Cross Product Calculator

Use this calculator to find the vector product of two three-dimensional vectors. The result is a new vector that is perpendicular to both inputs, which makes cross products important in physics, engineering, robotics, and 3D graphics.

How to use this calculator

Enter Vector 1 components X, Y, and Z, then enter Vector 2 components X, Y, and Z. The calculator returns v₁ × v₂, the individual i, j, and k components, and the magnitude of the result.

Cross product formula

For vectors A = (a1, a2, a3) and B = (b1, b2, b3), the cross product A x B is:

A x B = (a2 b3 - a3 b2, a3 b1 - a1 b3, a1 b2 - a2 b1)

The magnitude of the result is related to the angle between the vectors:

|A x B| = |A| |B| sin(theta)

For example, vector 1 (1, 0, 0) and vector 2 (0, 1, 0) give (0, 0, 1) with magnitude 1.

Interpreting the result

The direction follows the right-hand rule. A zero vector result means the inputs are parallel, anti-parallel, or one vector has zero length. For related math tools, try the slope calculator, matrix calculator, or relative velocity calculator.