# Cross Product

The **cross product** of two vectors is another perpendicular vector to the two vectors.

The direction of the resultant vector can be determined by the right-hand rule. The thumb (u) and index finger (v) held perpendicularly to one another represent the vectors and the middle finger held perpendicularly to the index and thumb indicates the direction of the cross vector.

The magnitude of the cross product is:

The cross product can be expressed by the determinant:

#### Examples

Calculate the cross product of the vectors = (1, 2, 3) and = (−1, 1, 2).

Find the cross product of the vectors and and check that the resultant vector is orthogonal to and .

The cross product of is orthogonal to the vectors and .

#### Area of a Parallelogram

Geometrically, the magnitude of the cross product of two vectors coincides with the area of the parallelogram whose sides are formed by those vectors.

#### Example

Find the area of the parallelogram which is formed by the vectors and .

#### Area of a Triangle

#### Example

Determine the area of the triangle whose vertices are the points A = (1, 1, 3), B = (2, −1, 5) and C = (−3, 3, 1).

### Cross Product Properties

1. Anticommutative.

x = − x

2. Compatible with scalar multiplication.

**k** ( x ) = (**k**) x = x (k)

3. Distributive over addition.

x ( + ) = x + x ·

4. The cross product of two parallel vectors is equal to the zero vector.

x =

5. The cross product x is perpendicular to and .