unit vector UnitVector 2001-11-15 12:53:04 2005-07-26 08:16:17 Definition normalize \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\bu}{\mathbf{u}} \newcommand{\bv}{\mathbf{v}} A \emph{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 $\Vert \bu\Vert=1$, where $\bu$ is the vector in question. Let $\bv$ be a non-zero vector. To \emph{normalize} $\bv$ is to find the unique unit vector with the same direction as $\bv$. This is done by multiplying $\bv$ by the reciprocal of its length; the corresponding unit vector is given by $\bu=\frac{\bv}{\Vert \bv\Vert}$. \paragraph{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. \paragraph{Example:} Consider $\mathbb{R}^3$ and the vector $\bv=(1,2,3)$. The norm (length) is $\sqrt{14}$. Normalizing, we obtain the unit vector $\bu$ pointing in the same direction, namely $\bu=\left(\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\right)$.