Cross Product Calculator
Compute the cross product of two 3D vectors. Enter the components of each vector to find the resulting vector, which is perpendicular to both inputs, along with its magnitude and the area of the parallelogram they form.
The cross product (or vector product) of two 3D vectors A and B produces a new vector that is perpendicular to both A and B. For A = (a1, a2, a3) and B = (b1, b2, b3), the cross product is A × B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).
The magnitude of the cross product equals |A| |B| sin(theta), where theta is the angle between the vectors. This magnitude also equals the area of the parallelogram formed by A and B. The direction of the resulting vector follows the right-hand rule: point your fingers along A, curl them toward B, and your thumb points in the direction of A × B.
Unlike the dot product, the cross product is anti-commutative: A × B = -(B × A). The cross product is zero when the vectors are parallel (sin(0) = 0). Applications include computing torque, magnetic force, surface normals in 3D graphics, and determining the orientation of three points in space.