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.

How do I find the area of a triangle with the cross product?

Take half the magnitude of the cross product of two edges that start at the same corner. For edges (1, 2, 3) and (4, 5, 6), the triangle area is 7.3485 ÷ 2 = 3.6742.

Can I take the cross product of 2D vectors?

Enter 0 for both z components. The result points straight along the z axis, and its z component, a₁b₂ − a₂b₁, is the value often called the 2D cross product.

Cross Product Calculator

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

The cross product of A = (1, 2, 3) and B = (4, 5, 6) is A × B = (−3, 6, −3), with a magnitude of 7.3485.

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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Quick answer

The cross product of A = (1, 2, 3) and B = (4, 5, 6) is A × B = (−3, 6, −3), with a magnitude of 7.3485. The components are (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). The result is perpendicular to both vectors, and its length equals the area of the parallelogram they span.

Examples

A	B	A × B	\|A × B\|
(1, 2, 3)	(4, 5, 6)	(−3, 6, −3)	7.3485
(1, 0, 0)	(0, 1, 0)	(0, 0, 1)	1
(3, −3, 1)	(4, 9, 2)	(−15, −2, 39)	41.833
(2, 4, 6)	(1, 2, 3)	(0, 0, 0)	0, parallel

Enter the two vectors.

Summary

The cross product of two 3D vectors A and B produces a third vector that is perpendicular to both, following the right-hand rule. This calculator instantly computes the resulting vector, its magnitude, the unit vector in that direction, and the angle between A and B. All computation is done in your browser with no data sent to any server.

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: 2026-05-29 · Reviewed by Nham Vu