# What is the cross product?

Computes the cross product A × B of two 3D vectors, a vector perpendicular to both, and its length, the area of the parallelogram they span.

- Page: https://www.acalculator.org/math/cross-product-calculator
- JSON spec: https://www.acalculator.org/math/cross-product-calculator.json
- Version: 24614ab6117a

## Default answer

Example with the default inputs (A: x 2, A: y 3, A: z 1, B: x 1, B: y 2, B: z 3): The cross product of A and B is (7, -5, 1), with length 8.660254.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| ax | A: x | The x component of vector A. |
| ay | A: y | The y component of vector A. |
| az | A: z | The z component of vector A. |
| bx | B: x | The x component of vector B. |
| by | B: y | The y component of vector B. |
| bz | B: z | The z component of vector B. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| vector | A × B | The cross product as (x, y, z), each part rounded to 10 significant figures. |
| x | x | The x component of A × B: a₂b₃ − a₃b₂. |
| y | y | The y component of A × B: a₃b₁ − a₁b₃. |
| z | z | The z component of A × B: a₁b₂ − a₂b₁. |
| magnitude | Length \|A × B\| | The length of A × B, which is also the area of the parallelogram spanned by A and B. |
| steps | Working | Each component of A × B worked out with the numbers. |

## Method

A × B = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁); |A × B| = √(x² + y² + z²).

## Assumptions

- A and B are vectors in 3D space with real components.
- The direction of A × B follows the right-hand rule, so B × A = −(A × B).

## Worked examples

1. ax = 2, ay = 3, az = 1, bx = 1, by = 2, bz = 3 gives x = 7, y = -5, z = 1, vector = (7, -5, 1), magnitude = 8.660254. Source: OpenStax, Calculus Volume 3, §2.4 The Cross Product. https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product.
2. ax = 1, ay = 0, az = 0, bx = 0, by = 1, bz = 0 gives x = 0, y = 0, z = 1, magnitude = 1. Source: OpenStax, Calculus Volume 3, §2.4 The Cross Product. https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product.
3. ax = 1, ay = 2, az = 3, bx = 4, by = 5, bz = 6 gives x = -3, y = 6, z = -3, magnitude = 7.348469. Source: OpenStax, Calculus Volume 3, §2.4 The Cross Product. https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product.
4. ax = 2, ay = 4, az = 6, bx = 1, by = 2, bz = 3 gives x = 0, y = 0, z = 0, magnitude = 0. Source: OpenStax, Calculus Volume 3, §2.4 The Cross Product. https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product.

## FAQ

### What is the cross product of two vectors?

The cross product (also called vector product) of two vectors is a vector that is perpendicular to both input vectors. It's calculated using the formula: A × B = |A| |B| sin(θ) n, where θ is the angle between the vectors and n is a unit vector perpendicular to both A and B. The direction follows the right-hand rule.

### How do I calculate the cross product of two 3D vectors?

For vectors A = [a₁, a₂, a₃] and B = [b₁, b₂, b₃], the cross product A × B = [a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁]. This can be remembered using the determinant formula with the unit vectors i, j, k.

### What is the geometric interpretation of the cross product?

The magnitude of the cross product |A × B| equals the area of the parallelogram formed by vectors A and B. The direction of A × B is perpendicular to the plane containing A and B, following the right-hand rule. This makes cross products useful in physics, computer graphics, and engineering.

### What is the right-hand rule?

The right-hand rule determines the direction of the cross product: Point your right hand's index finger in the direction of the first vector, your middle finger in the direction of the second vector, and your thumb will point in the direction of the cross product. This rule ensures consistent direction determination across all cross product calculations.

### Is the cross product commutative?

No, the cross product is not commutative. In fact, A × B = -(B × A). This means changing the order of the vectors changes the sign of the result. This property is called anti-commutativity and is important in many applications.

### What are the applications of cross products?

Cross products are used in physics for torque calculations, in computer graphics for surface normals, in engineering for moment calculations, in electromagnetism for magnetic force, and in robotics for angular velocity. They're essential for any calculation involving perpendicular vectors or rotational motion.

### Can I calculate cross products in 2D?

In 2D, the cross product is a scalar (single number) rather than a vector. It represents the signed area of the parallelogram formed by the two vectors. The sign indicates the orientation (clockwise or counterclockwise) of the angle from the first vector to the second.

### What is the relationship between dot product and cross product?

The dot product and cross product are complementary operations. The dot product gives a scalar result and measures how much two vectors point in the same direction. The cross product gives a vector result and measures how much two vectors are perpendicular. Together, they provide complete information about the relationship between two vectors.

### How do I find a vector perpendicular to two given vectors?

The cross product of two vectors automatically gives you a vector perpendicular to both. This is one of the most common uses of the cross product. The resulting vector will be perpendicular to the plane containing the two input vectors.

### What is the magnitude of a cross product?

The magnitude of A × B is |A × B| = |A| |B| sin(θ), where θ is the angle between vectors A and B. This equals the area of the parallelogram formed by the two vectors. When the vectors are parallel (θ = 0° or 180°), the cross product is zero.

## Sources

- OpenStax, Calculus Volume 3, §2.4 The Cross Product. https://openstax.org/books/calculus-volume-3/pages/2-4-the-cross-product