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# unit vector

A *unit vector* is a unit-length element of Euclidean space.
Equivalently, one may say that the norm of a unit vector is equal
to $1$, and write $\|\mathbf{u}\|=1$, where $\mathbf{u}$ is the vector in
question.

Let $\mathbf{v}$ be a non-zero vector. To *normalize* $\mathbf{v}$ is to find
the unique unit vector with the same direction as $\mathbf{v}$. This is done
by multiplying $\mathbf{v}$ by the reciprocal of its length; the
corresponding unit vector is given by $\mathbf{u}=\frac{\mathbf{v}}{\|\mathbf{v}\|}$.

# Note:

The concept of a unit vector and normalization makes sense in any vector space equipped with a real or complex norm. Thus, in quantum mechanics one represents states as unit vectors belonging to a (possibly) infinite-dimensional Hilbert space. To obtain an expression for such states one normalizes the results of a calculation.

# Example:

Consider $\mathbb{R}^{3}$ and the vector $\mathbf{v}=(1,2,3)$. The norm (length) is $\sqrt{14}$. Normalizing, we obtain the unit vector $\mathbf{u}$ pointing in the same direction, namely $\mathbf{u}=\left(\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\right)$.

## Mathematics Subject Classification

15A03*no label found*

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