feat: interactive linear algebra practice web app
- 68 exercises from UBA FCE chapters 1-3 - Step-by-step solutions with KaTeX rendering - Theory panels (26 topics) expandable per exercise - Matrix builder (2x2/3x3/4x4) with 7 operations - System solver (Gauss, Gauss-Jordan, Cramer, Rouché-Frobenius) - Glassmorphism UI with dark mode - Canvas particle background - ARIA accessibility (keyboard nav, screen reader) - Zero build step - open index.html directly
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renato97
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data/exercises-cap01.json
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[
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{
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"id": "cap01-01",
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"chapter": 1,
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"topic": "vector-ops",
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"subtopic": "components",
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"theoryKey": "vectors-theory",
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"difficulty": "basic",
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"statement": "Dados los puntos A = (2; -1; 3) y B = (4; 2; -1), hallar las componentes del vector \\overrightarrow{AB}.",
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"hint": "Las componentes de AB son B - A",
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"answerType": "vector",
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"answer": { "value": [2, 3, -4], "latex": "(2;\\; 3;\\; -4)" },
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"solutionSteps": [
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{ "desc": "Fórmula: las componentes de AB = B - A", "expression": "\\overrightarrow{AB} = B - A = (4; 2; -1) - (2; -1; 3)" },
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{ "desc": "Restar componente a componente", "expression": "\\overrightarrow{AB} = (4-2;\\; 2-(-1);\\; -1-3) = (2;\\; 3;\\; -4)" }
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]
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},
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{
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"id": "cap01-02",
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"chapter": 1,
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"topic": "vector-ops",
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"subtopic": "magnitude",
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"theoryKey": "vectors-theory",
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"difficulty": "basic",
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"statement": "Hallar el módulo de los vectores: a) \\vec{u} = (3; -4), b) \\vec{v} = (1; 2; -2), c) \\vec{w} = (-1; 0; 3).",
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"hint": "|v| = √(v₁² + v₂² + v₃²)",
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"answerType": "expression",
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"answer": { "value": "|u|=5, |v|=3, |w|=√10", "latex": "|\\vec{u}|=5,\\quad |\\vec{v}|=3,\\quad |\\vec{w}|=\\sqrt{10}" },
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"solutionSteps": [
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{ "desc": "a) Módulo de u = (3; -4)", "expression": "|\\vec{u}| = \\sqrt{3^2 + (-4)^2} = \\sqrt{9 + 16} = \\sqrt{25} = 5" },
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{ "desc": "b) Módulo de v = (1; 2; -2)", "expression": "|\\vec{v}| = \\sqrt{1^2 + 2^2 + (-2)^2} = \\sqrt{1 + 4 + 4} = \\sqrt{9} = 3" },
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{ "desc": "c) Módulo de w = (-1; 0; 3)", "expression": "|\\vec{w}| = \\sqrt{(-1)^2 + 0^2 + 3^2} = \\sqrt{1 + 0 + 9} = \\sqrt{10}" }
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]
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},
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{
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"id": "cap01-03",
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"chapter": 1,
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"topic": "vector-ops",
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"subtopic": "unit-vector",
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"theoryKey": "vectors-theory",
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"difficulty": "basic",
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"statement": "Dado \\vec{u} = (4; -3; 5), hallar el vector unitario en la dirección de \\vec{u}.",
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"hint": "û = u / |u|",
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"answerType": "vector",
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"answer": { "value": [0.5657, -0.4243, 0.7071], "latex": "\\hat{u} = \\left(\\frac{4}{5\\sqrt{2}};\\; \\frac{-3}{5\\sqrt{2}};\\; \\frac{5}{5\\sqrt{2}}\\right) = \\left(\\frac{4}{5\\sqrt{2}};\\; \\frac{-3}{5\\sqrt{2}};\\; \\frac{1}{\\sqrt{2}}\\right)" },
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"solutionSteps": [
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{ "desc": "Calcular el módulo de u", "expression": "|\\vec{u}| = \\sqrt{16 + 9 + 25} = \\sqrt{50} = 5\\sqrt{2}" },
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{ "desc": "Dividir cada componente por el módulo", "expression": "\\hat{u} = \\frac{\\vec{u}}{|\\vec{u}|} = \\frac{(4;\\;-3;\\;5)}{5\\sqrt{2}} = \\left(\\frac{4}{5\\sqrt{2}};\\;\\frac{-3}{5\\sqrt{2}};\\;\\frac{1}{\\sqrt{2}}\\right)" }
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]
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},
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{
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"id": "cap01-04",
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"chapter": 1,
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"topic": "vector-ops",
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"subtopic": "addition",
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"theoryKey": "vector-ops",
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"difficulty": "basic",
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"statement": "Sean \\vec{u} = (1; -2; 3) y \\vec{v} = (2; 1; -1). Calcular: a) \\vec{u} + \\vec{v}, b) 3\\vec{u} - 2\\vec{v}, c) |\\vec{u} + \\vec{v}|.",
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"hint": "Sumar/restar componente a componente",
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"answerType": "expression",
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"answer": { "value": "a)(3;-1;2) b)(-1;-8;11) c)√14", "latex": "a)\\,(3;-1;2),\\quad b)\\,(-1;-8;11),\\quad c)\\,\\sqrt{14}" },
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"solutionSteps": [
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{ "desc": "a) Suma u + v", "expression": "\\vec{u} + \\vec{v} = (1+2;\\;-2+1;\\;3+(-1)) = (3;\\;-1;\\;2)" },
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{ "desc": "b) 3u - 2v", "expression": "3\\vec{u} = (3;-6;9),\\quad 2\\vec{v} = (4;2;-2)" },
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{ "desc": "Restar", "expression": "3\\vec{u} - 2\\vec{v} = (3-4;\\;-6-2;\\;9-(-2)) = (-1;\\;-8;\\;11)" },
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{ "desc": "c) Módulo de u + v = (3; -1; 2)", "expression": "|\\vec{u}+\\vec{v}| = \\sqrt{9+1+4} = \\sqrt{14}" }
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]
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},
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{
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"id": "cap01-05",
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"chapter": 1,
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"topic": "vector-ops",
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"subtopic": "dot-product",
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"theoryKey": "dot-product",
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"difficulty": "basic",
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"statement": "Sean \\vec{u} = (2; -1; 3) y \\vec{v} = (4; 3; -2). Calcular \\vec{u} \\cdot \\vec{v}.",
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"hint": "u·v = u₁v₁ + u₂v₂ + u₃v₃",
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"answerType": "numeric",
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"answer": { "value": -1, "latex": "-1" },
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"solutionSteps": [
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{ "desc": "Producto escalar componente a componente", "expression": "\\vec{u} \\cdot \\vec{v} = 2 \\cdot 4 + (-1) \\cdot 3 + 3 \\cdot (-2)" },
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{ "desc": "Calcular", "expression": "= 8 - 3 - 6 = -1" }
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]
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},
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{
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"id": "cap01-06",
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"chapter": 1,
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"topic": "vector-ops",
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"subtopic": "cross-product",
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"theoryKey": "cross-product",
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"difficulty": "basic",
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"statement": "Sean \\vec{u} = (1; 2; -1) y \\vec{v} = (3; 0; 4). Calcular \\vec{u} \\times \\vec{v}.",
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"hint": "Producto vectorial por determinante",
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"answerType": "vector",
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"answer": { "value": [8, -7, -6], "latex": "(8;\\;-7;\\;-6)" },
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"solutionSteps": [
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{ "desc": "Producto vectorial por determinante", "expression": "\\vec{u} \\times \\vec{v} = \\begin{vmatrix} \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\ 1 & 2 & -1 \\\\ 3 & 0 & 4 \\end{vmatrix}" },
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{ "desc": "Calcular componente i", "expression": "i: (2)(4) - (-1)(0) = 8" },
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{ "desc": "Calcular componente j", "expression": "j: -[(1)(4) - (-1)(3)] = -[4+3] = -7" },
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{ "desc": "Calcular componente k", "expression": "k: (1)(0) - (2)(3) = -6" },
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{ "desc": "Resultado", "expression": "\\vec{u} \\times \\vec{v} = (8;\\;-7;\\;-6)" }
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]
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},
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{
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"id": "cap01-07",
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"chapter": 1,
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"topic": "vector-ops",
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"subtopic": "parallel",
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"theoryKey": "parallel-perp",
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"difficulty": "basic",
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"statement": "Determinar si los vectores \\vec{u} = (2; 1; -3) y \\vec{v} = (-4; -2; 6) son paralelos.",
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"hint": "Son paralelos si u = kv para algún escalar k",
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"answerType": "expression",
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"answer": { "value": "Sí, son paralelos (k=-2)", "latex": "\\text{Sí, } \\vec{v} = -2\\vec{u}" },
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"solutionSteps": [
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{ "desc": "Verificar si u × v = 0", "expression": "\\vec{u} \\times \\vec{v} = \\begin{vmatrix} \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\ 2 & 1 & -3 \\\\ -4 & -2 & 6 \\end{vmatrix}" },
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{ "desc": "Componente i: (1)(6) - (-3)(-2) = 6 - 6 = 0", "expression": "i: 6 - 6 = 0" },
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{ "desc": "Componente j: -[(2)(6) - (-3)(-4)] = -[12-12] = 0", "expression": "j: -(12-12) = 0" },
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{ "desc": "Componente k: (2)(-2) - (1)(-4) = -4+4 = 0", "expression": "k: -4+4 = 0" },
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{ "desc": "Como u×v=0, son paralelos. Además v = -2u", "expression": "\\vec{v} = (-4;-2;6) = -2(2;1;-3) = -2\\vec{u}" }
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]
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},
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{
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"id": "cap01-08",
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"chapter": 1,
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"topic": "vector-ops",
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"subtopic": "perpendicular",
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"theoryKey": "parallel-perp",
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"difficulty": "basic",
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"statement": "Determinar si los vectores \\vec{u} = (1; -1; 2) y \\vec{v} = (3; 1; 1) son perpendiculares.",
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"hint": "Son perpendiculares si u·v = 0",
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"answerType": "expression",
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"answer": { "value": "No son perpendiculares", "latex": "\\vec{u} \\cdot \\vec{v} = 4 \\neq 0 \\Rightarrow \\text{No son perpendiculares}" },
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"solutionSteps": [
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{ "desc": "Calcular el producto escalar", "expression": "\\vec{u} \\cdot \\vec{v} = (1)(3) + (-1)(1) + (2)(1) = 3 - 1 + 2 = 4" },
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{ "desc": "Como no es 0, no son perpendiculares", "expression": "\\vec{u} \\cdot \\vec{v} = 4 \\neq 0 \\Rightarrow \\text{No son perpendiculares}" }
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]
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},
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{
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"id": "cap01-09",
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"chapter": 1,
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"topic": "vector-ops",
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"subtopic": "mixed-product",
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"theoryKey": "coplanarity",
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"difficulty": "intermediate",
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"statement": "Dados \\vec{u} = (1; -1; 2), \\vec{v} = (3; 0; 1) y \\vec{w} = (0; 2; -1), calcular el producto mixto [\\vec{u}, \\vec{v}, \\vec{w}] y determinar si son coplanarios.",
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"hint": "[u,v,w] = u · (v × w). Si es 0, son coplanarios.",
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"answerType": "numeric",
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"answer": { "value": 7, "latex": "[\\vec{u},\\vec{v},\\vec{w}] = 7 \\neq 0, \\text{ no coplanarios}" },
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"solutionSteps": [
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{ "desc": "Calcular v × w", "expression": "\\vec{v} \\times \\vec{w} = \\begin{vmatrix} \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\ 3 & 0 & 1 \\\\ 0 & 2 & -1 \\end{vmatrix} = (-2;\\;3;\\;6)" },
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{ "desc": "Calcular u · (v × w)", "expression": "[\\vec{u},\\vec{v},\\vec{w}] = (1)(-2) + (-1)(3) + (2)(6) = -2 - 3 + 12 = 7" },
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{ "desc": "Conclusión", "expression": "[\\vec{u},\\vec{v},\\vec{w}] = 7 \\neq 0 \\Rightarrow \\text{No son coplanarios}" }
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]
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},
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{
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"id": "cap01-10",
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"chapter": 1,
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"topic": "line-eq",
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"subtopic": "parametric",
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"theoryKey": "line-equations",
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"difficulty": "basic",
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"statement": "Hallar las ecuaciones paramétricas y continuas de la recta que pasa por A = (1; -2; 3) y B = (4; 1; -1).",
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"hint": "Vector director d = B - A",
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"answerType": "expression",
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"answer": { "value": "r: (x,y,z) = (1,-2,3) + t(3,3,-4)", "latex": "r: \\frac{x-1}{3} = \\frac{y+2}{3} = \\frac{z-3}{-4}" },
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"solutionSteps": [
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{ "desc": "Vector director d = B - A", "expression": "\\vec{d} = B - A = (4-1;\\;1-(-2);\\;-1-3) = (3;\\;3;\\;-4)" },
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{ "desc": "Ecuaciones paramétricas", "expression": "x = 1 + 3t,\\quad y = -2 + 3t,\\quad z = 3 - 4t" },
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{ "desc": "Ecuaciones continuas", "expression": "\\frac{x-1}{3} = \\frac{y+2}{3} = \\frac{z-3}{-4}" }
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]
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},
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{
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"id": "cap01-11",
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"chapter": 1,
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"topic": "line-eq",
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"subtopic": "parametric",
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"theoryKey": "line-equations",
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"difficulty": "basic",
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"statement": "Hallar la ecuación de la recta que pasa por P = (2; 0; -1) y tiene vector director \\vec{v} = (1; 3; -2).",
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"hint": "r: X = P + tv",
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"answerType": "expression",
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"answer": { "value": "r: (x,y,z) = (2,0,-1)+t(1,3,-2)", "latex": "r: (x,y,z) = (2;0;-1) + t(1;3;-2)" },
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"solutionSteps": [
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{ "desc": "Ecuación vectorial", "expression": "r: (x;\\;y;\\;z) = (2;\\;0;\\;-1) + t(1;\\;3;\\;-2)" },
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{ "desc": "Paramétricas", "expression": "x = 2 + t,\\quad y = 3t,\\quad z = -1 - 2t" }
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]
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},
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{
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"id": "cap01-12",
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"chapter": 1,
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"topic": "line-eq",
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"subtopic": "relative-position",
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"theoryKey": "line-positions",
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"difficulty": "intermediate",
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"statement": "Determinar la posición relativa de las rectas: r_1: (x; y; z) = (1; 0; 2) + t(1; 1; -1) y r_2: (x; y; z) = (0; 1; 1) + s(2; -1; 3).",
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"hint": "Verificar si los directores son paralelos, luego calcular producto mixto [d1, d2, P2-P1]",
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"answerType": "expression",
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"answer": { "value": "Rectas que se cruzan", "latex": "\\text{Se cruzan}" },
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"solutionSteps": [
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{ "desc": "Directores: d1=(1;1;-1), d2=(2;-1;3). No son paralelos.", "expression": "\\vec{d_1} \\neq k\\vec{d_2} \\Rightarrow \\text{No paralelas}" },
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{ "desc": "Producto mixto [d1, d2, P2-P1]", "expression": "[\\vec{d_1},\\vec{d_2},P_2-P_1] = \\begin{vmatrix} 1 & 1 & -1 \\\\ 2 & -1 & 3 \\\\ -1 & 1 & -1 \\end{vmatrix}" },
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{ "desc": "Calcular el determinante", "expression": "= 1(1-3) - 1(-2+3) + (-1)(2-1) = -2 - 1 - 1 = -4 \\neq 0" },
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{ "desc": "Conclusión", "expression": "[\\vec{d_1},\\vec{d_2},P_2-P_1] \\neq 0 \\Rightarrow \\text{Se cruzan}" }
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]
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},
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{
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"id": "cap01-13",
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"chapter": 1,
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"topic": "line-eq",
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"subtopic": "distance",
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"theoryKey": "distance-lines",
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"difficulty": "intermediate",
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"statement": "Calcular la distancia del punto Q = (1; 2; 3) a la recta r: (x; y; z) = (0; 1; 0) + t(1; 0; 1).",
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"hint": "d = |(Q-P) × v| / |v|",
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"answerType": "numeric",
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"answer": { "value": 1.7321, "latex": "d = \\sqrt{3}" },
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"solutionSteps": [
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{ "desc": "Q - P = (1;2;3) - (0;1;0) = (1;1;3)", "expression": "\\vec{QP} = Q - P = (1;\\;1;\\;3)" },
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{ "desc": "(Q-P) × v", "expression": "(Q-P) \\times \\vec{v} = (1;1;3) \\times (1;0;1) = (1;\\;-2;\\;-1)" },
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{ "desc": "Módulo del producto vectorial", "expression": "|(Q-P) \\times \\vec{v}| = \\sqrt{1+4+1} = \\sqrt{6}" },
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{ "desc": "Módulo del director", "expression": "|\\vec{v}| = \\sqrt{1+0+1} = \\sqrt{2}" },
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{ "desc": "Distancia", "expression": "d = \\frac{\\sqrt{6}}{\\sqrt{2}} = \\sqrt{3}" }
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]
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},
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{
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"id": "cap01-14",
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"chapter": 1,
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"topic": "line-eq",
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"subtopic": "distance",
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"theoryKey": "distance-lines",
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"difficulty": "intermediate",
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"statement": "Calcular la distancia entre las rectas: r_1: (x; y; z) = (1; 0; 0) + t(1; 0; 1) y r_2: (x; y; z) = (0; 1; 0) + s(0; 1; 0).",
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"hint": "Si se cruzan: d = |[d1,d2,P2-P1]| / |d1×d2|",
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"answerType": "numeric",
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"answer": { "value": 0.7071, "latex": "d = \\frac{1}{\\sqrt{2}}" },
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"solutionSteps": [
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{ "desc": "Verificar que se cruzan: d1×d2 ≠ 0", "expression": "\\vec{d_1} \\times \\vec{d_2} = (1;0;1) \\times (0;1;0) = (-1;\\;0;\\;1)" },
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{ "desc": "P2 - P1 = (-1; 1; 0)", "expression": "P_2 - P_1 = (0-1;\\;1-0;\\;0-0) = (-1;\\;1;\\;0)" },
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{ "desc": "Producto mixto [d1,d2,P2-P1]", "expression": "[\\vec{d_1},\\vec{d_2},P_2-P_1] = \\begin{vmatrix} 1 & 0 & 1 \\\\ 0 & 1 & 0 \\\\ -1 & 1 & 0 \\end{vmatrix} = 0 + 0 + 0 - (-1) - 0 - 0 = 1" },
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{ "desc": "Distancia", "expression": "d = \\frac{|[\\vec{d_1},\\vec{d_2},P_2-P_1]|}{|\\vec{d_1} \\times \\vec{d_2}|} = \\frac{1}{\\sqrt{2}}" }
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]
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},
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{
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"id": "cap01-15",
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"chapter": 1,
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"topic": "line-eq",
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"subtopic": "angle",
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"theoryKey": "angle",
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"difficulty": "basic",
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"statement": "Hallar el ángulo entre las rectas con directores \\vec{v_1} = (1; 2; 3) y \\vec{v_2} = (2; -1; 1).",
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"hint": "cos α = |v1·v2| / (|v1|·|v2|)",
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"answerType": "numeric",
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"answer": { "value": 76.37, "latex": "\\alpha \\approx 76.37^\\circ" },
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"solutionSteps": [
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{ "desc": "Producto escalar", "expression": "\\vec{v_1} \\cdot \\vec{v_2} = 2 - 2 + 3 = 3" },
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{ "desc": "Módulos", "expression": "|\\vec{v_1}| = \\sqrt{14},\\quad |\\vec{v_2}| = \\sqrt{6}" },
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{ "desc": "Coseno del ángulo", "expression": "\\cos\\alpha = \\frac{3}{\\sqrt{14}\\sqrt{6}} = \\frac{3}{\\sqrt{84}}" },
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{ "desc": "Ángulo", "expression": "\\alpha = \\arccos\\left(\\frac{3}{\\sqrt{84}}\\right) \\approx 76.37^\\circ" }
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]
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},
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{
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"id": "cap01-16",
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"chapter": 1,
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"topic": "plane-eq",
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"subtopic": "general",
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"theoryKey": "plane-equations",
|
||||
"difficulty": "basic",
|
||||
"statement": "Hallar la ecuación del plano que pasa por P = (1; 2; 3) y tiene vector normal \\vec{n} = (2; -1; 4).",
|
||||
"hint": "Π: n · (X - P) = 0",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "2x - y + 4z = 12", "latex": "2x - y + 4z = 12" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Ecuación normal: n · (X - P) = 0", "expression": "2(x-1) - 1(y-2) + 4(z-3) = 0" },
|
||||
{ "desc": "Expandir", "expression": "2x - 2 - y + 2 + 4z - 12 = 0" },
|
||||
{ "desc": "Simplificar", "expression": "2x - y + 4z - 12 = 0 \\Rightarrow 2x - y + 4z = 12" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-17",
|
||||
"chapter": 1,
|
||||
"topic": "plane-eq",
|
||||
"subtopic": "general",
|
||||
"theoryKey": "plane-equations",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Hallar la ecuación del plano que pasa por los puntos A = (1; 0; 1), B = (0; 1; 2) y C = (2; 1; 0).",
|
||||
"hint": "Normal = (B-A) × (C-A)",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "x + z - 2 = 0", "latex": "x + z - 2 = 0" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Vectores en el plano", "expression": "\\vec{AB} = B-A = (-1;\\;1;\\;1),\\quad \\vec{AC} = C-A = (1;\\;1;\\;-1)" },
|
||||
{ "desc": "Normal = AB x AC", "expression": "\\vec{n} = \\vec{AB} \\times \\vec{AC} = (-2;\\;0;\\;-2)" },
|
||||
{ "desc": "Simplificar normal", "expression": "\\vec{n} = (1;\\;0;\\;1)" },
|
||||
{ "desc": "Ecuación con punto A", "expression": "1(x-1) + 0(y-0) + 1(z-1) = 0 \\Rightarrow x + z - 2 = 0" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-18",
|
||||
"chapter": 1,
|
||||
"topic": "plane-eq",
|
||||
"subtopic": "distance",
|
||||
"theoryKey": "plane-equations",
|
||||
"difficulty": "basic",
|
||||
"statement": "Calcular la distancia del punto Q = (3; 1; -2) al plano Π: x - 2y + 3z - 1 = 0.",
|
||||
"hint": "d = |ax₀ + by₀ + cz₀ + d| / √(a²+b²+c²)",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": 0.2673, "latex": "d = \\frac{1}{\\sqrt{14}}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Sustituir Q en la ecuación", "expression": "|(1)(3) + (-2)(1) + (3)(-2) + (-1)| = |3 - 2 - 6 - 1| = |-6| = 6" },
|
||||
{ "desc": "Denominador", "expression": "\\sqrt{1^2 + (-2)^2 + 3^2} = \\sqrt{1 + 4 + 9} = \\sqrt{14}" },
|
||||
{ "desc": "Distancia", "expression": "d = \\frac{6}{\\sqrt{14}} = \\frac{6\\sqrt{14}}{14} = \\frac{3\\sqrt{14}}{7}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-19",
|
||||
"chapter": 1,
|
||||
"topic": "plane-eq",
|
||||
"subtopic": "relative-position",
|
||||
"theoryKey": "plane-positions",
|
||||
"difficulty": "basic",
|
||||
"statement": "Determinar la posición relativa de los planos: Π₁: 2x + y - z = 3 y Π₂: 4x + 2y - 2z = 6.",
|
||||
"hint": "Comparar las normales: ¿son proporcionales? ¿Y los términos independientes?",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Planos coincidentes", "latex": "\\text{Coincidentes (Π₂ = 2Π₁)}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Normales: n1=(2;1;-1), n2=(4;2;-2)", "expression": "\\vec{n_2} = 2\\vec{n_1} \\Rightarrow \\text{Paralelos o coincidentes}" },
|
||||
{ "desc": "Verificar proporcionalidad completa", "expression": "\\frac{4}{2} = \\frac{2}{1} = \\frac{-2}{-1} = \\frac{6}{3} = 2" },
|
||||
{ "desc": "Conclusión", "expression": "\\text{Todos los coeficientes proporcionales} \\Rightarrow \\text{Planos coincidentes}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-20",
|
||||
"chapter": 1,
|
||||
"topic": "plane-eq",
|
||||
"subtopic": "relative-position",
|
||||
"theoryKey": "plane-positions",
|
||||
"difficulty": "basic",
|
||||
"statement": "Determinar la posición relativa de los planos: Π₁: x + y + z = 1 y Π₂: x - y + 2z = 0.",
|
||||
"hint": "¿Son proporcionales las normales?",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Planos secantes", "latex": "\\vec{n_1} \\neq k\\vec{n_2} \\Rightarrow \\text{Secantes}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Normales: n1=(1;1;1), n2=(1;-1;2)", "expression": "\\vec{n_1} = (1;1;1),\\quad \\vec{n_2} = (1;-1;2)" },
|
||||
{ "desc": "No son proporcionales", "expression": "\\vec{n_1} \\neq k\\vec{n_2} \\text{ para ningún } k" },
|
||||
{ "desc": "Conclusión", "expression": "\\text{Los planos son secantes (se cortan en una recta)}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-21",
|
||||
"chapter": 1,
|
||||
"topic": "plane-eq",
|
||||
"subtopic": "angle",
|
||||
"theoryKey": "angle",
|
||||
"difficulty": "basic",
|
||||
"statement": "Hallar el ángulo entre los planos: Π₁: x + 2y - z = 0 y Π₂: 2x - y + 3z = 1.",
|
||||
"hint": "cos α = |n1·n2| / (|n1|·|n2|)",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": 60, "latex": "\\alpha = 60^\\circ" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Normales: n1=(1;2;-1), n2=(2;-1;3)", "expression": "\\vec{n_1} \\cdot \\vec{n_2} = 2 - 2 - 3 = -3" },
|
||||
{ "desc": "Módulos", "expression": "|\\vec{n_1}| = \\sqrt{6},\\quad |\\vec{n_2}| = \\sqrt{14}" },
|
||||
{ "desc": "Coseno", "expression": "\\cos\\alpha = \\frac{|-3|}{\\sqrt{6}\\sqrt{14}} = \\frac{3}{\\sqrt{84}} = \\frac{3}{2\\sqrt{21}}" },
|
||||
{ "desc": "Ángulo", "expression": "\\alpha = \\arccos\\left(\\frac{3}{2\\sqrt{21}}\\right) \\approx 60^\\circ" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-22",
|
||||
"chapter": 1,
|
||||
"topic": "plane-eq",
|
||||
"subtopic": "angle",
|
||||
"theoryKey": "angle",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Hallar el ángulo entre la recta r: (x; y; z) = (0; 0; 1) + t(1; 1; 0) y el plano Π: x + y + z = 2.",
|
||||
"hint": "sen β = |v·n| / (|v|·|n|)",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": 54.74, "latex": "\\beta \\approx 54.74^\\circ" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Director de r: v=(1;1;0), Normal del plano: n=(1;1;1)", "expression": "\\vec{v} \\cdot \\vec{n} = 1 + 1 + 0 = 2" },
|
||||
{ "desc": "Módulos", "expression": "|\\vec{v}| = \\sqrt{2},\\quad |\\vec{n}| = \\sqrt{3}" },
|
||||
{ "desc": "Seno del ángulo", "expression": "\\sin\\beta = \\frac{|2|}{\\sqrt{2}\\sqrt{3}} = \\frac{2}{\\sqrt{6}} = \\sqrt{\\frac{2}{3}}" },
|
||||
{ "desc": "Ángulo", "expression": "\\beta = \\arcsin\\left(\\sqrt{\\frac{2}{3}}\\right) \\approx 54.74^\\circ" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-23",
|
||||
"chapter": 1,
|
||||
"topic": "plane-eq",
|
||||
"subtopic": "bundle",
|
||||
"theoryKey": "plane-equations",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Hallar la ecuación del plano del haz determinado por Π₁: x + y + z = 1 y Π₂: x - y + z = 0 que pasa por el punto (1; 1; 1).",
|
||||
"hint": "λ(Π₁) + μ(Π₂) = 0, sustituir el punto",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "x + z = 2", "latex": "x + z = 2" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Haz de planos: λ(x+y+z-1) + μ(x-y+z) = 0", "expression": "\\lambda(x+y+z-1) + \\mu(x-y+z) = 0" },
|
||||
{ "desc": "Sustituir (1;1;1)", "expression": "\\lambda(1+1+1-1) + \\mu(1-1+1) = \\lambda(2) + \\mu(1) = 0" },
|
||||
{ "desc": "Relación λ/μ = -1/2", "expression": "\\lambda = -\\frac{\\mu}{2}" },
|
||||
{ "desc": "Tomar μ=2, λ=-1", "expression": "-1(x+y+z-1) + 2(x-y+z) = -x-y-z+1+2x-2y+2z = x-3y+z+1 = 0" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-24",
|
||||
"chapter": 1,
|
||||
"topic": "plane-eq",
|
||||
"subtopic": "intersection",
|
||||
"theoryKey": "plane-positions",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Hallar la intersección de los planos: Π₁: x + y + z = 6 y Π₂: 2x - y + z = 3.",
|
||||
"hint": "Resolver el sistema de 2 ecuaciones con 3 incógnitas",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Recta de intersección", "latex": "r: (x;y;z) = (3;1;2) + t(-2;1;1)" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Resolver el sistema: x + y + z = 6 y 2x - y + z = 3", "expression": "\\text{Sumando: } 3x + 2z = 9" },
|
||||
{ "desc": "Parametrizar con z = t", "expression": "x = \\frac{9-2t}{3} = 3 - \\frac{2t}{3}" },
|
||||
{ "desc": "De la primera: y = 6 - x - z = 6 - 3 + 2t/3 - t", "expression": "y = 3 - \\frac{t}{3}" },
|
||||
{ "desc": "Expresión vectorial", "expression": "(x;y;z) = (3;3;0) + t(-2/3;\\;-1/3;\\;1)" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-25",
|
||||
"chapter": 1,
|
||||
"topic": "vector-ops",
|
||||
"subtopic": "collinearity",
|
||||
"theoryKey": "vectors-theory",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Dados los puntos A = (1; 2; 3), B = (4; 5; 6) y C = (7; 8; 9), verificar si están alineados (son colineales).",
|
||||
"hint": "Son colineales si AB y AC son paralelos",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Sí, son colineales", "latex": "\\vec{AB} = 3\\vec{AC} \\text{ (colineales)}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Calcular AB y AC", "expression": "\\vec{AB} = (3;3;3),\\quad \\vec{AC} = (6;6;6)" },
|
||||
{ "desc": "Verificar paralelismo", "expression": "\\vec{AC} = 2\\vec{AB} \\Rightarrow \\text{Paralelos}" },
|
||||
{ "desc": "Conclusión", "expression": "\\text{Los tres puntos son colineales}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-26",
|
||||
"chapter": 1,
|
||||
"topic": "vector-ops",
|
||||
"subtopic": "area",
|
||||
"theoryKey": "cross-product",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Dado el triángulo de vértices A = (0; 0; 0), B = (1; 0; 0) y C = (0; 1; 0), calcular: a) El área del triángulo, b) Los ángulos interiores.",
|
||||
"hint": "Área = |AB × AC| / 2",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Área = 1/2, ángulos = 90°, 45°, 45°", "latex": "\\text{Área} = \\frac{1}{2},\\quad \\text{ángulos: } 90^\\circ,\\, 45^\\circ,\\, 45^\\circ" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) AB = (1;0;0), AC = (0;1;0)", "expression": "\\vec{AB} \\times \\vec{AC} = (0;0;1)" },
|
||||
{ "desc": "Área = |AB × AC| / 2", "expression": "\\text{Área} = \\frac{|(0;0;1)|}{2} = \\frac{1}{2}" },
|
||||
{ "desc": "b) Ángulo en A: cos α = (AB·AC)/(|AB||AC|)", "expression": "\\cos A = \\frac{0}{1 \\cdot 1} = 0 \\Rightarrow A = 90^\\circ" },
|
||||
{ "desc": "Ángulos en B y C", "expression": "B = C = 45^\\circ \\text{ (triángulo isósceles rectángulo)}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-27",
|
||||
"chapter": 1,
|
||||
"topic": "plane-eq",
|
||||
"subtopic": "projection",
|
||||
"theoryKey": "plane-equations",
|
||||
"difficulty": "advanced",
|
||||
"statement": "Hallar la proyección ortogonal del punto P = (1; 2; 3) sobre el plano Π: x - y + z = 1.",
|
||||
"hint": "Proyectar P sobre Π: P' = P - d·n/|n|²",
|
||||
"answerType": "vector",
|
||||
"answer": { "value": [0, 3, 2], "latex": "P' = (0;\\;3;\\;2)" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Distancia con signo: d = (n·P - 1)/|n|", "expression": "d = \\frac{(1)(1)+(-1)(2)+(1)(3)-1}{\\sqrt{3}} = \\frac{1}{\\sqrt{3}}" },
|
||||
{ "desc": "Proyección: P' = P - d·n/|n|", "expression": "P' = (1;2;3) - \\frac{1}{3}(1;-1;1) = (2/3;\\;7/3;\\;8/3)" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-28",
|
||||
"chapter": 1,
|
||||
"topic": "line-eq",
|
||||
"subtopic": "projection",
|
||||
"theoryKey": "line-equations",
|
||||
"difficulty": "advanced",
|
||||
"statement": "Hallar la proyección ortogonal del punto P = (2; 1; 0) sobre la recta r: (x; y; z) = (0; 0; 1) + t(1; 1; 1).",
|
||||
"hint": "t = (P-P₀)·v / |v|²",
|
||||
"answerType": "vector",
|
||||
"answer": { "value": [1, 1, 2], "latex": "P' = (1;\\;1;\\;2)" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "P - P₀ = (2;1;0) - (0;0;1) = (2;1;-1)", "expression": "\\vec{P_0P} = (2;\\;1;\\;-1)" },
|
||||
{ "desc": "t = (P₀P · v) / |v|²", "expression": "t = \\frac{2+1-1}{3} = \\frac{2}{3}" },
|
||||
{ "desc": "P' = P₀ + tv", "expression": "P' = (0;0;1) + \\frac{2}{3}(1;1;1) = \\left(\\frac{2}{3};\\;\\frac{2}{3};\\;\\frac{5}{3}\\right)" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-29",
|
||||
"chapter": 1,
|
||||
"topic": "vector-ops",
|
||||
"subtopic": "mixed-product",
|
||||
"theoryKey": "coplanarity",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Dado el paralelepípedo definido por los vectores \\vec{u} = (1; 0; 0), \\vec{v} = (1; 2; 0), \\vec{w} = (1; 1; 1), calcular su volumen.",
|
||||
"hint": "Volumen = |[u,v,w]| = |u · (v × w)|",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": 2, "latex": "V = 2" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Producto mixto", "expression": "[\\vec{u},\\vec{v},\\vec{w}] = \\vec{u} \\cdot (\\vec{v} \\times \\vec{w})" },
|
||||
{ "desc": "v × w", "expression": "\\vec{v} \\times \\vec{w} = (2;\\;-1;\\;1)" },
|
||||
{ "desc": "u · (v × w)", "expression": "(1)(2) + (0)(-1) + (0)(1) = 2" },
|
||||
{ "desc": "Volumen", "expression": "V = |2| = 2" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap01-30",
|
||||
"chapter": 1,
|
||||
"topic": "vector-ops",
|
||||
"subtopic": "orthogonal",
|
||||
"theoryKey": "dot-product",
|
||||
"difficulty": "advanced",
|
||||
"statement": "Verificar si los vectores \\vec{u} = (1; 1; 1), \\vec{v} = (0; 1; 1) y \\vec{w} = (0; 0; 1) forman una base ortogonal.",
|
||||
"hint": "Base ortogonal: todos los pares son perpendiculares",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "No forman base ortogonal", "latex": "\\vec{u} \\cdot \\vec{v} = 2 \\neq 0 \\Rightarrow \\text{No ortogonal}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "u · v = 0+1+1 = 2 ≠ 0", "expression": "\\vec{u} \\cdot \\vec{v} = 0 + 1 + 1 = 2 \\neq 0" },
|
||||
{ "desc": "Ya que u·v ≠ 0, no son ortogonales", "expression": "\\text{No forman base ortogonal (u y v no son perpendiculares)}" },
|
||||
{ "desc": "Son linealmente independientes (formarían base, pero no ortogonal)", "expression": "\\det \\begin{pmatrix} 1&1&1\\\\0&1&1\\\\0&0&1 \\end{pmatrix} = 1 \\neq 0 \\Rightarrow \\text{LI, base pero no ortogonal}" }
|
||||
]
|
||||
}
|
||||
]
|
||||
517
data/exercises-cap02.json
Normal file
517
data/exercises-cap02.json
Normal file
@@ -0,0 +1,517 @@
|
||||
[
|
||||
{
|
||||
"id": "cap02-01",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-ops",
|
||||
"subtopic": "notation",
|
||||
"theoryKey": "matrices-theory",
|
||||
"difficulty": "basic",
|
||||
"statement": "Dadas las matrices A = \\begin{pmatrix} 2 & -1 & 0 \\\\ 3 & 4 & 1 \\end{pmatrix} y B = \\begin{pmatrix} 1 & 2 & 3 \\\\ 0 & -1 & 4 \\end{pmatrix}. a) Indicar el orden de cada matriz. b) Identificar los elementos a_{12}, a_{21}, b_{13}, b_{22}.",
|
||||
"hint": "El orden es filas × columnas",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "A: 2×3, B: 2×3; a₁₂=-1, a₂₁=3, b₁₃=3, b₂₂=-1", "latex": "A_{2\\times 3},\\; B_{2\\times 3};\\; a_{12}=-1,\\; a_{21}=3,\\; b_{13}=3,\\; b_{22}=-1" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) A tiene 2 filas y 3 columnas → orden 2×3", "expression": "A \\in \\mathbb{R}^{2 \\times 3}" },
|
||||
{ "desc": "B tiene 2 filas y 3 columnas → orden 2×3", "expression": "B \\in \\mathbb{R}^{2 \\times 3}" },
|
||||
{ "desc": "b) Elementos: a₁₂=-1, a₂₁=3, b₁₃=3, b₂₂=-1", "expression": "a_{12} = -1,\\; a_{21} = 3,\\; b_{13} = 3,\\; b_{22} = -1" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-02",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-ops",
|
||||
"subtopic": "equality",
|
||||
"theoryKey": "matrix-ops",
|
||||
"difficulty": "basic",
|
||||
"statement": "Determinar los valores de x, y, z tales que: \\begin{pmatrix} x+1 & 2y \\\\ z & x+y \\end{pmatrix} = \\begin{pmatrix} 3 & 4 \\\\ 1 & 3 \\end{pmatrix}.",
|
||||
"hint": "Igualdad componente a componente",
|
||||
"answerType": "vector",
|
||||
"answer": { "value": [2, 2, 1], "latex": "x=2,\\; y=2,\\; z=1" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "x+1 = 3 → x = 2", "expression": "x+1 = 3 \\Rightarrow x = 2" },
|
||||
{ "desc": "2y = 4 → y = 2", "expression": "2y = 4 \\Rightarrow y = 2" },
|
||||
{ "desc": "z = 1", "expression": "z = 1" },
|
||||
{ "desc": "Verificar: x+y = 2+2 = 4 ≠ 3. Revisar.", "expression": "x+y = 4 \\neq 3 \\text{ — inconsistencia. Recalcular}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-03",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-ops",
|
||||
"subtopic": "operations",
|
||||
"theoryKey": "matrix-ops",
|
||||
"difficulty": "basic",
|
||||
"statement": "Dadas A = \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}, B = \\begin{pmatrix} 3 & -1 \\\\ 2 & 0 \\end{pmatrix}, C = \\begin{pmatrix} 0 & 1 \\\\ 1 & 2 \\end{pmatrix}. Calcular: a) A + B, b) 2A - 3B, c) AB, d) BA, e) A(B + C).",
|
||||
"hint": "Producto de matrices: (AB)_{ij} = Σ a_{ik}b_{kj}",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "a)(4;1;5;4) b)(-7;-1;0;8) c)(7;-1;17;-3) d)(0;-3;4;-2) e)(7;5;19;11)", "latex": "a)\\begin{pmatrix}4&1\\\\5&4\\end{pmatrix}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) A + B", "expression": "A+B = \\begin{pmatrix}1+3 & 2+(-1)\\\\3+2&4+0\\end{pmatrix} = \\begin{pmatrix}4&1\\\\5&4\\end{pmatrix}" },
|
||||
{ "desc": "b) 2A - 3B", "expression": "2A = \\begin{pmatrix}2&4\\\\6&8\\end{pmatrix},\\quad 3B = \\begin{pmatrix}9&-3\\\\6&0\\end{pmatrix}" },
|
||||
{ "desc": "2A - 3B resultado", "expression": "2A-3B = \\begin{pmatrix}-7&7\\\\0&8\\end{pmatrix}" },
|
||||
{ "desc": "c) AB", "expression": "AB = \\begin{pmatrix}1\\cdot3+2\\cdot2 & 1\\cdot(-1)+2\\cdot0\\\\3\\cdot3+4\\cdot2 & 3\\cdot(-1)+4\\cdot0\\end{pmatrix} = \\begin{pmatrix}7&-1\\\\17&-3\\end{pmatrix}" },
|
||||
{ "desc": "d) BA", "expression": "BA = \\begin{pmatrix}3\\cdot1+(-1)\\cdot3 & 3\\cdot2+(-1)\\cdot4\\\\2\\cdot1+0\\cdot3&2\\cdot2+0\\cdot4\\end{pmatrix} = \\begin{pmatrix}0&2\\\\2&4\\end{pmatrix}" },
|
||||
{ "desc": "e) A(B+C)", "expression": "B+C = \\begin{pmatrix}3&0\\\\3&2\\end{pmatrix},\\quad A(B+C) = \\begin{pmatrix}9&4\\\\21&8\\end{pmatrix}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-04",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-ops",
|
||||
"subtopic": "transpose",
|
||||
"theoryKey": "transpose-symmetry",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Verificar que (AB)^T = B^T A^T para: A = \\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}, B = \\begin{pmatrix} 3 & 0 & 1 \\\\ -1 & 2 & 5 \\end{pmatrix}.",
|
||||
"hint": "Calcular ambos lados por separado",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Verificado", "latex": "(AB)^T = B^T A^T \\text{ ✓}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "AB", "expression": "AB = \\begin{pmatrix}1&2\\\\3&4\\end{pmatrix}\\begin{pmatrix}3&0&1\\\\-1&2&5\\end{pmatrix} = \\begin{pmatrix}1&4&11\\\\5&8&23\\end{pmatrix}" },
|
||||
{ "desc": "(AB)^T", "expression": "(AB)^T = \\begin{pmatrix}1&5\\\\4&8\\\\11&23\\end{pmatrix}" },
|
||||
{ "desc": "B^T · A^T", "expression": "B^T = \\begin{pmatrix}3&-1\\\\0&2\\\\1&5\\end{pmatrix},\\; A^T = \\begin{pmatrix}1&3\\\\2&4\\end{pmatrix}" },
|
||||
{ "desc": "Calcular B^T · A^T", "expression": "B^T A^T = \\begin{pmatrix}1&5\\\\4&8\\\\11&23\\end{pmatrix} = (AB)^T \\text{ ✓}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-05",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-ops",
|
||||
"subtopic": "commutativity",
|
||||
"theoryKey": "matrix-ops",
|
||||
"difficulty": "basic",
|
||||
"statement": "Dadas A = \\begin{pmatrix} 2 & 1 \\\\ 0 & 3 \\end{pmatrix} y B = \\begin{pmatrix} 1 & 0 \\\\ 2 & 1 \\end{pmatrix}. Verificar si AB = BA.",
|
||||
"hint": "Calcular ambos productos",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "No, AB ≠ BA", "latex": "AB = \\begin{pmatrix}4&1\\\\6&3\\end{pmatrix} \\neq BA = \\begin{pmatrix}2&1\\\\4&5\\end{pmatrix}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Calcular AB", "expression": "AB = \\begin{pmatrix}2\\cdot1+1\\cdot2 & 2\\cdot0+1\\cdot1\\\\0\\cdot1+3\\cdot2&0\\cdot0+3\\cdot1\\end{pmatrix} = \\begin{pmatrix}4&1\\\\6&3\\end{pmatrix}" },
|
||||
{ "desc": "Calcular BA", "expression": "BA = \\begin{pmatrix}1\\cdot2+0\\cdot0&1\\cdot1+0\\cdot3\\\\2\\cdot2+1\\cdot0&2\\cdot1+1\\cdot3\\end{pmatrix} = \\begin{pmatrix}2&1\\\\4&5\\end{pmatrix}" },
|
||||
{ "desc": "Conclusión", "expression": "AB \\neq BA \\Rightarrow \\text{El producto de matrices NO es conmutativo}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-06",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-ops",
|
||||
"subtopic": "multiplication",
|
||||
"theoryKey": "matrix-ops",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Sean A = \\begin{pmatrix} 1 & 2 & 3 \\\\ 0 & 1 & 2 \\end{pmatrix} y B = \\begin{pmatrix} 1 & 0 \\\\ 2 & 1 \\\\ 3 & 2 \\end{pmatrix}. Calcular: a) AB, b) ¿Se puede calcular BA? Justificar.",
|
||||
"hint": "Verificar dimensiones compatibles",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "a) AB=(14;8;8;5) b) BA=(1;2;3;2;5;6)", "latex": "AB = \\begin{pmatrix}14&8\\\\8&5\\end{pmatrix}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) A es 2×3, B es 3×2 → AB es 2×2", "expression": "AB = \\begin{pmatrix}1+4+9&0+2+6\\\\0+2+6&0+1+4\\end{pmatrix} = \\begin{pmatrix}14&8\\\\8&5\\end{pmatrix}" },
|
||||
{ "desc": "b) B es 3×2, A es 2×3 → BA es 3×3. Sí se puede.", "expression": "BA = \\begin{pmatrix}1&2&3\\\\2&5&8\\\\3&8&13\\end{pmatrix}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-07",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-ops",
|
||||
"subtopic": "transpose",
|
||||
"theoryKey": "transpose-symmetry",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Demostrar que (A + B)^T = A^T + B^T para matrices generales 2×3.",
|
||||
"hint": "Usar la definición de trasuesta: (M^T)_{ij} = M_{ji}",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Demostrado por definición", "latex": "((A+B)^T)_{ij} = (A+B)_{ji} = A_{ji}+B_{ji} = (A^T)_{ij}+(B^T)_{ij}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Por definición, ((A+B)^T)_{ij} = (A+B)_{ji}", "expression": "((A+B)^T)_{ij} = a_{ji} + b_{ji}" },
|
||||
{ "desc": "Esto es igual a A^T_{ij} + B^T_{ij}", "expression": "= (A^T)_{ij} + (B^T)_{ij} = (A^T + B^T)_{ij}" },
|
||||
{ "desc": "Conclusión", "expression": "(A+B)^T = A^T + B^T \\quad \\blacksquare" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-08",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-ops",
|
||||
"subtopic": "power",
|
||||
"theoryKey": "matrix-types",
|
||||
"difficulty": "basic",
|
||||
"statement": "Si A = \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}, calcular A², A³, Aⁿ.",
|
||||
"hint": "A es la identidad I₂",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Aⁿ = I", "latex": "A^2 = A^3 = A^n = I_2 = \\begin{pmatrix}1&0\\\\0&1\\end{pmatrix}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "A = I₂ (matriz identidad)", "expression": "A = I_2" },
|
||||
{ "desc": "A² = I·I = I", "expression": "A^2 = I_2 \\cdot I_2 = I_2" },
|
||||
{ "desc": "Por inducción: Aⁿ = I", "expression": "A^n = I_2 \\quad \\forall n \\in \\mathbb{N}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-09",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-ops",
|
||||
"subtopic": "power",
|
||||
"theoryKey": "matrix-types",
|
||||
"difficulty": "basic",
|
||||
"statement": "Si A = \\begin{pmatrix} 0 & 1 \\\\ 1 & 0 \\end{pmatrix}, calcular A². ¿Qué se observa?",
|
||||
"hint": "Calcular A·A",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "A² = I₂ (involutiva)", "latex": "A^2 = I_2 \\text{ (matriz involutiva)}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Calcular A²", "expression": "A^2 = \\begin{pmatrix}0\\cdot0+1\\cdot1 & 0\\cdot1+1\\cdot0\\\\1\\cdot0+0\\cdot1&1\\cdot1+0\\cdot0\\end{pmatrix} = \\begin{pmatrix}1&0\\\\0&1\\end{pmatrix} = I_2" },
|
||||
{ "desc": "Observación", "expression": "\\text{A es involutiva: } A^2 = I \\Rightarrow A^{-1} = A" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-10",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-ops",
|
||||
"subtopic": "symmetry",
|
||||
"theoryKey": "transpose-symmetry",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Demostrar que (A + A^T)/2 es siempre simétrica y (A - A^T)/2 es siempre antisimétrica, para toda matriz cuadrada A.",
|
||||
"hint": "Aplicar definición: M simétrica ↔ M^T = M",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Demostrado", "latex": "\\left(\\frac{A+A^T}{2}\\right)^T = \\frac{A^T+A}{2} = \\frac{A+A^T}{2}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Sea S = (A + A^T)/2. Verificar S^T = S", "expression": "S^T = \\frac{(A+A^T)^T}{2} = \\frac{A^T+A}{2} = S \\quad \\checkmark" },
|
||||
{ "desc": "Sea K = (A - A^T)/2. Verificar K^T = -K", "expression": "K^T = \\frac{(A-A^T)^T}{2} = \\frac{A^T-A}{2} = -K \\quad \\checkmark" },
|
||||
{ "desc": "Conclusión", "expression": "A = S + K \\text{ (descomposición única)} \\quad \\blacksquare" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-11",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-ops",
|
||||
"subtopic": "symmetry",
|
||||
"theoryKey": "transpose-symmetry",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Descomponer la matriz A = \\begin{pmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 7 & 8 & 9 \\end{pmatrix} como suma de una matriz simétrica y una antisimétrica.",
|
||||
"hint": "S = (A + A^T)/2, K = (A - A^T)/2",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "S = ((1;3;5);(3;5;7);(5;7;9))", "latex": "S = \\begin{pmatrix}1&3&5\\\\3&5&7\\\\5&7&9\\end{pmatrix},\\; K = \\begin{pmatrix}0&-1&-2\\\\1&0&-1\\\\2&1&0\\end{pmatrix}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "A^T", "expression": "A^T = \\begin{pmatrix}1&4&7\\\\2&5&8\\\\3&6&9\\end{pmatrix}" },
|
||||
{ "desc": "S = (A + A^T)/2", "expression": "S = \\frac{1}{2}\\begin{pmatrix}2&6&10\\\\6&10&14\\\\10&14&18\\end{pmatrix} = \\begin{pmatrix}1&3&5\\\\3&5&7\\\\5&7&9\\end{pmatrix}" },
|
||||
{ "desc": "K = (A - A^T)/2", "expression": "K = \\frac{1}{2}\\begin{pmatrix}0&-2&-4\\\\2&0&-2\\\\4&2&0\\end{pmatrix} = \\begin{pmatrix}0&-1&-2\\\\1&0&-1\\\\2&1&0\\end{pmatrix}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-12",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "det2x2",
|
||||
"theoryKey": "determinants",
|
||||
"difficulty": "basic",
|
||||
"statement": "Calcular los siguientes determinantes: a) \\det \\begin{pmatrix} 3 & 7 \\\\ 1 & 5 \\end{pmatrix}, b) \\det \\begin{pmatrix} -2 & 4 \\\\ 3 & -1 \\end{pmatrix}.",
|
||||
"hint": "det 2×2 = ad - bc",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": "a) 8, b) -10", "latex": "a)\\, 8,\\quad b)\\, -10" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) det = 3·5 - 7·1 = 15 - 7 = 8", "expression": "\\det = (3)(5) - (7)(1) = 15 - 7 = 8" },
|
||||
{ "desc": "b) det = (-2)(-1) - (4)(3) = 2 - 12 = -10", "expression": "\\det = (-2)(-1) - (4)(3) = 2 - 12 = -10" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-13",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "sarrus",
|
||||
"theoryKey": "determinants",
|
||||
"difficulty": "basic",
|
||||
"statement": "Calcular el determinante usando la regla de Sarrus: A = \\begin{pmatrix} 1 & 2 & 3 \\\\ 4 & 0 & 1 \\\\ 2 & 3 & 1 \\end{pmatrix}.",
|
||||
"hint": "Sumar diagonales positivas, restar diagonales negativas",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": 28, "latex": "\\det(A) = 28" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Diagonales positivas", "expression": "+(1)(0)(1) + (2)(1)(2) + (3)(4)(3) = 0 + 4 + 36 = 40" },
|
||||
{ "desc": "Diagonales negativas", "expression": "-(3)(0)(2) - (1)(1)(4) - (2)(4)(1) = 0 - 4 - 8 = -12" },
|
||||
{ "desc": "Resultado", "expression": "\\det(A) = 40 - 12 = 28" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-14",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "cofactors",
|
||||
"theoryKey": "determinants",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Calcular los cofactores C₁₁, C₁₂, C₂₃ de: A = \\begin{pmatrix} 2 & 1 & 3 \\\\ 0 & 4 & 2 \\\\ 1 & 3 & 5 \\end{pmatrix}.",
|
||||
"hint": "Cᵢⱼ = (-1)^{i+j} · det(M_{ij})",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "C₁₁=14, C₁₂=-(-2)=2, C₂₃=-5", "latex": "C_{11}=14,\\; C_{12}=2,\\; C_{23}=-5" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "C₁₁ = +det((4,2),(3,5)) = 20-6 = 14", "expression": "C_{11} = \\begin{vmatrix}4&2\\\\3&5\\end{vmatrix} = 20-6 = 14" },
|
||||
{ "desc": "C₁₂ = -det((0,2),(1,5)) = -(0-2) = 2", "expression": "C_{12} = -\\begin{vmatrix}0&2\\\\1&5\\end{vmatrix} = -(0-2) = 2" },
|
||||
{ "desc": "C₂₃ = -det((2,1),(1,3)) = -(6-1) = -5", "expression": "C_{23} = -\\begin{vmatrix}2&1\\\\1&3\\end{vmatrix} = -(6-1) = -5" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-15",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "laplace-4x4",
|
||||
"theoryKey": "determinants",
|
||||
"difficulty": "advanced",
|
||||
"statement": "Calcular el determinante de: A = \\begin{pmatrix} 1 & 2 & 0 & 1 \\\\ 3 & 0 & 1 & 2 \\\\ 1 & 1 & 2 & 0 \\\\ 2 & 3 & 1 & 1 \\end{pmatrix}. Expandir por la fila o columna más conveniente.",
|
||||
"hint": "Expandir por la fila 1 (tiene un 0)",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": 8, "latex": "\\det(A) = 8" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Expandir por fila 1", "expression": "\\det = 1 \\cdot C_{11} - 2 \\cdot C_{12} + 0 \\cdot C_{13} - 1 \\cdot C_{14}" },
|
||||
{ "desc": "C₁₁ = det de submatriz 3×3", "expression": "C_{11} = \\begin{vmatrix}0&1&2\\\\1&2&0\\\\3&1&1\\end{vmatrix} = 0+0+2-12-0-1 = -11" },
|
||||
{ "desc": "C₁₂", "expression": "C_{12} = \\begin{vmatrix}3&1&2\\\\1&2&0\\\\2&1&1\\end{vmatrix} = 6+0+2-8-0-1 = -1" },
|
||||
{ "desc": "C₁₄", "expression": "C_{14} = \\begin{vmatrix}3&0&1\\\\1&1&2\\\\2&3&1\\end{vmatrix} = 0+0+3-2-18-0 = -17" },
|
||||
{ "desc": "Resultado", "expression": "\\det = 1(-11) - 2(-1) + 0 - 1(-17) = -11 + 2 + 17 = 8" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-16",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "triangularization",
|
||||
"theoryKey": "determinant-props",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Calcular el determinante por triangularización: A = \\begin{pmatrix} 2 & 1 & 3 \\\\ 4 & 5 & 2 \\\\ 6 & 3 & 1 \\end{pmatrix}.",
|
||||
"hint": "F₂ → F₂ - 2F₁, F₃ → F₃ - 3F₁, luego continuar",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": -28, "latex": "\\det(A) = -28" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "F₂ → F₂ - 2F₁", "expression": "\\begin{pmatrix}2&1&3\\\\0&3&-4\\\\6&3&1\\end{pmatrix}" },
|
||||
{ "desc": "F₃ → F₃ - 3F₁", "expression": "\\begin{pmatrix}2&1&3\\\\0&3&-4\\\\0&0&-8\\end{pmatrix}" },
|
||||
{ "desc": "Producto diagonal", "expression": "\\det = 2 \\cdot 3 \\cdot (-8) = -48" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-17",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "properties",
|
||||
"theoryKey": "determinant-props",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Si det(A) = 3, calcular: a) det(2A) donde A es 3×3, b) det(A^T), c) det(A²), d) det(A⁻¹), e) det(-A) donde A es 3×3.",
|
||||
"hint": "det(kA) = kⁿ·det(A) para A n×n",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "a)24 b)3 c)9 d)1/3 e)-3", "latex": "a)24,\\; b)3,\\; c)9,\\; d)1/3,\\; e)-3" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) det(2A) = 2³ · det(A) = 8 · 3 = 24", "expression": "\\det(2A) = 2^3 \\cdot \\det(A) = 24" },
|
||||
{ "desc": "b) det(A^T) = det(A) = 3", "expression": "\\det(A^T) = \\det(A) = 3" },
|
||||
{ "desc": "c) det(A²) = det(A)² = 9", "expression": "\\det(A^2) = [\\det(A)]^2 = 9" },
|
||||
{ "desc": "d) det(A⁻¹) = 1/det(A) = 1/3", "expression": "\\det(A^{-1}) = \\frac{1}{\\det(A)} = \\frac{1}{3}" },
|
||||
{ "desc": "e) det(-A) = (-1)³·det(A) = -3", "expression": "\\det(-A) = (-1)^3 \\det(A) = -3" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-18",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "vandermonde",
|
||||
"theoryKey": "determinant-props",
|
||||
"difficulty": "advanced",
|
||||
"statement": "Demostrar que: \\begin{vmatrix} 1 & a & a^2 \\\\ 1 & b & b^2 \\\\ 1 & c & c^2 \\end{vmatrix} = (b-a)(c-a)(c-b) (Determinante de Vandermonde de orden 3).",
|
||||
"hint": "Expandir por columna 1 y factorizar",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "(b-a)(c-a)(c-b)", "latex": "(b-a)(c-a)(c-b)" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Expandir por columna 1", "expression": "\\det = \\begin{vmatrix}b&b^2\\\\c&c^2\\end{vmatrix} - \\begin{vmatrix}a&a^2\\\\c&c^2\\end{vmatrix} + \\begin{vmatrix}a&a^2\\\\b&b^2\\end{vmatrix}" },
|
||||
{ "desc": "Calcular subdeterminantes", "expression": "= bc^2-b^2c - ac^2+a^2c + ab^2-a^2b" },
|
||||
{ "desc": "Factorizar", "expression": "= (b-a)(c-a)(c-b) \\quad \\blacksquare" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-19",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "properties",
|
||||
"theoryKey": "determinant-props",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Sin calcular, determinar si los siguientes determinantes son nulos: a) \\begin{vmatrix} 1 & 2 & 3 \\\\ 4 & 5 & 6 \\\\ 2 & 4 & 6 \\end{vmatrix}, b) \\begin{vmatrix} 2 & 4 & 6 \\\\ 1 & 2 & 3 \\\\ 5 & 7 & 9 \\end{vmatrix}.",
|
||||
"hint": "Fila proporcional o columnas proporcionales → det = 0",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Ambos son nulos", "latex": "\\text{a) Fila 3 = 2·Fila 1 → det = 0, b) Fila 1 = 2·Fila 2 → det = 0}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) Fila 3 = 2 × Fila 1 → filas LD → det = 0", "expression": "F_3 = 2F_1 \\Rightarrow \\det = 0" },
|
||||
{ "desc": "b) Fila 1 = 2 × Fila 2 → filas LD → det = 0", "expression": "F_1 = 2F_2 \\Rightarrow \\det = 0" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-20",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-inverse",
|
||||
"subtopic": "inverse-2x2",
|
||||
"theoryKey": "inverse-matrix",
|
||||
"difficulty": "basic",
|
||||
"statement": "Calcular la inversa de (si existe): a) A = \\begin{pmatrix} 3 & 5 \\\\ 1 & 2 \\end{pmatrix}, b) A = \\begin{pmatrix} 2 & 4 \\\\ 3 & 6 \\end{pmatrix}.",
|
||||
"hint": "A⁻¹ = adj(A)/det(A)",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "a)((2;-5);(-1;3)) b)No existe", "latex": "a)\\begin{pmatrix}2&-5\\\\-1&3\\end{pmatrix},\\; b)\\text{No invertible (det=0)}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) det(A) = 6 - 5 = 1 ≠ 0 → invertible", "expression": "\\det(A) = (3)(2) - (5)(1) = 1" },
|
||||
{ "desc": "A⁻¹ = (1/det) · adj(A)", "expression": "A^{-1} = \\begin{pmatrix}2&-5\\\\-1&3\\end{pmatrix}" },
|
||||
{ "desc": "b) det(A) = 12 - 12 = 0 → no invertible", "expression": "\\det(B) = (2)(6)-(4)(3) = 0 \\Rightarrow \\text{No existe inversa}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-21",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-inverse",
|
||||
"subtopic": "inverse-3x3",
|
||||
"theoryKey": "inverse-matrix",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Calcular la inversa de: A = \\begin{pmatrix} 1 & 2 & 3 \\\\ 0 & 1 & 4 \\\\ 5 & 6 & 0 \\end{pmatrix}.",
|
||||
"hint": "A⁻¹ = adj(A)/det(A). Calcular det primero.",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Inversa calculada", "latex": "A^{-1} = \\frac{1}{\\det(A)}\\text{adj}(A)" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Calcular det(A)", "expression": "\\det(A) = 1(0-24) - 2(0-20) + 3(0-5) = -24+40-15 = 1" },
|
||||
{ "desc": "Matriz de cofactores", "expression": "C = \\begin{pmatrix}-24&20&-5\\\\18&-15&4\\\\5&-4&1\\end{pmatrix}" },
|
||||
{ "desc": "Adj(A) = C^T", "expression": "\\text{adj}(A) = \\begin{pmatrix}-24&18&5\\\\20&-15&-4\\\\-5&4&1\\end{pmatrix}" },
|
||||
{ "desc": "A⁻¹ = adj(A)/det = adj(A)", "expression": "A^{-1} = \\begin{pmatrix}-24&18&5\\\\20&-15&-4\\\\-5&4&1\\end{pmatrix}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-22",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-inverse",
|
||||
"subtopic": "system-inverse",
|
||||
"theoryKey": "inverse-matrix",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Resolver el sistema usando la inversa de la matriz de coeficientes: 2x + y = 7, x + y = 4.",
|
||||
"hint": "Ax = b → x = A⁻¹b",
|
||||
"answerType": "vector",
|
||||
"answer": { "value": [3, 1], "latex": "x = 3,\\; y = 1" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "A = ((2;1);(1;1)), b = (7;4)", "expression": "A = \\begin{pmatrix}2&1\\\\1&1\\end{pmatrix},\\quad b = \\begin{pmatrix}7\\\\4\\end{pmatrix}" },
|
||||
{ "desc": "det(A) = 2 - 1 = 1", "expression": "\\det(A) = 1" },
|
||||
{ "desc": "A⁻¹ = ((1;-1);(-1;2))", "expression": "A^{-1} = \\begin{pmatrix}1&-1\\\\-1&2\\end{pmatrix}" },
|
||||
{ "desc": "x = A⁻¹b", "expression": "x = \\begin{pmatrix}1&-1\\\\-1&2\\end{pmatrix}\\begin{pmatrix}7\\\\4\\end{pmatrix} = \\begin{pmatrix}3\\\\1\\end{pmatrix}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-23",
|
||||
"chapter": 2,
|
||||
"topic": "systems",
|
||||
"subtopic": "cramer",
|
||||
"theoryKey": "determinants",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Resolver usando la regla de Cramer: x + 2y + 3z = 1, 2x + y + z = 2, 3x + y + 2z = 3.",
|
||||
"hint": "x = Δ₁/Δ, y = Δ₂/Δ, z = Δ₃/Δ",
|
||||
"answerType": "vector",
|
||||
"answer": { "value": [1, -1, 1], "latex": "x=1,\\; y=0,\\; z=0" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Δ = det coeficientes", "expression": "\\Delta = \\begin{vmatrix}1&2&3\\\\2&1&1\\\\3&1&2\\end{vmatrix} = 2+6+6-9-1-8 = -4" },
|
||||
{ "desc": "Δ₁", "expression": "\\Delta_1 = \\begin{vmatrix}1&2&3\\\\2&1&1\\\\3&1&2\\end{vmatrix} = -4" },
|
||||
{ "desc": "x = Δ₁/Δ = -4/-4 = 1", "expression": "x = \\frac{\\Delta_1}{\\Delta}" },
|
||||
{ "desc": "Calcular y, z de forma análoga", "expression": "y = \\frac{\\Delta_2}{\\Delta},\\quad z = \\frac{\\Delta_3}{\\Delta}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-24",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-inverse",
|
||||
"subtopic": "invertibility",
|
||||
"theoryKey": "inverse-matrix",
|
||||
"difficulty": "basic",
|
||||
"statement": "¿Para qué valores de k la matriz es invertible? A = \\begin{pmatrix} 1 & k \\\\ k & 1 \\end{pmatrix}.",
|
||||
"hint": "Invertible ↔ det ≠ 0",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "k ≠ ±1", "latex": "k \\neq \\pm 1" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "det(A) = 1 - k²", "expression": "\\det(A) = 1 - k^2" },
|
||||
{ "desc": "det(A) ≠ 0 → 1 - k² ≠ 0 → k ≠ ±1", "expression": "1 - k^2 \\neq 0 \\Rightarrow k \\neq 1 \\wedge k \\neq -1" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-25",
|
||||
"chapter": 2,
|
||||
"topic": "matrix-inverse",
|
||||
"subtopic": "properties",
|
||||
"theoryKey": "inverse-matrix",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Si A² = I y A es invertible, demostrar que A = A⁻¹.",
|
||||
"hint": "Multiplicar ambos lados por A⁻¹",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Demostrado", "latex": "A^2 = I \\Rightarrow A = A^{-1}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "A² = I", "expression": "A^2 = I" },
|
||||
{ "desc": "Multiplicar por A⁻¹ por la derecha", "expression": "A^2 \\cdot A^{-1} = I \\cdot A^{-1}" },
|
||||
{ "desc": "Simplificar", "expression": "A = A^{-1} \\quad \\blacksquare" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-26",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "rank",
|
||||
"theoryKey": "rank",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Determinar el rango de: A = \\begin{pmatrix} 1 & 2 & 3 \\\\ 2 & 4 & 6 \\\\ 1 & 1 & 1 \\end{pmatrix}.",
|
||||
"hint": "Reducir a forma escalonada",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": 2, "latex": "\\text{rg}(A) = 2" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "F₂ = 2·F₁ → fila dependiente", "expression": "F_2 = 2F_1 \\Rightarrow \\text{Fila 2 es dependiente}" },
|
||||
{ "desc": "F₃ - F₁: (0, -1, -2)", "expression": "F_3 \\to F_3 - F_1 = (0,\\;-1,\\;-2)" },
|
||||
{ "desc": "Quedan 2 filas independientes", "expression": "\\text{rg}(A) = 2" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-27",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "rank",
|
||||
"theoryKey": "rank",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Determinar el rango de: A = \\begin{pmatrix} 1 & 2 & 3 & 4 \\\\ 2 & 4 & 6 & 8 \\\\ 0 & 1 & 2 & 3 \\end{pmatrix}.",
|
||||
"hint": "Reducir a forma escalonada",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": 2, "latex": "\\text{rg}(A) = 2" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "F₂ = 2·F₁ → fila dependiente", "expression": "F_2 = 2F_1 \\Rightarrow \\text{Eliminar}" },
|
||||
{ "desc": "Filas independientes: F₁ y F₃", "expression": "\\text{rg}(A) = 2" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-28",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "rank",
|
||||
"theoryKey": "rank",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Determinar el rango de: A = \\begin{pmatrix} 1 & 0 & 2 & 1 \\\\ 0 & 1 & 3 & 2 \\\\ 1 & 1 & 5 & 3 \\end{pmatrix}.",
|
||||
"hint": "F₃ = F₁ + F₂?",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": 2, "latex": "\\text{rg}(A) = 2" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "F₃ = F₁ + F₂ → fila dependiente", "expression": "F_3 = F_1 + F_2 \\Rightarrow (1,1,5,3) = (1,0,2,1)+(0,1,3,2) \\checkmark" },
|
||||
{ "desc": "Dos filas independientes", "expression": "\\text{rg}(A) = 2" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-29",
|
||||
"chapter": 2,
|
||||
"topic": "determinants",
|
||||
"subtopic": "rank",
|
||||
"theoryKey": "rank",
|
||||
"difficulty": "advanced",
|
||||
"statement": "¿Para qué valores de k el rango de la siguiente matriz es 2? A = \\begin{pmatrix} 1 & 2 & 3 \\\\ 2 & 5 & k \\\\ 1 & 1 & 0 \\end{pmatrix}.",
|
||||
"hint": "rg=2 ↔ todos los det 3×3 = 0",
|
||||
"answerType": "numeric",
|
||||
"answer": { "value": 4, "latex": "k = 4" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "det(A) = 1(0-k) - 2(0-k) + 3(2-5) = -k + 2k - 9 = k - 9", "expression": "\\det(A) = -k + 2k - 9 = k - 9" },
|
||||
{ "desc": "Para rg = 2: det(A) = 0 → k = 9", "expression": "k - 9 = 0 \\Rightarrow k = 9" },
|
||||
{ "desc": "Verificar que el menor 2×2 es ≠ 0", "expression": "\\begin{vmatrix}1&2\\\\2&5\\end{vmatrix} = 1 \\neq 0 \\Rightarrow \\text{rg} \\geq 2" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap02-30",
|
||||
"chapter": 2,
|
||||
"topic": "systems",
|
||||
"subtopic": "rouche-frobenius",
|
||||
"theoryKey": "determinants",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Clasificar el sistema según su compatibilidad usando rango: x + 2y + z = 3, 2x + 4y + 2z = 6, 3x + 6y + 3z = 9.",
|
||||
"hint": "Comparar rg(A) con rg(A|b)",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Compatible Indeterminado", "latex": "\\text{CI: rg(A) = RG(A|b) = 1 < n = 3}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Todas las ecuaciones son proporcionales", "expression": "E_2 = 2E_1,\\quad E_3 = 3E_1" },
|
||||
{ "desc": "rg(A) = 1, rg(A|b) = 1", "expression": "\\text{rg}(A) = \\text{rg}(A|b) = 1" },
|
||||
{ "desc": "rg = 1 < 3 = n → Compatible Indeterminado", "expression": "\\text{CI con } n - \\text{rg} = 2 \\text{ parámetros libres}" }
|
||||
]
|
||||
}
|
||||
]
|
||||
149
data/exercises-cap03.json
Normal file
149
data/exercises-cap03.json
Normal file
@@ -0,0 +1,149 @@
|
||||
[
|
||||
{
|
||||
"id": "cap03-01",
|
||||
"chapter": 3,
|
||||
"topic": "systems",
|
||||
"subtopic": "matrix-form",
|
||||
"theoryKey": "systems-theory",
|
||||
"difficulty": "basic",
|
||||
"statement": "Escribir en forma matricial los siguientes sistemas: a) x + y = 5, x - y = 1; b) 2x + y - z = 3, x - y + 2z = 1, 3x + 2y + z = 7; c) x + y + z + t = 4, 2x - y + z - t = 1.",
|
||||
"hint": "Ax = b donde A es la matriz de coeficientes",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "Formas matriciales escritas", "latex": "a)\\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix}\\begin{pmatrix}x\\\\y\\end{pmatrix}=\\begin{pmatrix}5\\\\1\\end{pmatrix}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) Sistema 2×2", "expression": "\\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix}\\begin{pmatrix}x\\\\y\\end{pmatrix}=\\begin{pmatrix}5\\\\1\\end{pmatrix}" },
|
||||
{ "desc": "b) Sistema 3×3", "expression": "\\begin{pmatrix}2&1&-1\\\\1&-1&2\\\\3&2&1\\end{pmatrix}\\begin{pmatrix}x\\\\y\\\\z\\end{pmatrix}=\\begin{pmatrix}3\\\\1\\\\7\\end{pmatrix}" },
|
||||
{ "desc": "c) Sistema 2×4", "expression": "\\begin{pmatrix}1&1&1&1\\\\2&-1&1&-1\\end{pmatrix}\\begin{pmatrix}x\\\\y\\\\z\\\\t\\end{pmatrix}=\\begin{pmatrix}4\\\\1\\end{pmatrix}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap03-02",
|
||||
"chapter": 3,
|
||||
"topic": "systems",
|
||||
"subtopic": "classify-solve",
|
||||
"theoryKey": "systems-theory",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Clasificar los siguientes sistemas y resolver cuando sea posible: a) x + 2y = 5, 3x - y = 1; b) x + y + z = 3, x - y + z = 1, x + y - z = 1; c) x + y + z = 1, 2x + 2y + 2z = 3, x - y + z = 0.",
|
||||
"hint": "Usar Rouche-Frobenius para clasificar",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "a)CD:(1,2) b)CD:(1,1,1) c)SI", "latex": "a)\\,CD:\\,(1,2),\\quad b)\\,CD:\\,(1,1,1),\\quad c)\\,SI" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) det = -1 - 6 = -7 ≠ 0 → CD", "expression": "\\det\\begin{pmatrix}1&2\\\\3&-1\\end{pmatrix}=-7\\neq 0 \\Rightarrow CD" },
|
||||
{ "desc": "Solución a)", "expression": "x=1,\\; y=2" },
|
||||
{ "desc": "b) Sistema 3×3", "expression": "\\det\\begin{pmatrix}1&1&1\\\\1&-1&1\\\\1&1&-1\\end{pmatrix}=1(1-1)-1(-1-1)+1(1+1)=4\\neq 0" },
|
||||
{ "desc": "Solución b)", "expression": "x=1,\\; y=1,\\; z=1" },
|
||||
{ "desc": "c) F₂ = 2F₁ pero 3 ≠ 2 → SI", "expression": "E_2 = 2E_1 \\Rightarrow 2(1)=2\\neq 3 \\Rightarrow \\text{Incompatible}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap03-03",
|
||||
"chapter": 3,
|
||||
"topic": "systems",
|
||||
"subtopic": "gauss",
|
||||
"theoryKey": "gauss-method",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Resolver por eliminación de Gauss: a) x + y + z = 6, 2x - y + z = 3, x + 2y - z = 5; b) x - y + z = 2, 2x + y - z = 3, 3x + 0y + 0z = 5.",
|
||||
"hint": "F₂ → F₂ - 2F₁, F₃ → F₃ - F₁, etc.",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "a)(2;1;3) b)(5/3;1/3;2/3)", "latex": "a)\\,(2;1;3),\\quad b)\\,(5/3;\\;-2/3;\\;-1/3)" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) Matriz aumentada", "expression": "\\begin{pmatrix}1&1&1&|&6\\\\2&-1&1&|&3\\\\1&2&-1&|&5\\end{pmatrix}" },
|
||||
{ "desc": "F₂ → F₂ - 2F₁", "expression": "\\begin{pmatrix}1&1&1&|&6\\\\0&-3&-1&|&-9\\\\0&1&-2&|&-1\\end{pmatrix}" },
|
||||
{ "desc": "F₃ → F₃ + F₂/3", "expression": "\\begin{pmatrix}1&1&1&|&6\\\\0&-3&-1&|&-9\\\\0&0&-7/3&|&-4\\end{pmatrix}" },
|
||||
{ "desc": "Sustitución regresiva", "expression": "z = 3,\\; y = 1,\\; x = 2" },
|
||||
{ "desc": "b) 3x = 5 → x = 5/3", "expression": "3x = 5 \\Rightarrow x = \\frac{5}{3}" },
|
||||
{ "desc": "Sustituir", "expression": "y = \\frac{1}{3},\\; z = \\frac{2}{3}" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap03-04",
|
||||
"chapter": 3,
|
||||
"topic": "systems",
|
||||
"subtopic": "gauss-jordan",
|
||||
"theoryKey": "gauss-jordan",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Resolver por Gauss-Jordan: a) 2x + y = 5, x + 3y = 10; b) x + y + z = 3, 2x - y + z = 2, x + 2y - z = 4.",
|
||||
"hint": "Reducir a forma reducida por filas (identidad)",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "a)(1;3) b)(2;1;0)", "latex": "a)\\,(1;3),\\quad b)\\,(2;1;0)" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) Matriz aumentada", "expression": "\\begin{pmatrix}2&1&|&5\\\\1&3&|&10\\end{pmatrix}" },
|
||||
{ "desc": "F₁ ↔ F₂, F₂ → F₂ - 2F₁", "expression": "\\begin{pmatrix}1&3&|&10\\\\0&-5&|&-15\\end{pmatrix}" },
|
||||
{ "desc": "F₂ → F₂/(-5), F₁ → F₁ - 3F₂", "expression": "\\begin{pmatrix}1&0&|&1\\\\0&1&|&3\\end{pmatrix}" },
|
||||
{ "desc": "Solución: x=1, y=3", "expression": "x = 1,\\; y = 3" },
|
||||
{ "desc": "b) Gauss-Jordan en 3×3", "expression": "x = 2,\\; y = 1,\\; z = 0" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap03-05",
|
||||
"chapter": 3,
|
||||
"topic": "systems",
|
||||
"subtopic": "cramer",
|
||||
"theoryKey": "cramer-rule",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Resolver por la regla de Cramer: a) 3x + 2y = 12, x - y = 1; b) x + y + z = 6, x - y + z = 2, x + y - z = 0.",
|
||||
"hint": "x = Δ₁/Δ, y = Δ₂/Δ, etc.",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "a)(2.8;1.8) b)(1;3;2)", "latex": "a)\\,(\\frac{14}{5};\\;\\frac{9}{5}),\\quad b)\\,(1;\\;3;\\;2)" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) Δ = (3)(-1)-(2)(1) = -5", "expression": "\\Delta = -5" },
|
||||
{ "desc": "Δ₁ = (12)(-1)-(2)(1) = -14", "expression": "\\Delta_1 = -14" },
|
||||
{ "desc": "Δ₂ = (3)(1)-(12)(1) = -9", "expression": "\\Delta_2 = -9" },
|
||||
{ "desc": "x = 14/5, y = 9/5", "expression": "x = \\frac{14}{5},\\; y = \\frac{9}{5}" },
|
||||
{ "desc": "b) Δ = det coeficientes", "expression": "\\Delta = \\begin{vmatrix}1&1&1\\\\1&-1&1\\\\1&1&-1\\end{vmatrix} = 1(1-1)-1(-1-1)+1(1+1) = 4" },
|
||||
{ "desc": "Δ₁, Δ₂, Δ₃", "expression": "x = \\frac{4}{4}=1,\\; y = \\frac{12}{4}=3,\\; z = \\frac{8}{4}=2" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap03-06",
|
||||
"chapter": 3,
|
||||
"topic": "systems",
|
||||
"subtopic": "rouche-frobenius",
|
||||
"theoryKey": "rouche-frobenius",
|
||||
"difficulty": "advanced",
|
||||
"statement": "Aplicar el teorema de Rouche-Frobenius para clasificar y resolver: a) x + y + z = 1, x - y + z = 0, 2x + 0y + 2z = 2; b) x + 2y + 3z = 1, 2x + 4y + 6z = 2, x + y + z = 1; c) x + y + z = 1, x + y + z = 2, 2x + 2y + 2z = 3.",
|
||||
"hint": "Calcular rg(A) y rg(A|b), comparar",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "a)CI b)CI c)SI", "latex": "a)\\,CI,\\quad b)\\,CI,\\quad c)\\,SI" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) F₃ = F₁ + F₂ → rg(A) = 2, rg(A|b) = 2", "expression": "\\text{rg}(A)=\\text{rg}(A|b)=2 < 3 \\Rightarrow CI" },
|
||||
{ "desc": "b) F₂ = 2F₁ → rg(A) = 2, rg(A|b) = 2", "expression": "\\text{rg}(A)=\\text{rg}(A|b)=2 < 3 \\Rightarrow CI" },
|
||||
{ "desc": "c) E₁: x+y+z=1, E₂: x+y+z=2 (contradicción)", "expression": "1 \\neq 2 \\Rightarrow \\text{rg}(A)=1,\\;\\text{rg}(A|b)=2 \\Rightarrow SI" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap03-07",
|
||||
"chapter": 3,
|
||||
"topic": "systems",
|
||||
"subtopic": "homogeneous",
|
||||
"theoryKey": "homogeneous",
|
||||
"difficulty": "intermediate",
|
||||
"statement": "Resolver los siguientes sistemas homogéneos: a) x + y = 0, x - y = 0; b) x + 2y - z = 0, 2x + 4y - 2z = 0, 3x + 6y - 3z = 0; c) x + y + z = 0, 2x - y + z = 0, x + 2y - z = 0.",
|
||||
"hint": "Ax = 0 siempre tiene solución trivial. No trivial ↔ rg(A) < n",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "a)trivial:(0,0) b)CI c)trivial:(0,0,0)", "latex": "a)\\,(0;0),\\quad b)\\,CI,\\quad c)\\,(0;0;0)" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "a) det = -2 ≠ 0 → solución trivial x=y=0", "expression": "\\det = -2 \\neq 0 \\Rightarrow x=y=0" },
|
||||
{ "desc": "b) Todas las ecuaciones proporcionales → rg=1 < 3 → CI", "expression": "E_2=2E_1,\\;E_3=3E_1 \\Rightarrow \\text{rg}=1,\\;\\dim(\\ker)=2" },
|
||||
{ "desc": "Solución paramétrica b)", "expression": "(x,y,z) = \\alpha(-2,1,0) + \\beta(1,0,1)" },
|
||||
{ "desc": "c) det ≠ 0 → trivial", "expression": "\\det = \\begin{vmatrix}1&1&1\\\\2&-1&1\\\\1&2&-1\\end{vmatrix} = 7 \\neq 0 \\Rightarrow (0;0;0)" }
|
||||
]
|
||||
},
|
||||
{
|
||||
"id": "cap03-08",
|
||||
"chapter": 3,
|
||||
"topic": "systems",
|
||||
"subtopic": "parameter",
|
||||
"theoryKey": "rouche-frobenius",
|
||||
"difficulty": "advanced",
|
||||
"statement": "Determinar el valor de k para que el sistema sea: a) Compatible determinado, b) Compatible indeterminado, c) Incompatible. Sistema: x + y + z = 1, 2x - y + z = k, x + 2y - z = 3.",
|
||||
"hint": "Calcular det de coeficientes. Si ≠ 0 → CD. Si = 0 → analizar según k.",
|
||||
"answerType": "expression",
|
||||
"answer": { "value": "k≠4:CD, k=4:CI, nunca SI", "latex": "k \\neq 4: CD,\\; k = 4: CI,\\; \\text{nunca SI}" },
|
||||
"solutionSteps": [
|
||||
{ "desc": "Calcular det(A)", "expression": "\\det(A) = \\begin{vmatrix}1&1&1\\\\2&-1&1\\\\1&2&-1\\end{vmatrix} = 1(1-2)-1(-2-1)+1(4+1) = -1+3+5 = 7" },
|
||||
{ "desc": "Como det(A) = 7 ≠ 0 siempre → siempre CD", "expression": "\\det(A) = 7 \\neq 0 \\Rightarrow \\text{Siempre CD para todo } k" },
|
||||
{ "desc": "Solución para cada k", "expression": "x = \\frac{7-2k}{7},\\; y = \\frac{k+2}{7},\\; z = \\frac{k+4}{7}" }
|
||||
]
|
||||
}
|
||||
]
|
||||
62
data/theory-cap01.json
Normal file
62
data/theory-cap01.json
Normal file
@@ -0,0 +1,62 @@
|
||||
{
|
||||
"vectors-theory": {
|
||||
"title": "Vectores en ℝ² y ℝ³",
|
||||
"content": "Un vector es un segmento orientado con origen y extremo. En $\\mathbb{R}^2$: $\\vec{u} = (u_x; u_y)$. En $\\mathbb{R}^3$: $\\vec{u} = (u_x; u_y; u_z)$. El módulo es $|\\vec{u}| = \\sqrt{u_x^2 + u_y^2 + u_z^2}$. Vector unitario: $\\hat{u} = \\vec{u}/|\\vec{u}|$.",
|
||||
"relatedExercises": ["cap01-01", "cap01-02", "cap01-03", "cap01-04"]
|
||||
},
|
||||
"vector-ops": {
|
||||
"title": "Operaciones con Vectores",
|
||||
"content": "Suma: $\\vec{u} + \\vec{v} = (u_x+v_x; u_y+v_y; u_z+v_z)$. Multiplicación por escalar: $k\\vec{u} = (k\\cdot u_x; k\\cdot u_y; k\\cdot u_z)$. Propiedades: conmutativa, asociativa, elemento neutro, elemento opuesto.",
|
||||
"relatedExercises": ["cap01-04"]
|
||||
},
|
||||
"dot-product": {
|
||||
"title": "Producto Escalar (Dot Product)",
|
||||
"content": "$\\vec{u} \\cdot \\vec{v} = u_x v_x + u_y v_y + u_z v_z = |\\vec{u}|\\cdot|\\vec{v}|\\, \\cos(\\alpha)$. Propiedades: conmutativa, distributiva. Condición de perpendicularidad: $\\vec{u} \\perp \\vec{v} \\Leftrightarrow \\vec{u} \\cdot \\vec{v} = 0$.",
|
||||
"relatedExercises": ["cap01-05"]
|
||||
},
|
||||
"cross-product": {
|
||||
"title": "Producto Vectorial (Cross Product)",
|
||||
"content": "$\\vec{u} \\times \\vec{v} = (u_y v_z - u_z v_y; u_z v_x - u_x v_z; u_x v_y - u_y v_x)$. Se calcula por determinante: $\\vec{u} \\times \\vec{v} = \\begin{vmatrix} \\mathbf{i} & \\mathbf{j} & \\mathbf{k} \\\\ u_x & u_y & u_z \\\\ v_x & v_y & v_z \\end{vmatrix}$. $\\vec{u} \\times \\vec{v} \\perp \\vec{u}$ y $\\vec{u} \\times \\vec{v} \\perp \\vec{v}$.",
|
||||
"relatedExercises": ["cap01-06"]
|
||||
},
|
||||
"parallel-perp": {
|
||||
"title": "Paralelismo y Perpendicularidad",
|
||||
"content": "Paralelos: $\\vec{u} \\parallel \\vec{v} \\Leftrightarrow \\vec{u} = k\\vec{v}$ (componentes proporcionales). Perpendiculares: $\\vec{u} \\perp \\vec{v} \\Leftrightarrow \\vec{u} \\cdot \\vec{v} = 0$.",
|
||||
"relatedExercises": ["cap01-07", "cap01-08"]
|
||||
},
|
||||
"coplanarity": {
|
||||
"title": "Producto Mixto y Coplanaridad",
|
||||
"content": "Producto mixto: $[\\vec{u},\\vec{v},\\vec{w}] = \\vec{u} \\cdot (\\vec{v} \\times \\vec{w}) = \\begin{vmatrix} u_x & u_y & u_z \\\\ v_x & v_y & v_z \\\\ w_x & w_y & w_z \\end{vmatrix}$. $\\vec{u},\\vec{v},\\vec{w}$ son coplanarios $\\Leftrightarrow [\\vec{u},\\vec{v},\\vec{w}] = 0$.",
|
||||
"relatedExercises": ["cap01-09", "cap01-29"]
|
||||
},
|
||||
"line-equations": {
|
||||
"title": "Ecuaciones de la Recta",
|
||||
"content": "Vectorial: $r: \\vec{X} = \\vec{P_0} + t\\vec{v}$. Paramétricas: $x = x_0 + t v_x, y = y_0 + t v_y, z = z_0 + t v_z$. Continuas: \\frac{x-x_0}{v_x} = \\frac{y-y_0}{v_y} = \\frac{z-z_0}{v_z}$.",
|
||||
"relatedExercises": ["cap01-10", "cap01-11"]
|
||||
},
|
||||
"line-positions": {
|
||||
"title": "Posición Relativa de Dos Rectas",
|
||||
"content": "Paralelas: $\\vec{v_1} \\parallel \\vec{v_2}$. Secantes: $\\vec{v_1} \\neq k\\vec{v_2}$ y existe solución a $P_1 + t\\vec{v_1} = P_2 + s\\vec{v_2}$. Se cruzan: $\\vec{v_1} \\neq k\\vec{v_2}$ y $[\\vec{v_1},\\vec{v_2}, P_2-P_1] \\neq 0$.",
|
||||
"relatedExercises": ["cap01-12"]
|
||||
},
|
||||
"distance-lines": {
|
||||
"title": "Distancia entre Rectas",
|
||||
"content": "Paralelas: $d(r_1,r_2) = |(P_2-P_1) \\times \\vec{v}| / |\\vec{v}|$. Si se cruzan: $d = |[\\vec{v_1},\\vec{v_2},P_2-P_1]| / |\\vec{v_1} \\times \\vec{v_2}|$.",
|
||||
"relatedExercises": ["cap01-13", "cap01-14"]
|
||||
},
|
||||
"plane-equations": {
|
||||
"title": "Ecuaciones del Plano",
|
||||
"content": "Vectorial: $\\Pi: \\vec{X} = \\vec{P_0} + s\\vec{v_1} + t\\vec{v_2}$. Normal: $\\vec{n} = (a;b;c)$, ecuación general: $ax + by + cz + d = 0$ donde $d = -\\vec{n} \\cdot \\vec{P_0}$.",
|
||||
"relatedExercises": ["cap01-16", "cap01-17"]
|
||||
},
|
||||
"plane-positions": {
|
||||
"title": "Posición Relativa de Planos",
|
||||
"content": "Paralelos: $\\vec{n_1} \\parallel \\vec{n_2}$. Si además $d_1/k = d_2$: coincidentes. Secantes: $\\vec{n_1} \\neq k\\vec{n_2}$ (se cortan en una recta). $\\Pi_1 \\perp \\Pi_2 \\Leftrightarrow \\vec{n_1} \\cdot \\vec{n_2} = 0$.",
|
||||
"relatedExercises": ["cap01-19", "cap01-20"]
|
||||
},
|
||||
"angle": {
|
||||
"title": "Ángulos entre Rectas, Planos y Recta-Plano",
|
||||
"content": "Entre rectas: $\\cos \\alpha = |\\vec{v_1} \\cdot \\vec{v_2}| / (|\\vec{v_1}|\\cdot|\\vec{v_2}|)$. Entre planos: $\\cos \\alpha = |\\vec{n_1} \\cdot \\vec{n_2}| / (|\\vec{n_1}|\\cdot|\\vec{n_2}|)$. Entre recta y plano: $\\sin \\beta = |\\vec{v} \\cdot \\vec{n}| / (|\\vec{v}|\\cdot|\\vec{n}|)$.",
|
||||
"relatedExercises": ["cap01-15", "cap01-21", "cap01-22"]
|
||||
}
|
||||
}
|
||||
42
data/theory-cap02.json
Normal file
42
data/theory-cap02.json
Normal file
@@ -0,0 +1,42 @@
|
||||
{
|
||||
"matrices-theory": {
|
||||
"title": "Definición de Matriz",
|
||||
"content": "Una matriz es un arreglo rectangular de números dispuestos en filas y columnas. Una matriz de orden $m \\times n$ tiene $m$ filas y $n$ columnas. Notación: $A = [a_{ij}]$ donde $i$ indica la fila y $j$ la columna.",
|
||||
"relatedExercises": ["cap02-01", "cap02-02"]
|
||||
},
|
||||
"matrix-ops": {
|
||||
"title": "Operaciones con Matrices",
|
||||
"content": "Suma: $(A+B)_{ij} = a_{ij} + b_{ij}$ (mismo orden). Multiplicación por escalar: $(cA)_{ij} = c \\cdot a_{ij}$. Producto: $(AB)_{ij} = \\sum_{k} a_{ik} \\cdot b_{kj}$ (columnas de A = filas de B).",
|
||||
"relatedExercises": ["cap02-03", "cap02-05", "cap02-06"]
|
||||
},
|
||||
"matrix-types": {
|
||||
"title": "Matrices Especiales",
|
||||
"content": "Cuadrada: $n \\times n$. Identidad $I_n$: $1$ en diagonal, $0$ en el resto. Diagonal: $a_{ij}=0$ para $i \\neq j$. Triangular superior: $a_{ij}=0$ para $i > j$. Traspuesta $A^T$: $(A^T)_{ij} = a_{ji}$.",
|
||||
"relatedExercises": ["cap02-07", "cap02-08", "cap02-09"]
|
||||
},
|
||||
"transpose-symmetry": {
|
||||
"title": "Traspuesta y Simetría",
|
||||
"content": "Traspuesta: $(A^T)^T = A$, $(AB)^T = B^T A^T$. Simétrica: $A = A^T \\Leftrightarrow a_{ij} = a_{ji}$. Antisimétrica: $A = -A^T \\Leftrightarrow a_{ii}=0$. Toda matriz: $A = \\frac{A+A^T}{2} + \\frac{A-A^T}{2}$ (simétrica + antisimétrica).",
|
||||
"relatedExercises": ["cap02-10", "cap02-11"]
|
||||
},
|
||||
"determinants": {
|
||||
"title": "Determinantes",
|
||||
"content": "Orden 2: $|\\begin{matrix} a & b \\\\ c & d \\end{matrix}| = ad - bc$. Orden 3 (Sarrus): suma de 3 diagonales principales menos 3 diagonales secundarias. Cofactor: $C_{ij} = (-1)^{i+j} M_{ij}$ (menor con signo).",
|
||||
"relatedExercises": ["cap02-12", "cap02-13", "cap02-14"]
|
||||
},
|
||||
"determinant-props": {
|
||||
"title": "Propiedades de los Determinantes",
|
||||
"content": "$|A^T| = |A|$. Si se intercambian dos filas, cambia signo. Fila de ceros $\\Rightarrow |A|=0$. Filas proporcionales $\\Rightarrow |A|=0$. $F_i \\rightarrow F_i + cF_j$ no cambia $|A|$. $|AB| = |A| \\cdot |B|$. $|cA| = c^n |A|$ para $n \\times n$.",
|
||||
"relatedExercises": ["cap02-17", "cap02-18", "cap02-19"]
|
||||
},
|
||||
"inverse-matrix": {
|
||||
"title": "Matriz Inversa",
|
||||
"content": "$A$ invertible $\\Leftrightarrow |A| \\neq 0 \\Leftrightarrow \\text{rg}(A) = n$. $A^{-1} = \\frac{1}{|A|} \\text{adj}(A)$. Para $2 \\times 2$: $A^{-1} = \\frac{1}{ad-bc} \\begin{matrix} d & -b \\\\ -c & a \\end{matrix}$.",
|
||||
"relatedExercises": ["cap02-20", "cap02-21", "cap02-22", "cap02-24", "cap02-25"]
|
||||
},
|
||||
"rank": {
|
||||
"title": "Rango de una Matriz",
|
||||
"content": "Rango = orden del menor no nulo mayor. Equivale al número de filas no nulas en forma escalonada. Operaciones elementales (permutar, multiplicar por $c \\neq 0$, $F_i + cF_j$) no cambian el rango.",
|
||||
"relatedExercises": ["cap02-26", "cap02-27", "cap02-28", "cap02-29"]
|
||||
}
|
||||
}
|
||||
32
data/theory-cap03.json
Normal file
32
data/theory-cap03.json
Normal file
@@ -0,0 +1,32 @@
|
||||
{
|
||||
"systems-theory": {
|
||||
"title": "Sistemas de Ecuaciones Lineales",
|
||||
"content": "Sistema de $m$ ecuaciones con $n$ incógnitas: $A \\cdot X = B$. $A$ (matriz coeficientes $m \\times n$), $X$ (incógnitas $n \\times 1$), $B$ (términos independientes $m \\times 1$). Matriz ampliada $A' = [A|B]$ de dimensión $m \\times (n+1)$.",
|
||||
"relatedExercises": ["cap03-01", "cap03-02"]
|
||||
},
|
||||
"gauss-method": {
|
||||
"title": "Método de Eliminación de Gauss",
|
||||
"content": "Transformar la matriz ampliada a forma triangular mediante operaciones elementales ($F_i \\leftrightarrow F_j$, $F_i \\rightarrow cF_i$, $F_i \\rightarrow F_i + cF_j$) y resolver por sustitución regresiva. Pivote: elemento $a_{ii} \\neq 0$.",
|
||||
"relatedExercises": ["cap03-03"]
|
||||
},
|
||||
"gauss-jordan": {
|
||||
"title": "Método de Gauss-Jordan",
|
||||
"content": "Llevar la matriz ampliada a forma escalonada reducida por filas: cada pivote es 1 y tiene ceros arriba y abajo. La solución se lee directamente sin sustitución regresiva.",
|
||||
"relatedExercises": ["cap03-04"]
|
||||
},
|
||||
"cramer-rule": {
|
||||
"title": "Regla de Cramer",
|
||||
"content": "Para sistemas $n \\times n$ con $|A| \\neq 0$: $x_j = \\frac{|A_j|}{|A|}$ donde $A_j$ es $A$ con columna $j$ reemplazada por $B$. Requiere $n+1$ determinantes — impracticable para sistemas grandes.",
|
||||
"relatedExercises": ["cap03-05"]
|
||||
},
|
||||
"rouche-frobenius": {
|
||||
"title": "Teorema de Rouché-Frobenius",
|
||||
"content": "Sea $r = \\text{rg}(A)$, $r' = \\text{rg}(A'|)$. Si $r \\neq r'$: SI (incompatible). Si $r = r' = n$: SCD (única solución). Si $r = r' < n$: SCI (infinitas soluciones, $n-r$ parámetros libres).",
|
||||
"relatedExercises": ["cap03-06", "cap03-08"]
|
||||
},
|
||||
"homogeneous": {
|
||||
"title": "Sistemas Homogéneos",
|
||||
"content": "Sistema $AX = 0$ con todos los términos independientes nulos. SIEMPRE tiene al menos la solución trivial $X = 0$. Tiene soluciones no triviales $\\Leftrightarrow \\text{rg}(A) < n \\Leftrightarrow |A| = 0$ (sistema cuadrado).",
|
||||
"relatedExercises": ["cap03-07"]
|
||||
}
|
||||
}
|
||||
Reference in New Issue
Block a user