Three Dimensional Geometry

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.

1

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.

1/√3

For parallel lines r = a₁ + λb and r = a₂ + μb, distance = |b × (a₂ - a₁)|/|b|. Here: a₁ = î + 2ĵ, a₂ = 2î + ĵ, b = î + ĵ + k̂. a₂ - a₁ = î - ĵ. b × (a₂ - a₁) = (î + ĵ + k̂) × (î - ĵ) = ĵ + k̂. |b × (a₂ - a₁)| = √2, |b| = √3. Distance = √2/√3 = √(2/3) = 1/√3.

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).