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 : \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} 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 \vec{x}) rather than standard unit vector notation (e.g.,
x̂). In most contexts it can be assumed that
x,
y, and
z, (or \vec{x}, \vec{y}, and \vec{z}) are versors of a 3-D
Cartesian coordinate system. The notations (
î,
ĵ,
k̂), (
x̂1,
x̂2,
x̂3), (
êx,
êy,
êz), or (
ê1,
ê2,
ê3), with or without
hat, are also used, is used. This leaves the
azimuthal angle \varphi defined the same as in cylindrical coordinates. The
Cartesian relations are: :\mathbf{\hat{r}} = \sin \theta \cos \varphi\mathbf{\hat{x}} + \sin \theta \sin \varphi\mathbf{\hat{y}} + \cos \theta\mathbf{\hat{z}} :\boldsymbol{\hat \theta} = \cos \theta \cos \varphi\mathbf{\hat{x}} + \cos \theta \sin \varphi\mathbf{\hat{y}} - \sin \theta\mathbf{\hat{z}} :\boldsymbol{\hat \varphi} = - \sin \varphi\mathbf{\hat{x}} + \cos \varphi\mathbf{\hat{y}} The spherical unit vectors depend on both \varphi and \theta, and hence there are 5 possible non-zero derivatives. For a more complete description, see
Jacobian matrix and determinant. The non-zero derivatives are: :\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} :\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} :\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} :\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}} :\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}}
General unit vectors Common themes of unit vectors occur throughout
physics and
geometry: ==Curvilinear coordinates==