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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.

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You're exploiting the fact that $\|c\mathbf v\|=c\|\mathbf v\|$... – J. M. Nov 14 '11 at 15:33
Beautiful, thanks! – Curiosity Nov 14 '11 at 15:37
Thus the words norm and length are interchangeable (in lower dimensions). – Joshua Shane Liberman Nov 14 '11 at 17:08

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

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

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