Ok, I have a question that probably has a very simple answer but for some reason I can't see it. Let $a$ and $r$ be two vectors of nonzero length with a common origin and let $\theta$ be the nonzero angle between them. Then, by definition of the cosine funtion, $ \cos \theta = \frac{|a|}{|r|} $ where $|\cdot|$ denotes the norm. On the other hand, the scalar product is given by
$ \langle a, r \rangle = |a|\cdot |r| \cos \theta. $
Putting these facts together we have $ \langle a, r \rangle = |a|\cdot |r| \cdot \frac{|a|}{|r|} = |a|\cdot |a| = |a|^2 = \langle a, a \rangle $ which is a result that is independent of $r$ and thus makes no sense. What is my error?