The Cross Product

u × v is a vector perpendicular to both inputs, following the right-hand rule. Its magnitude, |u||v| sin θ, is the area of the parallelogram the vectors span — zero when they are parallel, maximum when perpendicular.

Computation

Component-wise: u × v = (uy·vz − uz·vy, uz·vx − ux·vz, ux·vy − uy·vx). A convenient memory aid is the determinant of the matrix with unit vectors i, j, k in the first row and the vector components in the next two.

Where It Appears

Torque (τ = r × F), angular momentum (L = r × p), and the Lorentz force (F = qv × B) all use cross products. In graphics and geometry, cross products produce surface normals and compute signed areas — essential for lighting, collision, and mesh work.