What does the cross product represent geometrically?

The cross product A x B produces a vector perpendicular to both A and B. Its direction follows the right-hand rule, and its magnitude equals the area of the parallelogram spanned by A and B.

Is the cross product commutative?

No. A x B = -(B x A). Swapping the order reverses the direction of the result. The magnitude stays the same.

What does a zero cross product mean?

A zero result means the two vectors are parallel (or at least one is the zero vector). Parallel vectors span no area, so the perpendicular vector has zero length.

What is the unit vector shown?

The unit vector is the cross product divided by its magnitude, giving a vector of length 1 in the same direction. It is undefined when the cross product is the zero vector.

Does cross product work in 2D?

The classical cross product is defined only for 3D vectors. For 2D vectors you can set z = 0; the result's only non-zero component will be the z-component, which equals the scalar 2D cross product.

3D Cross Product Calculator

Name: 3D Cross Product Calculator
Availability: InStock
Author: Nham Vu

Enter two 3D vectors and instantly get their cross product, magnitude, and unit vector.

Vector Inputs

Vector A

Vector B

Formula

A × B = (Ay·Bz − Az·By,

Az·Bx − Ax·Bz,

Ax·By − Ay·Bx)

Results

Cross Product A × B

i component

j component

k component

( 0, 0, 1 )

Magnitude |A × B|

Unit Vector (normalized)

Parallelogram Area

Same as |A × B|

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Summary

Enter two 3D vectors and instantly get their cross product, magnitude, and unit vector.

How it works

Enter the x, y, and z components for vector A.
Enter the x, y, and z components for vector B.
The calculator applies the cross product formula: A x B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx).
The resulting perpendicular vector is displayed with each component.
The magnitude |A x B| is computed as the Euclidean length of the result vector.
The unit vector (normalized direction) is shown if the magnitude is non-zero.

Use cases

Find a vector perpendicular to two given vectors in 3D space.
Compute surface normals for 3D graphics and rendering.
Calculate torque or angular momentum in physics problems.
Determine the area of a parallelogram formed by two vectors.
Verify vector orthogonality in linear algebra coursework.
Cross-check hand calculations for engineering assignments.

Frequently Asked Questions

Last updated: 2026-06-09 · Reviewed by Nham Vu