Creating dot products of vectors is not only excellent entertainment, but also useful for calculating the angle between two vectors.

We can do this because there is this other formula for dot product:

where is the angle between the two vectors,

, in other words, the length of vector

, in other words, the length of vector

To calculate the angle between the two vectors, we write their dot product using both formulas.

Here is an example:

Dot product by the earlier formula:

Dot product by the new formula: