**c** is much longer vector than **b**;
it is likely that **a · c** is the larger.

The answer is not only likely, but correct.

**a**is (6, 0)^{T}**b**is (3, 2)^{T}**c**is (5, 9)^{T}

The dot products are:

**a · b**= 18.0**a · c**= 30.00

There are two properties tangled up in the dot product :

- the angle between the input vectors
- the length of the input vectors

We would like to get just the angle by somehow removing the influence of length.

What is the normal method for removing the influence of length?