{"id":420,"date":"2025-11-03T08:15:37","date_gmt":"2025-11-03T07:15:37","guid":{"rendered":"https:\/\/clases.jesussoto.es\/?p=420"},"modified":"2025-11-01T18:09:03","modified_gmt":"2025-11-01T17:09:03","slug":"alg-el-espacio-afin-euclideo","status":"publish","type":"post","link":"https:\/\/clases.jesussoto.es\/?p=420","title":{"rendered":"ALG: El espacio af\u00edn eucl\u00eddeo"},"content":{"rendered":"<p>En este tema nos proponemos a 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<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 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\\[\\begin{vmatrix} \\vec{i} &#038; \\vec{j} &#038;\\vec{k}\\\\ 1&#038;-2&#038;3 \\\\ 2&#038;1&#038;-1\\end{vmatrix}=\\begin{vmatrix}-2&#038;3\\\\ 1&#038;-1\\end{vmatrix}\\vec{i} &#8211; \\begin{vmatrix}1&#038;3\\\\ 2&#038;-1\\end{vmatrix}\\vec{j} +\\begin{vmatrix}1&#038;-2\\\\ 2&#038;1\\end{vmatrix}\\vec{k}\\] \\[=-\\vec{i} +7\\vec{j} +5\\vec{k}\\]\n<\/div>\n<hr \/>\n<p>Este resultado junto con el anterior nos permiten 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>\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;\">\nEl 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<p>\\[\\begin{vmatrix}\\mathbf{i}&#038;\\mathbf{j}&#038;\\mathbf{k}\\\\ 1&#038;9&#038;-6 \\\\ -4&#038;-18&#038;15\\end{vmatrix}=27\\mathbf{i}+9\\mathbf{j}+18\\mathbf{k}\\]<br \/>\nLuego el vector normal es [27,9,18]. Ahora lo normalizamos(hacerlo unitario) \\[\\frac{1}{\\|[27,9,18]\\|}[27,9,18]=\\left[ \\frac{3}{\\sqrt{14}},\\frac{1}{\\sqrt{14}},\\frac{2}{\\sqrt{14}}\\right]\\]<br \/>\nPor \u00faltimo,<br \/>\n\\[[1,2,4].\\left[ \\frac{3}{\\sqrt{14}},\\frac{1}{\\sqrt{14}},\\frac{2}{\\sqrt{14}}\\right]=\\frac{13}{\\sqrt{14}}\\approx 3.47\\]\n<\/p><\/div>\n<hr \/>\n<p>El producto escalar nos da pie a definir la norma de un vector como la ra\u00edz cuadrada de el 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<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 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;\">\nDados\\(\\vec{v}=(1,-2,3)\\) y \\(\\vec{u}=(2,1,-1)\\), \\(\\|(1,-2,3)\\|=\\sqrt{1^2+(-2)^2+3^2}=\\sqrt{14}\\), \\(\\|(2,1,-1)\\|=\\sqrt{6}\\). Luego \\[\\mathbf{cos(\\vec{v},\\vec{u})}=-\\frac{3}{\\sqrt{6} \\sqrt{14}}\\approx -0.3273\\]\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>  Cu\u00e1l es la proyecci\u00f3n de \\(\\vec{u}=(1,-2,3)\\) sobre \\(\\vec{v}=(2,1,-1)\\)?<\/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\\[\\textbf{proy}_\\vec{v}(\\vec{u})=\\frac{(1,-2,3)\\bullet (2,1,-1)}{\\|(2,1,-1)\\|^2}(2,1,-1)=-\\frac{3}{6}(2,1,-1)\\]\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 \/>\nPor la definici\u00f3n tendremos: \\[\\textbf{comp}_{[2,-1,0]}([3,2,1])=\\frac{[3,2,1]\\bullet[2,-1,0]}{\\|[2,-1,0]\\|}=\\frac{4}{\\sqrt{5}}\\]\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\\[[d(P,Q)=||\\vec{QP}||=\\sqrt{(0-1)^2 +(2-(-1))^2 + ((-2)-2)^2}=\\sqrt{26}\\]\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>  Cu\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\\[d(Q,r)=\\frac{\\|\\vec{PQ}\\times\\vec{v}\\|}{\\|(3,2,-1)\\|}=\\frac{\\|(5,-13,-11)\\|}{\\|(3,2,-1)\\|}= \\frac{3 \\sqrt{35}}{\\sqrt{14}}\\approx4.7434\\]<\/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>  Cu\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;\">\nLa ecuaci\u00f3n impl\u00edcita del plano vendr\u00e1 dada por: \\[\\begin{vmatrix}x-1 &#038; y+1 &#038; z-2\\\\<br \/>\n3 &#038; 2 &#038; -1\\\\<br \/>\n0 &#038; 1 &#038; 1\\end{vmatrix}=3z-3y+3x-12=0\\]<br \/>\n\\[d(Q,\\pi)=\\frac{|(3,-3,3).(0,2,-2)-12|}{\\|(3,-3,3)\\|}=\\frac{8}{\\sqrt{3}}\\approx 4.6188\\]<\/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 afin, 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<h2>\u00c1rea de un paralelogramo<\/h2>\n<p>Veamos una aplicaci\u00f3n del determinate en el caso de un paralelogramo en el plano. Cosideremos 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). Cu\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<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Producto Mixto. Ej.1 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/Xsl5yqu2Bcs?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>Otros ejercicios de inter\u00e9s<\/h2>\n<blockquote><p><strong>Ejercicio:<\/strong> Determina si O(2,4,-13) es coplanario con \\(P\\)(1,-3,-1), \\(Q\\)(2,-2,1) y \\(R\\)(3,2,-4)<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2dx2() {\n  var htmlShow2dx2 = document.getElementById(\"html-show2dx2\");\n  if (htmlShow2dx2.style.display === \"none\") {\n    htmlShow2dx2.style.display = \"block\";\n  } else {\n    htmlShow2dx2.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2dx2()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2dx2\" 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<blockquote><p><strong>Ejercicio:<\/strong> Sea S(-2,3,x). \u00bfqu\u00e9 valor de x hace que S sea coplanario con P(2,2,-1), Q(1,2,-3) y R(3,-2,1) <\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2dx4() {\n  var htmlShow2dx4 = document.getElementById(\"html-show2dx4\");\n  if (htmlShow2dx4.style.display === \"none\") {\n    htmlShow2dx4.style.display = \"block\";\n  } else {\n    htmlShow2dx4.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2dx4()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2dx4\" style=\"display: none;\">\n<iframe loading=\"lazy\" title=\"\u00c1lgebra Lineal - Puntos coplanarios. Ej.1 - Jes\u00fas Soto\" width=\"640\" height=\"360\" src=\"https:\/\/www.youtube.com\/embed\/LlkxBlb1PY0?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><p><strong>Ejercicio:<\/strong> Sean \\(\\vec{a}\\) y \\(\\vec{b}\\) dos vectores tales que \\(||\\vec{a}||=3\\), \\(||\\vec{b}||=\\sqrt{6}\\) y \\(\\theta=45\u00ba\\), \u00bfcu\u00e1l es su producto escalar?<\/p><\/blockquote>\n<p><script>\nfunction showHtmlDiv2dx5() {\n  var htmlShow2dx5 = document.getElementById(\"html-show2dx5\");\n  if (htmlShow2dx5.style.display === \"none\") {\n    htmlShow2dx5.style.display = \"block\";\n  } else {\n    htmlShow2dx5.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<p><button onclick=\"showHtmlDiv2dx5()\">Soluci\u00f3n:<\/button><\/p>\n<div id=\"html-show2dx5\" style=\"display: none;\">\nSabemos que \\(\\cos(45\u00ba)=\\cos(\\tfrac{\\pi}{4})=\\frac{1}{\\sqrt{2}}\\). Adem\u00e1s sabemos que \\[\\textbf{cos}(\\vec{v},\\vec{u})=\\frac{\\vec{v}\\bullet\\vec{u}}{||\\vec{v}||\\cdot ||\\vec{u}||}.\\] Luego, como el \u00e1ngulo de nuestro vectores es de \\(\\frac{\\pi}{\\sqrt{4}}\\), ser\u00e1 \\[\\frac{1}{\\sqrt{2}}=\\textbf{cos}(\\vec{a},\\vec{b})=\\frac{\\vec{a}\\bullet\\vec{b}}{3\\cdot \\sqrt{6}}\\Rightarrow \\vec{a}\\bullet\\vec{b}=\\frac{3\\sqrt{6}}{\\sqrt{2}}=3\\sqrt{3}\\]\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 \\(ax+by+c=0\\) en la ecuaci\u00f3n impl\u00edcita de la recta del plano af\u00edn que pasa por los puntos P(1,-1) y Q(2,3). Si \\(u\\) es el resultado de normalizar \\([a,b]\\)(vector unitario normal), \u00bfcu\u00e1l es el primer decimal de \\(|[1,1].u|\\)?<\/td>\n<\/tr>\n<tr>\n<td>\n<div id=\"menu-a\">\n<ul>\n<li>7<\/li>\n<li>1<\/li>\n<li>4<\/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>\n<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\/EjrALGrecta_afin01\" width=\"650\" height=\"300\" allow=\"fullscreen\"><\/iframe><br \/>\nSolo nos resta por normalizar el vector y realizar el producto escalar:<br \/>\n\\[\\left|[1,1].[-4,1]\\frac{1}{\\sqrt{17}}\\right|=\\frac{3}{\\sqrt{17}}\\approx 0.7276\\]\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>En este tema nos proponemos a 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&#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":[],"class_list":["post-420","post","type-post","status-publish","format-standard","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/420","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=420"}],"version-history":[{"count":20,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/420\/revisions"}],"predecessor-version":[{"id":450,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=\/wp\/v2\/posts\/420\/revisions\/450"}],"wp:attachment":[{"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=420"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=420"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/clases.jesussoto.es\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=420"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}