algebra-webapp/data/exercises-cap01.json

[
  {
    "id": "cap01-01",
    "chapter": 1,
    "topic": "vector-ops",
    "subtopic": "components",
    "theoryKey": "vectors-theory",
    "difficulty": "basic",
    "statement": "Dados los puntos A = (2; -1; 3) y B = (4; 2; -1), hallar las componentes del vector \\overrightarrow{AB}.",
    "hint": "Las componentes de AB son B - A",
    "answerType": "vector",
    "answer": { "value": [2, 3, -4], "latex": "(2;\\; 3;\\; -4)" },
    "solutionSteps": [
      { "desc": "Fórmula: las componentes de AB = B - A", "expression": "\\overrightarrow{AB} = B - A = (4; 2; -1) - (2; -1; 3)" },
      { "desc": "Restar componente a componente", "expression": "\\overrightarrow{AB} = (4-2;\\; 2-(-1);\\; -1-3) = (2;\\; 3;\\; -4)" }
    ]
  },
  {
    "id": "cap01-02",
    "chapter": 1,
    "topic": "vector-ops",
    "subtopic": "magnitude",
    "theoryKey": "vectors-theory",
    "difficulty": "basic",
    "statement": "Hallar el módulo de los vectores: a) \\vec{u} = (3; -4), b) \\vec{v} = (1; 2; -2), c) \\vec{w} = (-1; 0; 3).",
    "hint": "|v| = √(v₁² + v₂² + v₃²)",
    "answerType": "expression",
    "answer": { "value": "|u|=5, |v|=3, |w|=√10", "latex": "|\\vec{u}|=5,\\quad |\\vec{v}|=3,\\quad |\\vec{w}|=\\sqrt{10}" },
    "solutionSteps": [
      { "desc": "a) Módulo de u = (3; -4)", "expression": "|\\vec{u}| = \\sqrt{3^2 + (-4)^2} = \\sqrt{9 + 16} = \\sqrt{25} = 5" },
      { "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" },
      { "desc": "c) Módulo de w = (-1; 0; 3)", "expression": "|\\vec{w}| = \\sqrt{(-1)^2 + 0^2 + 3^2} = \\sqrt{1 + 0 + 9} = \\sqrt{10}" }
    ]
  },
  {
    "id": "cap01-03",
    "chapter": 1,
    "topic": "vector-ops",
    "subtopic": "unit-vector",
    "theoryKey": "vectors-theory",
    "difficulty": "basic",
    "statement": "Dado \\vec{u} = (4; -3; 5), hallar el vector unitario en la dirección de \\vec{u}.",
    "hint": "û = u / |u|",
    "answerType": "vector",
    "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)" },
    "solutionSteps": [
      { "desc": "Calcular el módulo de u", "expression": "|\\vec{u}| = \\sqrt{16 + 9 + 25} = \\sqrt{50} = 5\\sqrt{2}" },
      { "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)" }
    ]
  },
  {
    "id": "cap01-04",
    "chapter": 1,
    "topic": "vector-ops",
    "subtopic": "addition",
    "theoryKey": "vector-ops",
    "difficulty": "basic",
    "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}|.",
    "hint": "Sumar/restar componente a componente",
    "answerType": "expression",
    "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}" },
    "solutionSteps": [
      { "desc": "a) Suma u + v", "expression": "\\vec{u} + \\vec{v} = (1+2;\\;-2+1;\\;3+(-1)) = (3;\\;-1;\\;2)" },
      { "desc": "b) 3u - 2v", "expression": "3\\vec{u} = (3;-6;9),\\quad 2\\vec{v} = (4;2;-2)" },
      { "desc": "Restar", "expression": "3\\vec{u} - 2\\vec{v} = (3-4;\\;-6-2;\\;9-(-2)) = (-1;\\;-8;\\;11)" },
      { "desc": "c) Módulo de u + v = (3; -1; 2)", "expression": "|\\vec{u}+\\vec{v}| = \\sqrt{9+1+4} = \\sqrt{14}" }
    ]
  },
  {
    "id": "cap01-05",
    "chapter": 1,
    "topic": "vector-ops",
    "subtopic": "dot-product",
    "theoryKey": "dot-product",
    "difficulty": "basic",
    "statement": "Sean \\vec{u} = (2; -1; 3) y \\vec{v} = (4; 3; -2). Calcular \\vec{u} \\cdot \\vec{v}.",
    "hint": "u·v = u₁v₁ + u₂v₂ + u₃v₃",
    "answerType": "numeric",
    "answer": { "value": -1, "latex": "-1" },
    "solutionSteps": [
      { "desc": "Producto escalar componente a componente", "expression": "\\vec{u} \\cdot \\vec{v} = 2 \\cdot 4 + (-1) \\cdot 3 + 3 \\cdot (-2)" },
      { "desc": "Calcular", "expression": "= 8 - 3 - 6 = -1" }
    ]
  },
  {
    "id": "cap01-06",
    "chapter": 1,
    "topic": "vector-ops",
    "subtopic": "cross-product",
    "theoryKey": "cross-product",
    "difficulty": "basic",
    "statement": "Sean \\vec{u} = (1; 2; -1) y \\vec{v} = (3; 0; 4). Calcular \\vec{u} \\times \\vec{v}.",
    "hint": "Producto vectorial por determinante",
    "answerType": "vector",
    "answer": { "value": [8, -7, -6], "latex": "(8;\\;-7;\\;-6)" },
    "solutionSteps": [
      { "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}" },
      { "desc": "Calcular componente i", "expression": "i: (2)(4) - (-1)(0) = 8" },
      { "desc": "Calcular componente j", "expression": "j: -[(1)(4) - (-1)(3)] = -[4+3] = -7" },
      { "desc": "Calcular componente k", "expression": "k: (1)(0) - (2)(3) = -6" },
      { "desc": "Resultado", "expression": "\\vec{u} \\times \\vec{v} = (8;\\;-7;\\;-6)" }
    ]
  },
  {
    "id": "cap01-07",
    "chapter": 1,
    "topic": "vector-ops",
    "subtopic": "parallel",
    "theoryKey": "parallel-perp",
    "difficulty": "basic",
    "statement": "Determinar si los vectores \\vec{u} = (2; 1; -3) y \\vec{v} = (-4; -2; 6) son paralelos.",
    "hint": "Son paralelos si u = kv para algún escalar k",
    "answerType": "expression",
    "answer": { "value": "Sí, son paralelos (k=-2)", "latex": "\\text{Sí, } \\vec{v} = -2\\vec{u}" },
    "solutionSteps": [
      { "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}" },
      { "desc": "Componente i: (1)(6) - (-3)(-2) = 6 - 6 = 0", "expression": "i: 6 - 6 = 0" },
      { "desc": "Componente j: -[(2)(6) - (-3)(-4)] = -[12-12] = 0", "expression": "j: -(12-12) = 0" },
      { "desc": "Componente k: (2)(-2) - (1)(-4) = -4+4 = 0", "expression": "k: -4+4 = 0" },
      { "desc": "Como u×v=0, son paralelos. Además v = -2u", "expression": "\\vec{v} = (-4;-2;6) = -2(2;1;-3) = -2\\vec{u}" }
    ]
  },
  {
    "id": "cap01-08",
    "chapter": 1,
    "topic": "vector-ops",
    "subtopic": "perpendicular",
    "theoryKey": "parallel-perp",
    "difficulty": "basic",
    "statement": "Determinar si los vectores \\vec{u} = (1; -1; 2) y \\vec{v} = (3; 1; 1) son perpendiculares.",
    "hint": "Son perpendiculares si u·v = 0",
    "answerType": "expression",
    "answer": { "value": "No son perpendiculares", "latex": "\\vec{u} \\cdot \\vec{v} = 4 \\neq 0 \\Rightarrow \\text{No son perpendiculares}" },
    "solutionSteps": [
      { "desc": "Calcular el producto escalar", "expression": "\\vec{u} \\cdot \\vec{v} = (1)(3) + (-1)(1) + (2)(1) = 3 - 1 + 2 = 4" },
      { "desc": "Como no es 0, no son perpendiculares", "expression": "\\vec{u} \\cdot \\vec{v} = 4 \\neq 0 \\Rightarrow \\text{No son perpendiculares}" }
    ]
  },
  {
    "id": "cap01-09",
    "chapter": 1,
    "topic": "vector-ops",
    "subtopic": "mixed-product",
    "theoryKey": "coplanarity",
    "difficulty": "intermediate",
    "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.",
    "hint": "[u,v,w] = u · (v × w). Si es 0, son coplanarios.",
    "answerType": "numeric",
    "answer": { "value": 7, "latex": "[\\vec{u},\\vec{v},\\vec{w}] = 7 \\neq 0, \\text{ no coplanarios}" },
    "solutionSteps": [
      { "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)" },
      { "desc": "Calcular u · (v × w)", "expression": "[\\vec{u},\\vec{v},\\vec{w}] = (1)(-2) + (-1)(3) + (2)(6) = -2 - 3 + 12 = 7" },
      { "desc": "Conclusión", "expression": "[\\vec{u},\\vec{v},\\vec{w}] = 7 \\neq 0 \\Rightarrow \\text{No son coplanarios}" }
    ]
  },
  {
    "id": "cap01-10",
    "chapter": 1,
    "topic": "line-eq",
    "subtopic": "parametric",
    "theoryKey": "line-equations",
    "difficulty": "basic",
    "statement": "Hallar las ecuaciones paramétricas y continuas de la recta que pasa por A = (1; -2; 3) y B = (4; 1; -1).",
    "hint": "Vector director d = B - A",
    "answerType": "expression",
    "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}" },
    "solutionSteps": [
      { "desc": "Vector director d = B - A", "expression": "\\vec{d} = B - A = (4-1;\\;1-(-2);\\;-1-3) = (3;\\;3;\\;-4)" },
      { "desc": "Ecuaciones paramétricas", "expression": "x = 1 + 3t,\\quad y = -2 + 3t,\\quad z = 3 - 4t" },
      { "desc": "Ecuaciones continuas", "expression": "\\frac{x-1}{3} = \\frac{y+2}{3} = \\frac{z-3}{-4}" }
    ]
  },
  {
    "id": "cap01-11",
    "chapter": 1,
    "topic": "line-eq",
    "subtopic": "parametric",
    "theoryKey": "line-equations",
    "difficulty": "basic",
    "statement": "Hallar la ecuación de la recta que pasa por P = (2; 0; -1) y tiene vector director \\vec{v} = (1; 3; -2).",
    "hint": "r: X = P + tv",
    "answerType": "expression",
    "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)" },
    "solutionSteps": [
      { "desc": "Ecuación vectorial", "expression": "r: (x;\\;y;\\;z) = (2;\\;0;\\;-1) + t(1;\\;3;\\;-2)" },
      { "desc": "Paramétricas", "expression": "x = 2 + t,\\quad y = 3t,\\quad z = -1 - 2t" }
    ]
  },
  {
    "id": "cap01-12",
    "chapter": 1,
    "topic": "line-eq",
    "subtopic": "relative-position",
    "theoryKey": "line-positions",
    "difficulty": "intermediate",
    "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).",
    "hint": "Verificar si los directores son paralelos, luego calcular producto mixto [d1, d2, P2-P1]",
    "answerType": "expression",
    "answer": { "value": "Rectas que se cruzan", "latex": "\\text{Se cruzan}" },
    "solutionSteps": [
      { "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}" },
      { "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}" },
      { "desc": "Calcular el determinante", "expression": "= 1(1-3) - 1(-2+3) + (-1)(2-1) = -2 - 1 - 1 = -4 \\neq 0" },
      { "desc": "Conclusión", "expression": "[\\vec{d_1},\\vec{d_2},P_2-P_1] \\neq 0 \\Rightarrow \\text{Se cruzan}" }
    ]
  },
  {
    "id": "cap01-13",
    "chapter": 1,
    "topic": "line-eq",
    "subtopic": "distance",
    "theoryKey": "distance-lines",
    "difficulty": "intermediate",
    "statement": "Calcular la distancia del punto Q = (1; 2; 3) a la recta r: (x; y; z) = (0; 1; 0) + t(1; 0; 1).",
    "hint": "d = |(Q-P) × v| / |v|",
    "answerType": "numeric",
    "answer": { "value": 1.7321, "latex": "d = \\sqrt{3}" },
    "solutionSteps": [
      { "desc": "Q - P = (1;2;3) - (0;1;0) = (1;1;3)", "expression": "\\vec{QP} = Q - P = (1;\\;1;\\;3)" },
      { "desc": "(Q-P) × v", "expression": "(Q-P) \\times \\vec{v} = (1;1;3) \\times (1;0;1) = (1;\\;-2;\\;-1)" },
      { "desc": "Módulo del producto vectorial", "expression": "|(Q-P) \\times \\vec{v}| = \\sqrt{1+4+1} = \\sqrt{6}" },
      { "desc": "Módulo del director", "expression": "|\\vec{v}| = \\sqrt{1+0+1} = \\sqrt{2}" },
      { "desc": "Distancia", "expression": "d = \\frac{\\sqrt{6}}{\\sqrt{2}} = \\sqrt{3}" }
    ]
  },
  {
    "id": "cap01-14",
    "chapter": 1,
    "topic": "line-eq",
    "subtopic": "distance",
    "theoryKey": "distance-lines",
    "difficulty": "intermediate",
    "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).",
    "hint": "Si se cruzan: d = |[d1,d2,P2-P1]| / |d1×d2|",
    "answerType": "numeric",
    "answer": { "value": 0.7071, "latex": "d = \\frac{1}{\\sqrt{2}}" },
    "solutionSteps": [
      { "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)" },
      { "desc": "P2 - P1 = (-1; 1; 0)", "expression": "P_2 - P_1 = (0-1;\\;1-0;\\;0-0) = (-1;\\;1;\\;0)" },
      { "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" },
      { "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}}" }
    ]
  },
  {
    "id": "cap01-15",
    "chapter": 1,
    "topic": "line-eq",
    "subtopic": "angle",
    "theoryKey": "angle",
    "difficulty": "basic",
    "statement": "Hallar el ángulo entre las rectas con directores \\vec{v_1} = (1; 2; 3) y \\vec{v_2} = (2; -1; 1).",
    "hint": "cos α = |v1·v2| / (|v1|·|v2|)",
    "answerType": "numeric",
    "answer": { "value": 76.37, "latex": "\\alpha \\approx 76.37^\\circ" },
    "solutionSteps": [
      { "desc": "Producto escalar", "expression": "\\vec{v_1} \\cdot \\vec{v_2} = 2 - 2 + 3 = 3" },
      { "desc": "Módulos", "expression": "|\\vec{v_1}| = \\sqrt{14},\\quad |\\vec{v_2}| = \\sqrt{6}" },
      { "desc": "Coseno del ángulo", "expression": "\\cos\\alpha = \\frac{3}{\\sqrt{14}\\sqrt{6}} = \\frac{3}{\\sqrt{84}}" },
      { "desc": "Ángulo", "expression": "\\alpha = \\arccos\\left(\\frac{3}{\\sqrt{84}}\\right) \\approx 76.37^\\circ" }
    ]
  },
  {
    "id": "cap01-16",
    "chapter": 1,
    "topic": "plane-eq",
    "subtopic": "general",
    "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}" }
    ]
  }
]