## Vectors and their Operations: Cross product

### The cross product and its properties

The cross product of two vectors is an operation that takes two vectors and and returns a vector . This operation is denoted by and is read “C equals A cross B”. The magnitude of is defined as,

(2.22)

in which is the angle between the two vectors.

The definition of the magnitude of the cross product implies that, if two non-zero vectors are parallel.

By definition, is perpendicular to both and , which are not parallel, and its sense of direction is specified by the right-hand rule. In other words, is **perpendicular** to the **plane** containing the non-parallel vectors and such that (or ) is in the direction of the thumb following the **right-hand rule** (Fig. 2.26a). The right-hand rule can also be equivalently demonstrated by the **right-hand three-finger rule **as demonstrated in Fig. 2.26b.

If is the unit vector of , the resultant of a cross product can be written as,

(2.23)

The following interactive tool illustrates . Use the sliders to change the angles of and with the Cartesian axes, and observe the resultant of their cross product.

##### Properties of Cross product

- Anti-commutativity:
- Associativity: for being a scalar,
- Distributivity:

### Cross product in CVN

In a Cartesian frame the following relationships are readily obtained for the mutually perpendicular unit vectors , and .

(2.24)

Based on the above results and the properties of the cross product, the cross product of two vectors in their CVN is,

leading to,

(2.25)

This result need not to be memorized and can be obtained by calculating the following determinant symbolically.

(2.26)

As a reminder, the determinant of a two-by-two matrix is a scalar defined as,

Consequently, the determinant of a three-by-three matrix is a scalar defined as

The following tool calculates the cross product . The vector is drawn in green.