= Vector normalization
{title2=$\widehat v=v/\|v\|,\quad v\ne0$}
<Vector normalization> converts a nonzero <vector> in a <normed vector space> into a <unit vector>. <Absolute homogeneity of a norm> gives
$$
\left\|\frac{v}{\|v\|}\right\|=\frac{\|v\|}{\|v\|}=1.
$$
The direction is unchanged by multiplying by a positive real <scalar>. For $c>0$, $\|cv\|=c\|v\|$ and $\widehat{cv}=\widehat v$; for a negative real <scalar>, $\widehat{cv}=-\widehat v$. Thus $(3,4)$ and $(6,8)$ normalize to the same <unit vector>, while $(-3,-4)$ normalizes to its negative. This decomposition is useful for specifying a direction independently of a <force>, <velocity> or <displacement> magnitude.
In a complex <normed vector space>, a nonzero complex <scalar> $c$ instead gives $\widehat{cv}=(c/|c|)\widehat v$. The remaining <complex argument> is not removed by normalization. For a pure <quantum state>, vectors differing only by this overall <global phase> represent the same physical state; <vector normalization> alone does not identify them.
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