# Cross Product Calculator

URL: https://speedycalculator.com/calculators/math/cross-product/
Last updated: 2026-10-30

> The cross product calculator multiplies two 3D vectors a × b, giving a vector perpendicular to both, its magnitude (the area of the parallelogram) and the angle between a and b.

## Inputs

- a – x
- a – y
- a – z
- b – x
- b – y
- b – z

## Results

- c = a × b: x
- c = a × b: y
- c = a × b: z
- |a × b| (parallelogram area)
- Triangle area
- Angle between a and b

## Formula

```
a × b = (a_y b_z − a_z b_y, a_z b_x − a_x b_z, a_x b_y − a_y b_x)
|a × b| = |a||b| sin θ
```

## Explanation

The cross product gives the axis of rotation for torque (τ = r × F), the normal vector of a plane, the area of a triangle in 3D and the direction of magnetic forces. It follows the right-hand rule.

### How to use the Cross Product Calculator

- **a – x**
- **a – y**
- **a – z**
- **b – x**
- **b – y**
- **b – z**

Results update as you type, and everything is calculated in your browser.

### How it works

Each component is a 2 × 2 determinant of the other two coordinates. The magnitude equals the area of the parallelogram spanned by a and b; half of it is the triangle area.

### Inputs explained

FieldWhat to entera – xA numbera – yA numbera – zA numberb – xA numberb – yA numberb – zA number

### What each result means

- **c = a × b: x** = [a – y]×[b – z] - [a – z]×[b – y]
- **c = a × b: y** = [a – z]×[b – x] - [a – x]×[b – z]
- **c = a × b: z** = [a – x]×[b – y] - [a – y]×[b – x]
- **|a × b| (parallelogram area)** = sqrt(([a – y]×[b – z] - [a – z]×[b – y])^2 + ([a – z]×[b – x] - [a – x]×[b – z])^2 + ([a – x]×[b – y] - [a – y]×[b – x])^2)
- **Triangle area** = sqrt(([a – y]×[b – z] - [a – z]×[b – y])^2 + ([a – z]×[b – x] - [a – x]×[b – z])^2 + ([a – x]×[b – y] - [a – y]×[b – x])^2)÷2

### Example values

How the main result changes when **a – x** changes and everything else stays at the example values above:

a – xc = a × b: x1-31.5-32-32.5-33-34-3

## Worked example

Inputs — a – x: 2; a – y: 3; a – z: 4; b – x: 5; b – y: 6; b – z: 7.

Results — **c = a × b: x = -3**; c = a × b: y = 6; c = a × b: z = -3; |a × b| (parallelogram area) = 7.34846923; Triangle area = 3.67423461; Angle between a and b = 7.4758 °.

## Key terms

- **Cross product**: A vector perpendicular to two given 3D vectors.

## FAQ

**Is a × b the same as b × a?**
No: b × a = −(a × b).

**What does a zero cross product mean?**
The vectors are parallel (or one is zero).

**Does the cross product exist in 2D?**
Not as a vector; the scalar a_x b_y − a_y b_x gives the signed area.

**What is the right-hand rule?**
Point your fingers along a, curl them towards b, and your thumb points along a × b.


## Sources

- [OpenStax – Calculus Volume 3](https://openstax.org/details/books/calculus-volume-3)
- [Wolfram MathWorld – Cross product](https://mathworld.wolfram.com/CrossProduct.html)