Vector product gives perpendicular vector. Magnitude equals parallelogram area. Direction by right-hand rule.
Physics — Torque: τ = r × F.
Physics — Magnetic force: F = qv × B.
Physics — Angular momentum: L = r × p.
Computer graphics — Surface normal calculation for lighting.
3D modeling — Determining face orientations.
Robotics — Rotation axis calculations.
Navigation — Course correction in 3D.
Engineering — Moment calculations.
Gaming — Determining "inside" vs "outside" of polygons.
Fluid dynamics — Vorticity calculations.
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