# Cross Product Calculator

Calculate the cross product of two 3D vectors. Find the resulting perpendicular vector, its magnitude, and the parallelogram area.

## What this calculates

Compute the cross product of two 3D vectors. Enter the components of each vector to find the resulting vector, which is perpendicular to both inputs, along with its magnitude and the area of the parallelogram they form.

## Inputs

- **V1 x** — X component of vector 1.
- **V1 y** — Y component of vector 1.
- **V1 z** — Z component of vector 1.
- **V2 x** — X component of vector 2.
- **V2 y** — Y component of vector 2.
- **V2 z** — Z component of vector 2.

## Outputs

- **Result x** — X component of the cross product vector.
- **Result y** — Y component of the cross product vector.
- **Result z** — Z component of the cross product vector.
- **Magnitude** — The magnitude of the cross product vector, equal to |A||B|sin(θ).
- **Result Vector** — formatted as text — The cross product vector in angle-bracket notation.
- **Parallelogram Area** — The area of the parallelogram formed by the two vectors.

## Details

The cross product (or vector product) of two 3D vectors A and B produces a new vector that is perpendicular to both A and B. For A = (a1, a2, a3) and B = (b1, b2, b3), the cross product is A × B = (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1).

The magnitude of the cross product equals |A| |B| sin(theta), where theta is the angle between the vectors. This magnitude also equals the area of the parallelogram formed by A and B. The direction of the resulting vector follows the right-hand rule: point your fingers along A, curl them toward B, and your thumb points in the direction of A × B.

Unlike the dot product, the cross product is anti-commutative: A × B = -(B × A). The cross product is zero when the vectors are parallel (sin(0) = 0). Applications include computing torque, magnetic force, surface normals in 3D graphics, and determining the orientation of three points in space.

## Frequently Asked Questions

**Q: What is the cross product?**

A: The cross product of two 3D vectors produces a new vector perpendicular to both inputs. For A = (a1, a2, a3) and B = (b1, b2, b3), it is (a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1). It is defined only for 3D vectors.

**Q: Why is the cross product magnitude equal to the parallelogram area?**

A: The magnitude |A × B| = |A||B|sin(θ), where θ is the angle between the vectors. This is exactly the formula for the area of a parallelogram with sides |A| and |B| and included angle θ. Half this value gives the area of the triangle formed by the vectors.

**Q: What is the right-hand rule?**

A: The right-hand rule determines the direction of the cross product. Point your right hand's fingers along vector A, then curl them toward vector B. Your thumb points in the direction of A × B. Reversing the order (B × A) reverses the direction.

**Q: When is the cross product zero?**

A: The cross product is zero when the two vectors are parallel (or anti-parallel), because sin(0°) = sin(180°) = 0. It is also zero if either vector is the zero vector. A zero cross product means the vectors do not span a plane.

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Source: https://vastcalc.com/calculators/math/cross-product
Category: Math
Last updated: 2026-04-21