Editing Rotation matrix (section)

==In two dimensions==
[[File:Counterclockwise rotation.png|thumb|A counterclockwise rotation of a vector through angle {{mvar|θ}}. The vector is initially aligned with the {{mvar|x}}-axis.]]

In two dimensions, the standard rotation matrix has the following form:
:<math>R(\theta) = \begin{bmatrix}
  \cos\theta & -\sin\theta \\
  \sin\theta &  \cos\theta \\
\end{bmatrix}.</math><!-- These signs are correct.  Please do not change them.  Do not confound with the matrix that rotate a coordinate system. DO check calculations with a test vector, i.e. (1, 0) or (0, 1) over 45 degrees and try to understand the results, why is the rotation *clockwise*? Review the results. Then ask yourself 'what is rotated from which coordinate system?' It would be helpfull if such an detailed example is included in the text itself -->

This rotates [[column vector]]s by means of the following [[matrix multiplication]],
:<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>

Thus, the new coordinates {{math|(''x''′, ''y''′)}} of a point {{math|(''x'', ''y'')}} after rotation are
:<math>\begin{align}
  x' &= x \cos\theta - y \sin\theta\, \\
  y' &= x \sin\theta + y \cos\theta\,
\end{align}.</math>

=== Examples ===
For example, when the vector (initially aligned with the ''x''-axis of the [[Cartesian coordinate system]])
:<math>
  \mathbf{\hat{x}} = \begin{bmatrix}
    1 \\
    0 \\
  \end{bmatrix}
</math>
is rotated by an angle {{mvar|θ}}, its new coordinates are
:<math>
\begin{bmatrix}
  \cos\theta \\
  \sin\theta \\
\end{bmatrix},
</math>

and when the vector (initially aligned with the ''y''-axis of the coordinate system)
:<math>
  \mathbf{\hat{y}} = \begin{bmatrix}
    0 \\
    1 \\
  \end{bmatrix}
</math>
is rotated by an angle {{mvar|θ}}, its new coordinates are
:<math>
  \begin{bmatrix}
    -\sin\theta \\
     \cos\theta \\
  \end{bmatrix}.
</math>

=== Direction ===
The direction of vector rotation is counterclockwise if {{mvar|θ}} is positive (e.g. 90°), and clockwise if {{mvar|θ}} is negative (e.g. −90°) for <math>
  R(\theta)</math>. Thus the clockwise rotation matrix is found as (by replacing {{mvar|θ}} with {{mvar|-θ}} and using the trigonometric symmetry of <math display="inline">\sin(-\theta) = - \sin(\theta)</math> and <math display="inline">\cos( - \theta) = \cos(\theta)</math>)
:<math>
  R(-\theta) = \begin{bmatrix}
     \cos\theta & \sin\theta \\
     -\sin\theta & \cos\theta \\
  \end{bmatrix}.</math>

An alternative convention uses rotating axes (instead of rotating a vector),<ref>{{cite book |last=Swokowski |first=Earl |url=https://archive.org/details/studentsupplemen00bron |title=Calculus with Analytic Geometry |publisher=Prindle, Weber, and Schmidt |year=1979 |isbn=0-87150-268-2 |edition=Second |location=Boston |url-access=registration}}</ref> and the above matrices also represent a rotation of the ''axes clockwise'' through an angle {{mvar|θ}}.

The two-dimensional case is the only non-trivial case where the rotation matrices group is commutative; it does not matter in which order rotations are multiply performed. For the 3-dimensional case, for example, a different order of multiple rotations gives a different result. (E.g., rotating a cell phone along ''z''-axis then ''y''-axis is not equal to rotations along the ''y''-axis then ''z''-axis.)

===Non-standard orientation of the coordinate system===

[[File:Clockwise rotation.png|thumb|A rotation through angle {{mvar|θ}} with non-standard axes.]]
If a standard [[Orientation (mathematics)|right-handed]] [[Cartesian coordinate system]] is used, with the {{nowrap|{{mvar|x}}-axis}} to the right and the {{nowrap|{{mvar|y}}-axis}} up, the rotation {{math|''R''(''θ'')}} is counterclockwise. If a left-handed Cartesian coordinate system is used, with  {{mvar|x}} directed to the right but {{mvar|y}} directed down, {{math|''R''(''θ'')}} is clockwise. Such non-standard orientations are rarely used in mathematics but are common in [[2D computer graphics]], which often have the origin in the top left corner and the {{nowrap|{{mvar|y}}-axis}} down the screen or page.<ref>{{Cite web |url=http://www.w3.org/TR/SVG/coords.html#InitialCoordinateSystem |title=Scalable Vector Graphics – the initial coordinate system |author=W3C recommendation |year=2003 }}</ref>

See [[#Ambiguities|below]] for other alternative conventions which may change the sense of the rotation produced by a rotation matrix.

===Common 2D rotations===
Matrices
:<math>\begin{bmatrix}
   0 & -1 \\[3pt]
   1 &  0 \\
\end{bmatrix}, \quad
\begin{bmatrix}
  -1 &  0 \\[3pt]
   0 & -1 \\
\end{bmatrix}, \quad
\begin{bmatrix}
   0 &  1 \\[3pt]
  -1 &  0 \\
\end{bmatrix}</math>
are 2D rotation matrices corresponding to counter-clockwise rotations of respective angles of 90°, 180°, and 270°.

===Relationship with complex plane===

The matrices of the shape
<math display=block>\begin{bmatrix} x & -y \\ y & x \end{bmatrix}</math> form a [[ring (mathematics)|ring]], since their set is closed under addition and multiplication.
Since
<math display=block>\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}^2 \ =\  \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \ = -I</math>
(where <math display="inline">I</math> is the [[identity matrix]]), the map
:<math>\begin{bmatrix} x & -y \\ y & x \end{bmatrix} = x\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + y \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \mapsto x+iy</math>
(where <math>i = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}</math>) is a [[ring isomorphism]] from this ring to the [[field (mathematics)|field]] of the [[complex number]]s {{tmath|\C}} (incidentally, this shows that this ring is a field). Under this isomorphism, the rotation matrices correspond to the [[circle]] of the [[unit complex number]]s, the complex numbers of modulus {{math|1}}.

If one identifies <math>\mathbb R^2</math> with <math>\mathbb C</math> through the [[linear isomorphism]] <math>(a,b)\mapsto a+ib</math>, where <math>(a,b) \in \mathbb R^2</math> and <math>a+ib \in \mathbb C</math>, the action of a matrix <math>\begin{bmatrix} x & -y \\ y & x \end{bmatrix}</math> on a vector <math>(a,b)</math> corresponds to multiplication on the complex number <math>a+ib</math> by {{math|''x'' + ''iy''}}, and a rotation correspond to multiplication by a complex number of modulus {{math|1}}.

As every 2-dimensional rotation matrix can be written
:<math>\begin{pmatrix}\cos t&-\sin t\\ \sin t&\cos t\end{pmatrix},</math>
the above correspondence associates such a matrix with the complex number
:<math>e^{it} = \cos t + i\sin t = \cos t \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \sin t \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}</math>
where the first equality is [[Euler's formula]], the matrix <math>I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}</math>corresponds to 1, and the matrix <math>\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}</math> corresponds to the [[imaginary unit]] <math display="inline">i</math>.