Math

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Cross Product Calculator.

Calculate the cross product of two three-dimensional vectors.

On-device calculationNo signup
01

Set your values

Results update as you type.

x: 0

x

0.000000
y: 0

y

0.000000
z: 1

z

0.000000
Magnitude: 1

Magnitude

0.000000

Results update automatically as you type.

Use Cases

Find a vector perpendicular to two given vectors

In 3D geometry and physics, the cross product yields a vector orthogonal to both input vectors, useful for constructing normals to planes or axes of rotation.

Example: Given a = (1, 0, 0) and b = (0, 1, 0), the cross product is (0, 0, 1), the z-axis.

Compute torque or angular momentum

In physics, torque is the cross product of the position vector and force vector. This calculator helps quickly find the resulting torque vector.

Example: If r = (2, 0, 0) m and F = (0, 3, 0) N, torque = (0, 0, 6) N·m.

Frequently Asked Questions

What is the cross product of two vectors?
The cross product of two 3D vectors a and b is a new vector that is perpendicular to both a and b. Its magnitude equals the area of the parallelogram formed by the vectors. The direction follows the right-hand rule.
How is the cross product calculated?
Given vectors a = (a1, a2, a3) and b = (b1, b2, b3), the cross product a × b = (a2*b3 - a3*b2, a3*b1 - a1*b3, a1*b2 - a2*b1). This calculator uses these formulas with your input components.
What is the result of the cross product of two parallel vectors?
If two vectors are parallel (or one is zero), their cross product is the zero vector (0, 0, 0). This indicates that the vectors do not form a parallelogram with nonzero area.

Tips & Common Mistakes

Tips

  • Double-check that you enter the components in the correct order: x, y, z for each vector.
  • Remember that the cross product is not commutative: a × b = -(b × a). Swapping the vectors reverses the direction.
  • If you need the magnitude of the cross product, you can calculate it from the result using the Pythagorean theorem in 3D.

Common Mistakes to Avoid

  • Entering the components of the two vectors in the wrong order, which reverses the direction of the result.
  • Forgetting that the cross product is only defined for 3D vectors; this calculator expects exactly three components for each vector.
  • Assuming the cross product of two parallel vectors is nonzero; it is always the zero vector.

Last updated: August 13, 2026