derivation of rotation matrix using polar coordinates

We derive formally the expression for the rotation of a two-dimensional vector $\boldsymbol{v}=a\boldsymbol{x}+b\boldsymbol{y}$ by an angle $\phi$ counter-clockwise. Here $\boldsymbol{x}$ and $\boldsymbol{y}$ are perpendicular unit vectors that are oriented counter-clockwise (the usual orientation).

In terms of polar coordinates, $\boldsymbol{v}$ may be rewritten:

	$\displaystyle\boldsymbol{v}$	$\displaystyle=r(\cos\theta\,\boldsymbol{x}+\sin\theta\,\boldsymbol{y})\,,\quad a% =r\cos\theta\,;b=r\sin\theta\,,$
for some angle $\theta$ and radius $r\geq 0$ . To rotate a vector $\boldsymbol{v}$ by $\phi$ really means to shift its polar angle by a constant amount $\phi$ but leave its polar radius fixed. Therefore, the result of the rotation must be:
	$\displaystyle\boldsymbol{v}^{\prime}$	$\displaystyle=r\bigl{(}\cos(\theta+\phi)\,\boldsymbol{x}+\sin(\theta+\phi)\,% \boldsymbol{y}\bigr{)}$
Expanding using the angle addition formulae, we obtain
	$\displaystyle\boldsymbol{v}^{\prime}$	$\displaystyle=r\bigl{(}\cos\theta\cos\phi-\sin\theta\sin\phi)\,\boldsymbol{x}+% (\sin\theta\cos\phi+\cos\theta\sin\phi)\,\boldsymbol{y}\bigr{)}$
		$\displaystyle=(a\cos\phi-b\sin\phi)\,\boldsymbol{x}+(b\cos\phi+a\sin\phi)\,% \boldsymbol{y}\,.$

When this transformation is written out in $[\boldsymbol{x},\boldsymbol{y}]$ -coordinates, we obtain the formula for the rotation matrix:

\boldsymbol{v}^{\prime}=\begin{bmatrix}\cos\phi&-\sin\phi\\ \sin\phi&\cos\phi\end{bmatrix}\begin{bmatrix}a\\ b\end{bmatrix}\,.

Title	derivation of rotation matrix using polar coordinates
Canonical name	DerivationOfRotationMatrixUsingPolarCoordinates
Date of creation	2013-03-22 15:25:02
Last modified on	2013-03-22 15:25:02
Owner	stevecheng (10074)
Last modified by	stevecheng (10074)
Numerical id	9
Author	stevecheng (10074)
Entry type	Derivation
Classification	msc 15-00
Related topic	RotationMatrix
Related topic	PolarCoordinates
Related topic	DecompositionOfOrthogonalOperatorsAsRotationsAndReflections
Related topic	DerivationOf2DReflectionMatrix