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

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What is associative property of an operator `***`?

- `(x *** y) *** z = x *** (y *** z)`
- `(x *** y) *** z = x *** (y *** z)`
- `x *** y = y *** x`
- `x *** y = - y *** x`

The answer is '`(x *** y) *** z = x *** (y *** z)`'. The element `y` in the equation is associated with `x` first in the left-hand-side, where as it is associated with `z` first in the right-hand-side.

The vector cross product is not associative. It is proven with one simple example.

Consider two non-zero vectors `vec p` and `vec q`.

`vec p xx vec p = 0`

and so `(vec p xx vec p) xx vec q = 0`

But, `vec p xx vec q` is perpendicular to `vec p`

So `vec p xx (vec p xx vec q) !=0`

It is proven that (vec p xx vec p) xx vec q !=vec p xx (vec p xx vec q)

• Cross product is not associative.

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

`(vec p xx vec q) xx vec r != vec p xx (vec q xx vec r)`

*Solved Exercise Problem: *

Consider `(vec x xx vec y) xx vec z` `= vec p xx (vec q xx vec z)`. Can the vector `vec z` be canceled on either side of the equation?

- Yes
- No
- No

The answer is 'No'.

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