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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

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 `***`?associative property illustration

  • `(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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