In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in <math>\hat{\mathbf{v</math> (pronounced "v-hat"). The term normalized vector is sometimes used as a synonym for unit vector.
The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e.,
:<math alt="u-hat equals the vector u divided by its length">\mathbf{\hat{u = \frac{\mathbf{u{\|\mathbf{u}\|}=\left(\frac{u_1}{\|\mathbf{u}\|}, \frac{u_2}{\|\mathbf{u}\|}, ... , \frac{u_n}{\|\mathbf{u}\|}\right)</math>
where ‖u‖ is the norm (or length) of u and <math display="inline">\mathbf{u} = (u_1, u_2, ..., u_n)</math>.
The proof is the following: <math alt="u-hat equals the vector u divided by its length" display="inline">\|\mathbf{\hat{u\|=\sqrt{\frac{u_1}{\sqrt{u_1^2+...+u_n^2^2+...+\frac{u_n}{\sqrt{u_1^2+...+u_n^2^2}=\sqrt{\frac{u_1^2+...+u_n^2}{u_1^2+...+u_n^2=\sqrt{1}=1</math>
A unit vector is often used to represent directions, such as normal directions.
Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination form of unit vectors.
Orthogonal coordinates
Cartesian coordinates
Unit vectors may be used to represent the axes of a Cartesian coordinate system. For instance, the standard unit vectors in the direction of the x, y, and z axes of a three dimensional Cartesian coordinate system are
:<math alt="i-hat equals the 3 by 1 matrix 1,0,0; j-hat equals the 3 by 1 matrix 0,1,0; k-hat equals the 3 by 1 matrix 0,0,1">
\mathbf{\hat{x = \begin{bmatrix}1\\0\\0\end{bmatrix}, \,\, \mathbf{\hat{y = \begin{bmatrix}0\\1\\0\end{bmatrix}, \,\, \mathbf{\hat{z = \begin{bmatrix}0\\0\\1\end{bmatrix}</math>
They form a set of mutually orthogonal unit vectors, typically referred to as a standard basis in linear algebra.
They are often denoted using common vector notation (e.g., x or <math alt="vector i">\vec{x}</math>) rather than standard unit vector notation (e.g., x̂). In most contexts it can be assumed that x, y, and z, (or <math alt="vector i">\vec{x},</math> <math alt="vector j">\vec{y},</math> and <math alt="vector k"> \vec{z}</math>) are versors of a 3-D Cartesian coordinate system. The notations (î, ĵ, k̂), (x̂<sub>1</sub>, x̂<sub>2</sub>, x̂<sub>3</sub>), (ê<sub>x</sub>, ê<sub>y</sub>, ê<sub>z</sub>), or (ê<sub>1</sub>, ê<sub>2</sub>, ê<sub>3</sub>), with or without hat, are also used, is used. This leaves the azimuthal angle <math alt="phi">\varphi</math> defined the same as in cylindrical coordinates. The Cartesian relations are:
:<math alt="r-hat equals sin of theta times cosine of phi in the x-hat direction plus sine of theta times sine of phi in the y-hat direction plus cosine of theta in the z-hat direction">\mathbf{\hat{r = \sin \theta \cos \varphi\mathbf{\hat{x + \sin \theta \sin \varphi\mathbf{\hat{y + \cos \theta\mathbf{\hat{z</math>
:<math alt="theta-hat equals cosine of theta times cosine of phi in the x-hat direction plus cosine of theta times sine of phi in the y-hat direction minus sine of theta in the z-hat direction">\boldsymbol{\hat \theta} = \cos \theta \cos \varphi\mathbf{\hat{x + \cos \theta \sin \varphi\mathbf{\hat{y - \sin \theta\mathbf{\hat{z</math>
:<math alt="phi-hat equals minus sine of phi in the x-hat direction plus cosine of phi in the y-hat direction">\boldsymbol{\hat \varphi} = - \sin \varphi\mathbf{\hat{x + \cos \varphi\mathbf{\hat{y</math>
The spherical unit vectors depend on both <math alt="phi">\varphi</math> and <math alt="theta">\theta</math>, and hence there are 5 possible non-zero derivatives. For a more complete description, see Jacobian matrix and determinant. The non-zero derivatives are:
:<math alt="partial derivative of r-hat with respect to phi equals minus sine of theta times sine of phi in the x-hat direction plus sine of theta times cosine of phi in the y-hat direction equals sine of theta in the phi-hat direction">\frac{\partial \mathbf{\hat{r} {\partial \varphi} = -\sin \theta \sin \varphi\mathbf{\hat{x + \sin \theta \cos \varphi\mathbf{\hat{y = \sin \theta\boldsymbol{\hat \varphi}</math>
:<math alt="partial derivative of r-hat with respect to theta equals cosine of theta times cosine of phi in the x-hat direction plus cosine of theta times sine of phi in the y-hat direction minus sine of theta in the z-hat direction equals theta-hat">\frac{\partial \mathbf{\hat{r} {\partial \theta} =\cos \theta \cos \varphi\mathbf{\hat{x + \cos \theta \sin \varphi\mathbf{\hat{y - \sin \theta\mathbf{\hat{z= \boldsymbol{\hat \theta}</math>
:<math alt="partial derivative of theta-hat with respect to phi equals minus cosine of theta times sine of phi in the x-hat direction plus cosine of theta times cosine of phi in the y-hat direction equals cosine of theta in the phi-hat direction">\frac{\partial \boldsymbol{\hat{\theta} {\partial \varphi} =-\cos \theta \sin \varphi\mathbf{\hat{x + \cos \theta \cos \varphi\mathbf{\hat{y = \cos \theta\boldsymbol{\hat \varphi}</math>
:<math alt="partial derivative of theta-hat with respect to theta equals minus sine of theta times cosine of phi in the x-hat direction minus sine of theta times sine of phi in the y-hat direction minus cosine of theta in the z-hat direction equals minus r-hat">\frac{\partial \boldsymbol{\hat{\theta} {\partial \theta} = -\sin \theta \cos \varphi\mathbf{\hat{x - \sin \theta \sin \varphi\mathbf{\hat{y - \cos \theta\mathbf{\hat{z = -\mathbf{\hat{r</math>
:<math alt="partial derivative of phi-hat with respect to phi equals minus cosine of phi in the x-hat direction minus sine of phi in the y-hat direction equals minus sine of theta in the r-hat direction minus cosine of theta in the theta-hat direction">\frac{\partial \boldsymbol{\hat{\varphi} {\partial \varphi} = -\cos \varphi\mathbf{\hat{x - \sin \varphi\mathbf{\hat{y = -\sin \theta\mathbf{\hat{r -\cos \theta\boldsymbol{\hat{\theta</math>
General unit vectors
Common themes of unit vectors occur throughout physics and geometry:
{| class="wikitable"
|-
! scope="col" width="200" | Unit vector
! scope="col" width="150" | Nomenclature
! scope="col" width="410" | Diagram
|-
| Tangent vector to a curve/flux line || <math> \mathbf{\hat{t</math> || rowspan="3" | 200px|"200px" 200px|"200px"
A normal vector <math> \mathbf{\hat{n </math> to the plane containing and defined by the radial position vector <math> r \mathbf{\hat{r </math> and angular tangential direction of rotation <math> \theta \boldsymbol{\hat{\theta </math> is necessary so that the vector equations of angular motion hold.
|-
|Normal to a surface tangent plane/plane containing radial position component and angular tangential component
|| <math> \mathbf{\hat{n</math>
In terms of polar coordinates;
<math> \mathbf{\hat{n = \mathbf{\hat{r \times \boldsymbol{\hat{\theta </math>
|-
| Binormal vector to tangent and normal
|| <math> \mathbf{\hat{b = \mathbf{\hat{t \times \mathbf{\hat{n </math>
|-
| Parallel to some axis/line || <math> \mathbf{\hat{e_{\parallel} </math> || rowspan="2" | 200px|"200px"
One unit vector <math> \mathbf{\hat{e_{\parallel}</math> aligned parallel to a principal direction (red line), and a perpendicular unit vector <math> \mathbf{\hat{e_{\bot}</math> is in any radial direction relative to the principal line.
|-
| Perpendicular to some axis/line in some radial direction
|| <math> \mathbf{\hat{e_{\bot} </math>
|-
| Possible angular deviation relative to some axis/line
|| <math> \mathbf{\hat{e_{\angle} </math>
|| 200px|"200px"
Unit vector at acute deviation angle φ (including 0 or π/2 rad) relative to a principal direction.
|-
|}
Curvilinear coordinates
In general, a coordinate system may be uniquely specified using a number of linearly independent unit vectors <math alt="e-hat sub n">\mathbf{\hat{e_n</math>