Why is the angle a well-defined quantity in a Euclidean space?

Question

I am having a problem with an exercise. In a certain euclidean space, the angle between two vectors $u$ and $v$ is said to be $\theta$ which satisfies the equation:

$$\cos\theta=\frac{(u,v)}{\|u\|\|v\|}$$

Why is the angle a well-defined quantity if $\theta \in [0;\pi]$?

Answer 1

It's not.

But suppose you add the restriction that $\|u\| >0, \|v\|>0$. By the Cauchy-Schwarz inequality, $|(u,v)| \leq \|u\|\|v\|$, so $$-1 \leq \frac{(u,v)}{\|u\|\|v\|} \leq 1.$$

It follows that $\theta\in [0,\pi]$ exists and is unique, since $\cos$ is a bijection from $[0,\pi]$ to $[-1,1]$.

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