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mathsVector AlgebraProperties of Cross Product

### Not Associative

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