# 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 $\parallel \mathbf{u}\parallel =1$, 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 $\mathbf{u}=\frac{\mathbf{v}}{\parallel \mathbf{v}\parallel}$.

## 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 $\mathbf{u}=(\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}})$.

Title | unit vector |
---|---|

Canonical name | UnitVector |

Date of creation | 2013-03-22 11:58:50 |

Last modified on | 2013-03-22 11:58:50 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 16 |

Author | rmilson (146) |

Entry type | Definition |

Classification | msc 15A03 |

Related topic | VectorNorm |

Related topic | NormedVectorSpace |

Defines | normalize |