The Planes

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General equation: ax + by + cz + d = 0 Geometrically: This represents a flat surface extending infinitely in all directions. Key properties: • If any two points lie on the plane, the entire line join

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Step-by-step solution: Step 1: Use the general form for a plane through point (x₁, y₁, z₁) a(x - x₁) + b(y - y₁) + c(z - z₁) = 0 Step 2: Substitute (x₁, y₁, z₁) = (2, -1, 3) a(x - 2) + b(y - (-1)) +

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Step-by-step derivation: Step 1: Start with plane through A(x₁, y₁, z₁) a(x - x₁) + b(y - y₁) + c(z - z₁) = 0 ... (1) Step 2: Since B and C lie on the plane: a(x₂ - x₁) + b(y₂ - y₁) + c(z₂ - z₁) = 0

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Intercept Form: x/a + y/b + z/c = 1 where a, b, c are x, y, z intercepts respectively. Derivation: • Plane passes through (a,0,0), (0,b,0), (0,0,c) • Using three-point formula and simplifying gives x

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Normal Form: lx + my + nz = p where l, m, n are direction cosines of normal and p is perpendicular distance from origin. Alternative form: x cos α + y cos β + z cos γ = p Conversion Process: Given:

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Formula: cos θ = |a₁a₂ + b₁b₂ + c₁c₂|/[√(a₁² + b₁² + c₁²) × √(a₂² + b₂² + c₂²)] Step 1: Identify coefficients Plane 1: a₁ = 3, b₁ = -2, c₁ = 6 Plane 2: a₂ = 2, b₂ = -1, c₂ = 2 Step 2: Calculate nume

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Conditions for planes a₁x + b₁y + c₁z + d₁ = 0 and a₂x + b₂y + c₂z + d₂ = 0: (a) PARALLEL: a₁/a₂ = b₁/b₂ = c₁/c₂ Example: 2x + 3y - z + 5 = 0 and 4x + 6y - 2z + 7 = 0 Check: 2/4 = 3/6 = -1/(-2) = 1/2

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Formula: Distance = |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²) where (x₁, y₁, z₁) is the given point. Step 1: Identify values Plane: 2x - 3y + 6z - 14 = 0 So a = 2, b = -3, c = 6, d = -14 Point: (x₁, y₁,