= Unit vector
{title2=$\lVert v\rVert=1$}
{wiki}
= Unit direction
{synonym}
A <unit vector> is a <vector> of <norm> one in a <normed vector space>. For a nonzero <vector> $v$, <vector normalization> produces one by dividing by $\|v\|$. In a real <Euclidean space>, this keeps the direction and removes the magnitude. The <zero vector> cannot be normalized because its <norm> is zero.
For $\mathbf v=(3,4)$ with <Euclidean norm> $5$, the corresponding <unit vector> is
$$
\widehat{\mathbf v}=\left(\frac35,\frac45\right),\qquad
\left\|\widehat{\mathbf v}\right\|_2^2=\frac9{25}+\frac{16}{25}=1.
$$
The <Euclidean vector> can therefore be reconstructed as $\mathbf v=5\widehat{\mathbf v}$: size and direction are distinct pieces of information. The <standard basis> vectors are familiar <unit vectors>, but a <unit vector> need not point along a coordinate axis.
\Image[/euclidean-vector-normalization-unit-circle.png]
{title=Normalization keeps direction and places the endpoint on the unit circle}
{description=The original <Euclidean vector> $(3,4)$ has magnitude $5$. Its <vector normalization> $(3/5,4/5)$ lies on the <unit circle>. The enlarged right panel also shows its negative, which has the same <Euclidean norm> and the opposite direction.}
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