Why is $\frac{x}{\| x \|}$ a unit vector? Let $x$ be a vector in $\mathbb{R}^n$. Why is $\frac{x}{\| x \|}$ a unit vector, for $x\neq 0$?
If I try to simplify it, I get the following: $\frac{x}{\sqrt{x \cdot x}}$, and I'm not sure how to proceed.
So I'm looking for a proof or explanation as to why it is a unit vector.

Hint: Show that for any $\lambda\in\mathbb{R}$ and any $x\in\mathbb{R}^n$, we have
$$\|\lambda x\|=|\lambda|\cdot\|x\|.$$
What does this become when $x\neq 0$ and $\lambda=\frac{1}{\|x\|}$?
