un punto \\(P(p_1,p_2,p_3)\\) y que tiene por subespacio director el generado por los vectores \\(\\vec{v}=(v_1,v_2,v_3)\\) y \\(\\vec{u}=(u_1,u_2,u_3)\\), vendr\u00e1 determinado por el determinante \\[\\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>\n<blockquote>\n<p><strong>Definici\u00f3n:<\/strong> Si \\[Ax+By+Cz+D=0\\] es la ecuaci\u00f3n impl\u00edcita de un plano en el espacio af\u00edn, se denomina vector normal al plano al vector \\([A,B,C]\\).<\/p>\n<p>Si al vector normal lo dividimos por su norma, se denomina vector unitario normal.<\/p>\n<\/blockquote>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> \u00bfCu\u00e1l es el valor del producto escalar del vector \\([3,2,1]\\) por el vector unitario normal de la ecuaci\u00f3n impl\u00edcita del plano que pasa por los puntos \\(P(1,2,3)\\), \\(Q(-1,0,2)\\) y \\(R(4,-2,0)\\)?  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv72() {\n  var htmlShow72 = document.getElementById(\"html-show72\");\n  if (htmlShow72.style.display === \"none\") {\n    htmlShow72.style.display = \"block\";\n  } else {\n    htmlShow72.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv72()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show72\" style=\"display: none;\">\nSi aplicamos lo anterior, fijado \\(P\\), tendremos \\(\\overrightarrow{PQ}=[-2,-2,-1]\\) y \\(\\overrightarrow{PR}=[3,-4,-3]\\). Luego la ecuaci\u00f3n vendr\u00e1 dada por el determinante:<br \/>\n\\[\\begin{vmatrix}x-1 &#038; y-2 &#038; z-3\\\\<br \/>\n-2 &#038; -2 &#038; -1\\\\<br \/>\n3 &#038; -4 &#038; -3\\end{vmatrix}=0\\Rightarrow 14z-9y+2x-26=0\\]<br \/>\nLa norma del vector normal del plano es: \\[\\parallel[2,-9,14]\\parallel\\approx  16.76\\]<br \/>\nPor tanto, el producto escalar del vector \\([3,2,1]\\) por el vector unitario normal es \\[\\frac{3\\cdot 2-2\\cdot 9+1\\cdot 14}{16.76}\\approx 0.119\\]\n<\/div>\n<hr \/>\n<h2>Aplicaciones geom\u00e9tricas de los determinantes<\/h2>\n<p>Con la definici\u00f3n de determinante podemos definir una operaci\u00f3n especial entre vectores de \\(\\mathbb{R}^3\\): el producto vectorial. El producto vectorial de dos vectores es a su vez un vector de \\(\\mathbb{R}^3\\). 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}=\\begin{vmatrix} v_2 &#038; v_3 \\\\ u_2 &#038; u_3\\end{vmatrix}\\vec{i}- \\begin{vmatrix}v_1 &#038; v_3 \\\\ u_1 &#038; u_3\\end{vmatrix}\\vec{j} +\\begin{vmatrix}v_1 &#038; v_2 \\\\ u_1 &#038; u_2 \\end{vmatrix}\\vec{k}\\]<\/p>\n<blockquote>\n<p><strong>Propiedad:<\/strong> El producto vectorial es perpendicular a los vectores que lo forman:\\[\\vec{v}\\perp  (\\vec{v}\\times\\vec{u}) \\wedge \\vec{u}\\perp  (\\vec{v}\\times\\vec{u})\\]  <\/p>\n<\/blockquote>\n<blockquote>\n<p><strong>Ejercicio:<\/strong> \u00bfCu\u00e1l es la norma del vector normal al subespacio vectorial \\(\\mbox{Gen}\\{(1,-1,2),(0,-1,3)\\}\\)?  <\/p>\n<\/blockquote>\n<p><script>\nfunction showHtmlDiv721() {\n  var htmlShow721 = document.getElementById(\"html-show721\");\n  if (htmlShow721.style.display === \"none\") {\n    htmlShow721.style.display = \"block\";\n  } else {\n    htmlShow721.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv721()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show721\" style=\"display: none;\">\nPor lo anterior, el vector normal vendr\u00e1 dado por el determinante:<br \/>\n\\[\\begin{vmatrix} \\vec{i} &#038; \\vec{j} &#038;\\vec{k}\\\\<br \/>\n1&#038;-1&#038;2\\\\<br \/>\n0&#038;-1&#038;3\\end{vmatrix}=\\begin{vmatrix}-1&#038;2\\\\ -1&#038;3\\end{vmatrix}\\vec{i} -\\begin{vmatrix}1&#038;2\\\\ 0&#038;3\\end{vmatrix} \\vec{j} +\\begin{vmatrix}1&#038;-1\\\\ 0&#038;-1\\end{vmatrix}\\vec{k}\\]<br \/>\nPor tanto, la norma del vector normal del plano es: \\[\\left \\| \\left[\\begin{vmatrix}-1&#038;2\\\\ -1&#038;3\\end{vmatrix},-\\begin{vmatrix}1&#038;2\\\\ 0&#038;3\\end{vmatrix},\\begin{vmatrix}1&#038;-1\\\\ 0&#038;-1\\end{vmatrix}\\right]\\right \\| =\\parallel[-1,-3,-1]\\parallel\\approx 3.31\\]\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 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 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<p>Adem\u00e1s, tenemos una f\u00f3rmula que relaci\u00f3n producto vectorial con el seno del \u00e1ngulo que forman:\\[\\left \\| \\vec{v}\\times\\vec{u} \\right \\|=\\left \\|\\vec{v} \\right \\|\\, \\left \\| \\vec{u} \\right \\| \\, |\\sin(\\widehat{\\vec{v}\\vec{u}})|\\]  <\/p>\n<p>El producto vectorial permite una definici\u00f3n muy \u00fatil: el producto mixto, que se define como:<br \/>\n\\[[\\vec{v},\\vec{u},\\vec{w}]=\\vec{v}\\bullet(\\vec{u}\\times\\vec{w})\\]<br \/>\ndado tres vectores \\(\\vec{v},\\vec{u},\\vec{w}\\in\\mathbb{R}^3\\)<\/p>\n<p>El producto mixto de tres vectores cumple una propiedad geom\u00e9trica muy curiosa: es el volumen de un paralep\u00edpedo que tiene por lados los vectores indicados. As\u00ed se cumple que este volumen es:<br \/>\n\\[[\\vec{v},\\vec{u},\\vec{w}]=\\begin{vmatrix} v_1 &#038; v_2 &#038; v_3 \\\\ u_1 &#038; u_2 &#038; u_3\\\\ w_1 &#038; w_2 &#038; w_3\\end{vmatrix}\\]<\/p>\n<hr \/>\n<h3>Bibliograf\u00eda<\/h3>\n<ul>\n<li>Cap\u00edtulo 4 de \u00c1lgebra lineal y sus aplicaciones. 5\u00ba edici\u00f3n, David C. Lay. Pearson. 2016.<\/li>\n<\/ul>\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><br \/>\nDada la matriz \\(\\begin{bmatrix}1&#038;0&#038;0&#038;1\\\\0&#038;1&#038;2&#038;-1\\\\ 0&#038;1&#038;-1&#038;0\\\\ 0&#038;2&#038;0&#038;-1\\end{bmatrix}\\), \u00bfcu\u00e1l es la traza de su inversa?<\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>-1<\/li>\n<li>1<\/li>\n<li>0<\/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>C.)<\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Ejercicio: Dada la matriz \\[A=\\begin{bmatrix} 2 &#038; 4 &#038; 1 &#038; 12 \\\\ -1 &#038; 1 &#038; 0 &#038; 3 \\\\ 0 &#038; -1 &#038; 9 &#038; -3 \\\\ 7 &#038; 3&#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":[],"class_list":["post-144","post","type-post","status-publish","format-standard","hentry","category-mathbio"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/144","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=144"}],"version-history":[{"count":3,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/144\/revisions"}],"predecessor-version":[{"id":207,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/144\/revisions\/207"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=144"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=144"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=144"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}