Why does dividing a vector by its norm give a unit vector?

Question

Straightforward question, so if it is applied to every element of a vector that means that every one of them is scaled down exactly length times. How did people come up with this, to make it exactly 1? I am sorry if this is fairly obvious, but I just don't see it.

Answer 1

By definition of norm we have $\| \alpha x \| = |\alpha|\cdot \|x\|$, so:

$$\left\|\frac{x}{\|x\|}\right\| = \frac1{\|x\|} \|x\| = 1$$