Unit vector ($\Vert v\Vert =1$)

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 $\Vert v\Vert$. 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 isThe Euclidean vector can therefore be reconstructed as $\mathbf{v}=5\hat{\mathbf{v}}$: 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.

Vector normalization converts a nonzero vector in a normed vector space into a unit vector. Absolute homogeneity of a norm givesThe direction is unchanged by multiplying by a positive real scalar. For $c>0$, $\Vert cv\Vert =c\Vert v\Vert$ and $\widehat{cv}=\hat{v}$; for a negative real scalar, $\widehat{cv}=-\hat{v}$. 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 $\widehat{cv}=(c/|c|)\hat{v}$. 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.

The zero vector cannot be normalized this way. The construction also applies in a function space: an eigenfunction of a Sturm-Liouville problem can be scaled to unit weighted norm byThe positive weight $w$ specifies the weighted inner product. Multiplication by this nonzero constant preserves the eigenvalue equation and its homogeneous boundary conditions.

Vector normalization is not a linear map. With ${\mathbf{e}}_1=(1,0)$ and ${\mathbf{e}}_2=(0,1)$,Consequently, adding individually normalized directions does not generally give the direction of the original vector sum. Compute the intended vector sum first, then normalize it if it is nonzero. Cancellation is a further boundary case: $v+(-v)=0$ has no normalized direction even though both nonzero summands do. A numerical algorithm must check that case before dividing by the norm.