Just a little problem with geometric intuition here (or perhaps I just haven't slept in far too long!).
I know that the scalar projection of vectors $ \vec{u} $ and $ \vec{v} $ is defined as
$ \displaystyle proj_{\vec{v}}\vec{u} = \frac{\vec{u} \cdot \vec{v}}{\| \vec{v} \|} $
I read in another thread that the dot product represents the "amount one vector goes in the direction of another". However, I thought that was the definition of the scalar projection?
They are obviously not the same thing (however I can see that the scalar projection is derived from the geometric definition of the dot product); the scalar projection is the dot product divided by the magnitude of one of the vectors:
$ \displaystyle \vec{u} \cdot \vec{v} = \| \vec{u} \| \| \vec{v} \| \cos{\theta} $
$ \displaystyle \frac{\vec{u} \cdot \vec{v}}{\| \vec{v} \|} = \| \vec{u} \| \cos{\theta} = proj_{\vec{v}}\vec{u} $
What am I missing, in general, about the differences or similarities between these two things?