Editing Rotation matrix (section)

{{Short description|Matrix representing a Euclidean rotation}}
In [[linear algebra]], a '''rotation matrix''' is a [[transformation matrix]] that is used to perform a [[rotation (mathematics)|rotation]] in [[Euclidean space]]. For example, using the convention below, the matrix
<!-- The following formula has been changed back and forth on a regular basis. Thus: PLEASE READ FOLLOWING COMMENT BEFORE COMMITTING ANY CHANGE. (Then think about it...) -->

:<math>R = \begin{bmatrix}
\cos \theta & -\sin \theta \\  
\sin \theta & \cos \theta 
\end{bmatrix} 
</math>
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These signs are correct. THE MINUS SIGN GOES IN THE UPPER RIGHT CORNER. Please do not change them.  
-->

rotates points in the {{mvar|xy}} plane counterclockwise through an angle {{mvar|θ}} about the origin of a two-dimensional [[Cartesian coordinate system]]. To perform the rotation on a plane point with standard coordinates {{math|1='''v''' = (''x'', ''y'')}}, it should be written as a [[column vector]], and [[matrix multiplication|multiplied]] by the matrix {{mvar|R}}:

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

If {{mvar|x}} and {{mvar|y}} are the coordinates of the endpoint of a vector with the length ''r'' and the angle <math>\phi</math> with respect to the {{mvar|x}}-axis, so that <math display="inline">x = r \cos \phi</math> and <math>y = r \sin \phi</math>, then the above equations become the [[List of trigonometric identities#Angle sum and difference identities|trigonometric summation angle formulae]]:<math display="block">R\mathbf{v} =
r\begin{bmatrix} 
\cos\phi\cos\theta-\sin\phi\sin\theta \\
\cos\phi\sin\theta+\sin\phi\cos\theta
\end{bmatrix}
=
r\begin{bmatrix} 
\cos(\phi+\theta)\\
\sin(\phi+\theta)
\end{bmatrix}.</math>Indeed, this is the trigonometric summation angle formulae in matrix form. One way to understand this is to say we have a vector at an angle 30° from the {{mvar|x}}-axis, and we wish to rotate that angle by a further 45°. We simply need to compute the vector endpoint coordinates at 75°.

The examples in this article apply to ''[[Active and passive transformation#Active transformation|active]] rotations'' of vectors ''counterclockwise'' in a ''right-handed coordinate system'' ({{mvar|y}} counterclockwise from {{mvar|x}}) by ''pre-multiplication'' (the rotation matrix {{mvar|R}} applied on the left of the column vector {{math|1='''v'''}} to be rotated). If any one of these is changed (such as rotating axes instead of vectors, a ''[[Active and passive transformation#Passive transformation|passive]] transformation''), then the [[Matrix inverse|inverse]] of the example matrix should be used, which coincides with its [[transpose]].

Since matrix multiplication has no effect on the [[zero vector]] (the coordinates of the origin), rotation matrices describe rotations about the origin. Rotation matrices provide an algebraic description of such rotations, and are used extensively for computations in [[geometry]], [[physics]], and [[computer graphics]]. In some literature, the term ''rotation'' is generalized to include [[improper rotation]]s, characterized by orthogonal matrices with a [[determinant]] of −1 (instead of +1). An improper rotation combines a ''proper'' rotation with ''reflections'' (which invert [[orientation (mathematics)|orientation]]). In other cases, where reflections are not being considered, the label ''proper'' may be dropped. The latter convention is followed in this article.

Rotation matrices are [[square matrix|square matrices]], with [[real number|real]] entries. More specifically, they can be characterized as [[orthogonal matrix|orthogonal matrices]] with [[determinant]] 1; that is, a square matrix {{math|''R''}} is a rotation matrix if and only if {{math|''R''<sup>T</sup> {{=}} ''R''<sup>−1</sup>}} and {{math|det ''R'' {{=}} 1}}. The [[set (mathematics)|set]] of all orthogonal matrices of size  {{mvar|n}} with determinant +1 is a [[Group representation|representation]] of a [[group (mathematics)|group]] known as the [[special orthogonal group]] {{math|SO(''n'')}}, one example of which is the [[rotation group SO(3)]]. The set of all orthogonal matrices of size {{mvar|n}} with determinant +1 or −1 is a representation of the (general) [[orthogonal group]] {{math|O(''n'')}}.