What does the cross product represent geometrically?

The cross product A × B produces a vector that is perpendicular (orthogonal) to both A and B. Its direction follows the right-hand rule, and its magnitude equals the area of the parallelogram formed by A and B.

What does a zero cross product mean?

A zero cross product means the two vectors are parallel (or anti-parallel), including the case where one or both vectors are the zero vector. Parallel vectors span no area, so the perpendicular is undefined.

Is the cross product commutative?

No. The cross product is anti-commutative: A × B = −(B × A). Switching the order reverses the direction of the resulting vector.

What is the unit vector?

The unit vector is the cross product divided by its own magnitude, giving a vector of length 1 pointing in the same direction. It represents only the direction of the cross product, not its scale.

How is the angle between A and B calculated?

The angle θ is derived from the dot product formula: cos(θ) = (A · B) / (|A| × |B|). The result is in degrees. If either vector is zero, the angle is undefined.

Cross Product Calculator

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

Enter two 3D vectors to instantly compute their cross product, magnitude, unit vector, and the angle between them.

Enter Vectors

A Vector A

A_x

A_y

A_z

B Vector B

B_x

B_y

B_z

Formula

A × B = (A_yB_z − A_zB_y, A_zB_x − A_xB_z, A_xB_y − A_yB_x)

Cross Product A × B

— i + — j + — k

Magnitude |A × B|

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Angle Between A & B

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Parallelogram Area

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Vectors Parallel?

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Unit Vector (A × B) / |A × B|

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Summary

Enter two 3D vectors to instantly compute their cross product, magnitude, unit vector, and the angle between them.

How it works

Enter the X, Y, and Z components of the first vector (A).
Enter the X, Y, and Z components of the second vector (B).
Click "Calculate" (or results update automatically).
Read the resulting cross product vector A × B.
Review the magnitude, unit vector, and angle between A and B.

Use cases

Find a vector perpendicular to a surface defined by two edge vectors.
Compute torque or angular momentum in physics problems.
Determine surface normals for 3D graphics and lighting calculations.
Verify solutions to linear algebra homework involving vector operations.
Check if two vectors are parallel (cross product equals zero).
Calculate the area of a parallelogram spanned by two vectors.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu