You have 2 vectors $\vec{a}$ and $\vec{b}$, with an angle of $\theta$ between them. You get:
cos $\theta = \dfrac{adj}{||\vec{a}||}$ -> $||\vec{a}|| cos\theta$ = adj. But then, as I noted in my notebook:
$\vec{a} . \vec{b}$ = $||\vec{b}|| . adj $
Sadly, I forgot how my professor came to that conclusion, can someone enlighten me?
Also:
You AGAIN have 2 vectors $\vec{a}$ and $\vec{b}$, with an angle of $\theta$ between them. You can make a parallelogram with Area = bh.
Area = $\vec{b} . ||\vec{a}|| sin \theta $ = $||\vec{a}$ x $\vec{b} || $
I also don't understand this at all, so, if again someone could enlighten me.