Editing Rotation (mathematics) (section)

====Two dimensions====<!-- caution: an internal #-link -->
{{main|Rotations and reflections in two dimensions}}
{{see also|Rotation of axes in two dimensions}}

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

:<math> \begin{bmatrix} x' \\ y' \end{bmatrix} =
 \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}</math>.

The coordinates of the point after rotation are {{math|''x′'', ''y′''}}, and the formulae for {{mvar|x′}} and {{mvar|y′}} are

:<math>\begin{align}
x'&=x\cos\theta-y\sin\theta\\
y'&=x\sin\theta+y\cos\theta.
\end{align}</math>

The vectors <math> \begin{bmatrix} x \\ y \end{bmatrix} </math> and <math> \begin{bmatrix} x' \\ y' \end{bmatrix} </math> have the same magnitude and are separated by an angle {{mvar|θ}} as expected.

{{anchor|Complex numbers}}
Points on the {{math|'''R'''<sup>2</sup>}} plane can be also presented as [[complex number]]s: the point {{math|(''x'', ''y'')}} in the plane is represented by the complex number

:<math> z = x + iy </math>

This can be rotated through an angle {{mvar|θ}} by multiplying it by  {{math|''e''<sup>''iθ''</sup>}}, then expanding the product using [[Euler's formula]] as follows:

:<math>\begin{align}
e^{i \theta} z &= (\cos \theta + i \sin \theta) (x + i y) \\
               &= x \cos \theta + i y \cos \theta + i x \sin \theta - y \sin \theta \\
               &= (x \cos \theta - y \sin \theta) + i ( x \sin \theta + y \cos \theta) \\
               &= x' + i y' ,
\end{align}</math>

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

:<math>\begin{align}
x'&=x\cos\theta-y\sin\theta\\
y'&=x\sin\theta+y\cos\theta.
\end{align}</math>

Since complex numbers form a [[commutative ring]], vector rotations in two dimensions are commutative, unlike in higher dimensions. They have only one [[degrees of freedom (mechanics)|degree of freedom]], as such rotations are entirely determined by the angle of rotation.<ref>Lounesto 2001, p. 30.</ref>