**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, 1.9)^{T}**c**is (5, 9)^{T}

The dot products are:

**a · b**is 15.0**a · c**is 30.00

So there are two properties tangled up in the dot product: the angle between the input vectors, and the length of the input vectors. We would like to get at just the angle alone by somehow removing the influence of length.