Three Dimensional Geometry

Direction cosines (l

Direction cosines (l, m, n) are cosines of angles a line makes with positive x, y, z axes respectively, where l² + m² + n² = 1. Direction ratios (a, b

Vector form

Vector form: r⃗ = a⃗ + λb⃗ where a⃗ is position vector of a point on line and b⃗ is direction vector. Cartesian form: (x-x₁)/a = (y-y₁)/b = (z-z₁)/c w

For lines with direction ratios (a₁

For lines with direction ratios (a₁,b₁,c₁) and (a₂,b₂,c₂), the acute angle θ satisfies: cos θ = |a₁a₂ + b₁b₂ + c₁c₂|/[√(a₁² + b₁² + c₁²) × √(a₂² + b₂²

Lines that are neither parallel nor

Lines that are neither parallel nor intersecting, lying in different planes. Most pairs of lines in 3D space are skew lines. Examples include diagonal

For skew lines r⃗ = a⃗₁

For skew lines r⃗ = a⃗₁ + λb⃗₁ and r⃗ = a⃗₂ + μb⃗₂, shortest distance = |(b⃗₁ × b⃗₂) · (a⃗₂ - a⃗₁)|/|b⃗₁ × b⃗₂|. For parallel lines r⃗ = a⃗₁ + λb⃗ and