## Angle Between Two Vectors

How do we find the angle between two vectors?

Recall the **dot product** of vectors and in the real number space, is in the form of:

There are two dependencies which lie within the vector dot product:

- The
*angle*between the two vectors. - The
*length*of the input vectors.

Suppose we just want to find the angle of the two vectors using just the dot product. We'll need to find a way to remove the dependency of *length* within the dot product.

The easiest way to do this is to *unitize* the vectors, or *normalize* them. Or, simply, just convert them into unit vectors.

We can find the unit vector from by

Where in this case .

We can then unitize two vectors , , to , and . Then find use the *alternate* form of the dot product:

Since we know that , and are unit vectors, then both lengths of these vectors are . Which then the expression can be simplified to:

Using this expression, the angle between , and is now the of the dot product.

### Notes

- We take advantage of the fact that the range of can only range from to .