Cross Product Calculator

The cross product of two 3D vectors A and B produces a third vector perpendicular to both. The formula is A × B = (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1). The magnitude of the cross product equals |A| * |B| * sin(θ), where θ is the angle between them.

The right-hand rule determines the direction of the cross product. Point your right hand's fingers in the direction of A, curl them toward B, and your thumb points in the direction of A × B. This is crucial because B × A points in the opposite direction.

A zero cross product means the vectors are parallel (or one is zero). Parallel vectors are either in the same direction or opposite directions. Geometrically, they lie on the same line.

The cross product produces a vector perpendicular to both input vectors. In 3D space, there is exactly one perpendicular direction (up to orientation). In 2D, perpendicularity doesn't work the same way. In higher dimensions, multiple perpendicular directions exist.

Cross products are used in physics (torque = force × position), computer graphics (computing surface normals), engineering (angular momentum), and electromagnetism (Lorentz force). They are essential for 3D geometric calculations.