Two dimensions

In two dimensions, to carry out a rotation using a matrix, the point (x, y) to be rotated counterclockwise is written as a column vector, then multiplied by a rotation matrix calculated from the angle θ:

${\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}={\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}}$.

The coordinates of the point after rotation are x′, y′, and the formulae for x′ and y′ are

${\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \\y'&=x\sin \theta +y\cos \theta .\end{aligned}}}$

The vectors ${\displaystyle {\begin{bmatrix}x\\y\end{bmatrix}}}$ and ${\displaystyle {\begin{bmatrix}x'\\y'\end{bmatrix}}}$ have the same magnitude and are separated by an angle θ as expected.

Points on the R² plane can be also presented as complex numbers: the point (x, y) in the plane is represented by the complex number

${\displaystyle z=x+iy}$

This can be rotated through an angle θ by multiplying it by e^iθ, then expanding the product using Euler's formula as follows:

${\displaystyle {\begin{aligned}e^{i\theta }z&=(\cos \theta +i\sin \theta )(x+iy)\\&=x\cos \theta +iy\cos \theta +ix\sin \theta -y\sin \theta \\&=(x\cos \theta -y\sin \theta )+i(x\sin \theta +y\cos \theta )\\&=x'+iy',\end{aligned}}}$

and equating real and imaginary parts gives the same result as a two-dimensional matrix:

${\displaystyle {\begin{aligned}x'&=x\cos \theta -y\sin \theta \\y'&=x\sin \theta +y\cos \theta .\end{aligned}}}$

Since complex numbers form a commutative ring, vector rotations in two dimensions are commutative, unlike in higher dimensions. They have only one degree of freedom, as such rotations are entirely determined by the angle of rotation.^[3]

Three dimensions

As in two dimensions, a matrix can be used to rotate a point (x, y, z) to a point (x′, y′, z′). The matrix used is a 3×3 matrix,

${\displaystyle \mathbf {A} ={\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}}$

This is multiplied by a vector representing the point to give the result

${\displaystyle \mathbf {A} {\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}a&b&c\\d&e&f\\g&h&i\end{pmatrix}}{\begin{pmatrix}x\\y\\z\end{pmatrix}}={\begin{pmatrix}x'\\y'\\z'\end{pmatrix}}}$

The set of all appropriate matrices together with the operation of matrix multiplication is the rotation group SO(3). The matrix A is a member of the three-dimensional special orthogonal group, SO(3), that is it is an orthogonal matrix with determinant 1. That it is an orthogonal matrix means that its rows are a set of orthogonal unit vectors (so they are an orthonormal basis) as are its columns, making it simple to spot and check if a matrix is a valid rotation matrix.

Above-mentioned Euler angles and axis–angle representations can be easily converted to a rotation matrix.

Another possibility to represent a rotation of three-dimensional Euclidean vectors are quaternions described below.

Quaternions

Unit quaternions, or versors, are in some ways the least intuitive representation of three-dimensional rotations. They are not the three-dimensional instance of a general approach. They are more compact than matrices and easier to work with than all other methods, so are often preferred in real-world applications.

A versor (also called a rotation quaternion) consists of four real numbers, constrained so the norm of the quaternion is 1. This constraint limits the degrees of freedom of the quaternion to three, as required. Unlike matrices and complex numbers two multiplications are needed:

${\displaystyle \mathbf {x'} =\mathbf {qxq} ^{-1},}$

where q is the versor, q⁻¹ is its inverse, and x is the vector treated as a quaternion with zero scalar part. The quaternion can be related to the rotation vector form of the axis angle rotation by the exponential map over the quaternions,

${\displaystyle \mathbf {q} =e^{\mathbf {v} /2},}$

where v is the rotation vector treated as a quaternion.

A single multiplication by a versor, either left or right, is itself a rotation, but in four dimensions. Any four-dimensional rotation about the origin can be represented with two quaternion multiplications: one left and one right, by two different unit quaternions.

Further notes

More generally, coordinate rotations in any dimension are represented by orthogonal matrices. The set of all orthogonal matrices in n dimensions which describe proper rotations (determinant = +1), together with the operation of matrix multiplication, forms the special orthogonal group SO(n).

Matrices are often used for doing transformations, especially when a large number of points are being transformed, as they are a direct representation of the linear operator. Rotations represented in other ways are often converted to matrices before being used. They can be extended to represent rotations and transformations at the same time using homogeneous coordinates. Projective transformations are represented by 4×4 matrices. They are not rotation matrices, but a transformation that represents a Euclidean rotation has a 3×3 rotation matrix in the upper left corner.

The main disadvantage of matrices is that they are more expensive to calculate and do calculations with. Also in calculations where numerical instability is a concern matrices can be more prone to it, so calculations to restore orthonormality, which are expensive to do for matrices, need to be done more often.

More alternatives to the matrix formalism

As was demonstrated above, there exist three multilinear algebra rotation formalisms: one with U(1), or complex numbers, for two dimensions, and two others with versors, or quaternions, for three and four dimensions.

In general (even for vectors equipped with a non-Euclidean Minkowski quadratic form) the rotation of a vector space can be expressed as a bivector. This formalism is used in geometric algebra and, more generally, in the Clifford algebra representation of Lie groups.

In the case of a positive-definite Euclidean quadratic form, the double covering group of the isometry group ${\displaystyle \mathrm {SO} (n)}$ is known as the Spin group, ${\displaystyle \mathrm {Spin} (n)}$. It can be conveniently described in terms of a Clifford algebra. Unit quaternions give the group ${\displaystyle \mathrm {Spin} (3)\cong \mathrm {SU} (2)}$.