Is it true that: $\| a \| \| b \| \cos \alpha = \langle a,b\rangle$

Question

Let $a , b \in \mathbb{R}^n$ and let $\alpha$ be the angle formed between $a$ and $b$. Is it true that:

$$ \| a \| \| b \| \cos \alpha = \langle a,b\rangle $$

($\langle\cdot,\cdot\rangle$ being the dot product)

If so why?

Answer 1

That's taken as the definition of $\cos \alpha$ in order to define the angle between two vectors in $\mathbb R^n$. Note that it is well defined since $|<a,b>| \le ||a|| ||b||$ by C-S inequality.