Ex5 21

$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 5.21}$

a) In the Euclidean plane, show that a rotation of angle $\theta$ may be achieved by composing two reflections.

b) Use part a) to show that a clockwise rotation of angle $\theta$ about a point $P$ followed by a clockwise rotation of angle $\phi$ about a point $Q$ results in a clockwise rotation of angle $\theta + \phi$ around the point $R$, where $R$ is the intersection point of the two rays, one through $P$ at angle $\theta/2$ from the line between $P$ and $Q$ and the other through point $Q$ at an angle of $\phi/2$ from the line between $P$ and $Q$.

c) Show that the product of any two rational rotations of the Euclidean plane is also rational.

d) On a sphere of radius $1$, a triangle with angles $\theta$, $\phi$, and $\eta$ has area $\theta + \phi + \eta$ (where $\theta$, $\phi$, and $\eta$ are in radians).

Use this fact to describe the result of rotating clockwise by angle $\theta$ around a point $P$ followed by rotating clockwise by angle $\phi$ around a point $Q$ in terms of the area of a triangle.

e) Prove that on the sphere the product of two rational rotations may be an irrational rotation.