__maths__>__Properties of Vector Arithmetics__>__Properties of Dot Product__### Dot Product : No Multiplicative Inverse

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

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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

`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`.

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