Cross Product Calculator

Enter two 3-D vectors to get their cross product — the vector perpendicular to both — and its magnitude.

Cross product (a × b) —
Magnitude —

How to Calculate a Cross Product

Enter two 3-D vectors.

1

Enter the two vectors

Type three components for each vector, separated by commas or spaces. The cross product is only defined in three dimensions.

2

Read the resulting vector

The cross product is a new vector perpendicular to both inputs, following the right-hand rule.

3

See the magnitude

The magnitude equals the area of the parallelogram the two vectors span; it is zero when they are parallel.

Frequently Asked Questions

How do you calculate the cross product?
Each component of a × b is a difference of products: a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁). For (1, 2, 3) × (4, 5, 6) that gives (2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4) = (−3, 6, −3). The calculator shows this expansion term by term. Swapping the order flips every sign, because b × a = −(a × b).
What does the cross product represent?
The cross product returns a vector perpendicular to both inputs, with a length equal to the area of the parallelogram they span. That length is |a||b| sin θ, so parallel vectors give the zero vector and perpendicular unit vectors give length 1. The direction follows the right-hand rule, which is why the order of the operands matters. It is used for surface normals, torque and angular momentum.
Why is the cross product only defined in 3-D?
The familiar cross product that returns a perpendicular vector exists only in three dimensions, plus a special case in seven. In two dimensions there is no third axis for the result to point along, so the analogous operation returns a single scalar, a₁b₂ − a₂b₁. In four dimensions or more, a pair of vectors leaves more than one perpendicular direction, so no single answer is determined. The general replacement is the wedge product, which returns an oriented area rather than a vector.
Is my data sent anywhere?
Nothing you type is sent anywhere. The cross product calculator runs entirely in your browser with JavaScript, so both vectors never leave your device. There is no account, no upload step and nothing is logged. The page keeps working offline once it has loaded.

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