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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.

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In this page, you will learn that the vector dot 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)`'. In the left hand side of the equation, `y` is associated with `x` first, where in the right hand side, y is associated with `z` first.

Dot product cannot be considered for associative property. Consider the two `(vec p cdot vec q)` and `vec r`. The first term in parenthesis is a scalar and the second term is a vector.

The dot product is not defined between a scalar and a vector.

 •  Dot product is not associative.

Not Associative: For any vector `vec p, vec q, vec r in bbb V`
`(vec p cdot vec q) vec r != vec p (vec q cdot vec r)`

Solved Exercise Problem:

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

  • Yes
  • No
  • No

The answer is 'No'. The `vec z` can be canceled in `(vec x cdot vec y) vec z = (vec p cdot vec q) vec z`. Since associative property is not defined for dot product, the equation given in the question can not be equivalently expressed in this form.

                            
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