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Thought-Process to Discover Knowledge

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mathsVector AlgebraProperties of Dot Product

Not Associative

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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