Three Dimensional Geometry

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Direction cosines are the cosines of the angles made by a line with the positive directions of the x, y, and z axes. If a line makes angles α, β, and γ with the x, y, and z axes respectively, then cos…

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For any line in space, if l, m, n are the direction cosines, then: l² + m² + n² = 1. This is because the sum of squares of direction cosines of any line is always equal to 1.

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Given P(1, 2, 3) and Q(4, 6, 8) Direction ratios = (4-1, 6-2, 8-3) = (3, 4, 5) PQ = √(3² + 4² + 5²) = √(9 + 16 + 25) = √50 = 5√2 Direction cosines = (3/5√2, 4/5√2, 5/5√2) = (3/5√2, 4/5√2, 1/√2)…

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Direction ratios are any three numbers which are proportional to the direction cosines of a line. If l, m, n are direction cosines and a, b, c are direction ratios, then a = λl, b = λm, c = λn for som…

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If a, b, c are direction ratios, then direction cosines are: l = ±a/√(a² + b² + c²) m = ±b/√(a² + b² + c²) n = ±c/√(a² + b² + c²) The sign depends on the orientation of the line.

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The vector equation of the line is: r⃗ = a⃗ + λb⃗ where r⃗ is the position vector of any point on the line, a⃗ is the position vector of the given point A, b⃗ is the parallel vector, and λ is a parame…

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The Cartesian equation is: (x - x₁)/a = (y - y₁)/b = (z - z₁)/c This is also called the symmetric form of the equation of a line.

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Given point: (2, -1, 3), Direction vector: 3î + 4ĵ - 5k̂ Direction ratios: 3, 4, -5 Cartesian equation: (x - 2)/3 = (y + 1)/4 = (z - 3)/(-5) Vector equation: r⃗ = (2î - ĵ + 3k̂) + λ(3î + 4ĵ - 5k̂)…