What does the cross product return?

The cross product returns a vector that is perpendicular to both input vectors.

When is the cross product useful?

It is useful in physics, engineering, geometry, and any situation where you need an area or normal vector.

Can I use 2D vectors?

This calculator is optimized for 3D vectors, which is the standard form for the cross product.

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Cross Product Calculator

Calculate the vector cross product of two 3D vectors.

Vector A

Vector B

Cross Product

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Magnitude

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Comprehensive Guide

The cross product takes two 3D vectors and produces a new vector that is perpendicular to both. That makes it useful whenever direction matters, not just size. In geometry, physics, and engineering, the result helps describe areas, rotation, torque, and surface normals.

The formula is deterministic: if you know the x, y, and z components of both vectors, the output vector is fixed. That makes a cross product calculator valuable for students learning linear algebra and for technical work where vector math must be quick and accurate.

Real-World Examples

In physics, a force applied away from a pivot creates torque, and torque is often expressed with a cross product. The direction of the resulting vector tells you the axis of rotation, while the magnitude reflects how strongly the force twists the object. A larger angle between vectors generally creates a larger result.

In computer graphics and geometry, the cross product helps determine a surface normal, which is how software knows which way a polygon faces. That normal affects lighting, shading, and collision calculations. If the vectors are parallel, the result is zero because there is no perpendicular direction to extract.

Reference Table

Input A	Input B	Output
(1, 0, 0)	(0, 1, 0)	(0, 0, 1)
(2, 0, 0)	(3, 0, 0)	(0, 0, 0)
(1, 2, 3)	(4, 5, 6)	(-3, 6, -3)

Frequently Asked Questions

What does the result mean?

The result is a new vector perpendicular to both inputs. It is not a single number like a dot product. The direction and magnitude both matter.

What if the vectors are parallel?

Parallel vectors produce a zero vector because there is no unique perpendicular direction. In that case, the magnitude is zero and the cross product carries no area information.

Why is the sign important?

The sign reflects orientation. Swapping the order of the vectors flips the direction of the result. That is why a × b is the negative of b × a.

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