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Thought-Process to Discover Knowledge

Welcome to nubtrek.

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
mathsProperties of Vector ArithmeticsProperties of Dot Product

Dot Product : No Multiplicative Inverse

In this page, you will learn that the dot product does not have a multiplicative inverse.

click on the content to continue..

What does 'inverse' mean?

  • same; identical; similar
  • opposite; converse
  • opposite; converse

The answer is 'opposite; converse'.

For two vectors `vec p` and `vec q`, the vector dot product is given by multiplication of
 •  magnitude of one vector and
 •  the magnitude of the component of second vector in parallel to the first vector

vector dot product illustration
`vec p cdot vec q = |p| |a|`
where `|a|` is the component of `vec q` in the direction of `vec p`.

Given `vec p cdot vec q` and `vec p` can `vec q` be calculated?

What is division?

  • inverse of multiplication
  • inverse of multiplication
  • does not have any relation to multiplication

The answer is 'inverse of multiplication'.

Since dot product is not invertible, the inverse of dot product -- a division -- is not defined.

 •  Dot product is not invertible - No multiplicative inverse and no definition of division.

Dot Product - No Multiplicative Inverse: For any vector `vec p, vec q in bbb V` it is not possible to calculate `vec q` using the values `vec p` and `vec p cdot vec q`.

slide-show version coming soon