A turntable can be rotated by any angle about its spindle axis

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Group theory and symmetry

Invariance, Ch Groups: the algebra of symmetry operations

Problem

A turntable can be rotated by any angle θ\theta about its spindle axis. Represent this active rotation of the plane of the turntable, about the spindle, by the matrix R(θ)=(cos⁡θ−sin⁡θsin⁡θcos⁡θ)R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}, for θ\theta taken modulo 2π2\pi radians, the rotation-matrix form held from the linear algebra course. Let G={R(θ):0≤θ<2π}G=\{R(\theta):0\le\theta<2\pi\} with matrix multiplication as the group operation. ((a)) Using the compound-angle (addition) formulae for cos⁡\cos and sin⁡\sin, prove that R(θ1)R(θ2)=R(θ1+θ2 mod 2π)R(\theta_1)R(\theta_2)=R(\theta_1+\theta_2 \bmod 2\pi), and hence that GG is closed under matrix multiplication. ((b)) Explain why associativity needs no fresh proof here, and give the identity element and the inverse of R(θ)R(\theta). ((c)) As a check on part (a), compute R(40∘)R(50∘)R(40^\circ)R(50^\circ) directly by matrix multiplication and confirm the four entries of the product match R(90∘)R(90^\circ).

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