__maths__>__Properties of Vector Arithmetics__>__Properties of Cross Product__### Cross Product : Not Commutative

In this page, you will learn that the vector cross product is not commutative.

*click on the content to continue..*

What does 'commute' mean?

- to go to and fro on a regular basis
- to go to and fro on a regular basis
- it does not mean anything

The answer is 'to go to and fro between two places on a regular basis'.

Given the definition of cross product as

`vec p xx vec q = |vec p||vec q|sin theta hat n`

What is `vec q xx vec p`?

- `vec p xx vec q`
- `|p||q|sin(theta) hat n`
- `|p||q|sin(-theta) hat n`
- `|p||q|sin(-theta) hat n`
- all the above

The answer is '`|p||q|sin(-theta) hat n`'. The angle traversed from `vec q` to `vec p` is `-theta`.

Given the definition of cross product as

`vec p xx vec q = |(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)|`

What is `vec q xx vec p`?

- `-vec p xx vec q`
- `|(i, j, k),(q_x, q_y, q_z),(p_x, p_y, p_z)|`
- `-|(i, j, k),(p_x, p_y, p_z),(q_x, q_y, q_z)|`
- all the above
- all the above

The answer is 'All the above'

• In a cross product if the vectors are swapped, the result is negative of the product. - not commutative.

**Not Commutative: ** For any vector `vec p, vec q in bbb V`

`vec p xx vec q = - vec q xx vec p`

*Solved Exercise Problem: *

Given that `vec p xx vec q = 2.1i-1.7j+6k`, find the `vec q xx vec p`.

- `-2.1i+1.7j-6k`
- `-2.1i+1.7j-6k`
- `2.1i-1.7j+6k`
- `sqrt(2.1^2+1.7^2+6^2)`
- `2.1^2+1.7^2+6^2`

The answer is '`-2.1i+1.7j-6k`'.

*slide-show version coming soon*