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 `***`?

- `(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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