derivation of rotation matrix using polar coordinates

We derive formally the expression for the rotationMathworldPlanetmath of a two-dimensional vector 𝒗=a𝒙+b𝒚 by an angle ϕ counter-clockwise. Here 𝒙 and 𝒚 are perpendicularMathworldPlanetmathPlanetmathPlanetmathPlanetmath unit vectorsMathworldPlanetmath that are oriented counter-clockwise (the usual orientation).

In terms of polar coordinates, 𝒗 may be rewritten:

𝒗 =r(cosθ𝒙+sinθ𝒚),a=rcosθ;b=rsinθ,
for some angle θ and radius r0. To rotate a vector 𝒗 by ϕ really means to shift its polar angleMathworldPlanetmath by a constant amount ϕ but leave its polar radius fixed. Therefore, the result of the rotation must be:
𝒗 =r(cos(θ+ϕ)𝒙+sin(θ+ϕ)𝒚)
Expanding using the angle addition formulae, we obtain
𝒗 =r(cosθcosϕ-sinθsinϕ)𝒙+(sinθcosϕ+cosθsinϕ)𝒚)

When this transformation is written out in [𝒙,𝒚]-coordinatesMathworldPlanetmathPlanetmath, we obtain the formula for the rotation matrixMathworldPlanetmath:

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