Angle between two lines

The angle between two lines is the angle between their directions; if the lines do not meet, use parallels through one point.

Write the direction ratios of the two lines as (p₁, q₁, r₁) and (p₂, q₂, r₂). The acute angle θ has cos θ = |p₁p₂ + q₁q₂ + r₁r₂| divided by the product of the two lengths √(p² + q² + r²); in vector form, |b₁·b₂|/(|b₁||b₂|).

With direction cosines the denominators are 1: cos θ = |l₁l₂ + m₁m₂ + n₁n₂|.

sin θ uses the cross product instead: sin θ = |b₁ × b₂|/(|b₁||b₂|), where the top is the length of the vector with components q₁r₂ − q₂r₁, r₁p₂ − r₂p₁ and p₁q₂ − p₂q₁.

Perpendicular lines: a₁a₂ + b₁b₂ + c₁c₂ = 0. Parallel lines: a₁/a₂ = b₁/b₂ = c₁/c₂.

Worked: directions î + 2ĵ + 2k̂ and 3î + 2ĵ + 6k̂ have dot product 19 and lengths 3 and 7, so cos θ = 19/21.

Worked: ratios 3, 5, 4 and 1, 1, 2 give cos θ = 16/(√50 · √6) = 16/(10√3) = 8√3/15.