Introduction To Three- Dimensional Geometry

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A point P in 3D space is represented as P(x, y, z) where: • x = perpendicular distance from YZ-plane • y = perpendicular distance from ZX-plane • z = perpendicular distance from XY-plane These coordi

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Step-by-step identification: (a) (2, -3, 4): x > 0, y < 0, z > 0 → Octant XY'Z (b) (-1, -2, -3): x < 0, y < 0, z < 0 → Octant X'Y'Z' (c) (5, 6, -2): x > 0, y > 0, z < 0 → Octant XYZ' Rule: Check sign

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Distance Formula: |PQ| = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] Special Case - Distance from origin: |OP| = √[x² + y² + z²] This extends the 2D distance formula by adding the z-component difference

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Step-by-step solution: Given: A(2, -1, 3) and B(-1, 2, 1) Step 1: Apply distance formula |AB| = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] Step 2: Substitute values |AB| = √[(-1 - 2)² + (2 - (-1))² + (

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Internal Division Formula: If point R divides line segment PQ internally in ratio m:n, then: R = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)) Where P(x₁, y₁, z₁) and Q(x₂, y₂, z₂)

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Step-by-step solution: Given: A(1, -2, 3), B(3, 4, -1), ratio = 2:1 Step 1: Apply internal section formula P = ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)) Step 2: Substitute m =

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External Division Formula: If point R divides line segment PQ externally in ratio m:n, then: R = ((mx₂ - nx₁)/(m - n), (my₂ - ny₁)/(m - n), (mz₂ - nz₁)/(m - n)) Where P(x₁, y₁, z₁) and Q(x₂, y₂, z₂)

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Direction Cosines: If a line makes angles α, β, γ with positive X, Y, Z axes respectively, then: l = cos α, m = cos β, n = cos γ are called direction cosines. Fundamental Relationship: l² + m² + n² =