Kinda late and my brain is glitching.
I know this is wrong, but I can't give a good reason why. I gave a counterexample to prove it is false, but I don't understand why this is wrong.
$\frac{u \cdot v}{|u|} = \frac{v \cdot u}{|v|}$
$\hat{u} \cdot v \neq v \cdot \hat{u}$
$u$ and $v$ are nonzero.