Three Dimensional Geometry — Important Questions

a₁a₂ + b₁b₂ + c₁c₂ = 0, The angle between them is 90°, cos θ = 0

Two lines are perpendicular when the angle between them is 90°, which means cos θ = 0. This occurs when the dot product of their direction ratios equals zero: a₁a₂ + b₁b₂ + c₁c₂ = 0. The condition a₁/a₂ = b₁/b₂ = c₁/c₂ is for parallel lines.

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By definition, direction cosines are the cosines of angles made by a line with coordinate axes. Since they form a unit vector, the sum of squares of direction cosines is always equal to 1: l² + m² + n² = 1.

r = (î + 2ĵ + 3k̂) + λ(3î + 3ĵ + 3k̂)

The vector equation is r = a + λ(b - a) where a and b are position vectors of the given points. Here: a = î + 2ĵ + 3k̂, b = 4î + 5ĵ + 6k̂. Direction vector = b - a = 3î + 3ĵ + 3k̂. So equation is r = (î + 2ĵ + 3k̂) + λ(3î + 3ĵ + 3k̂).

cos²α + cos²β + cos²γ = 1, Direction cosines are cos α, cos β, cos γ

By definition, direction cosines are cos α, cos β, cos γ where α, β, γ are angles with coordinate axes. The fundamental property is cos²α + cos²β + cos²γ = 1. The angles α, β, γ don't necessarily sum to 180°, and the sum of squares of sines is not always 2.