AIMS Lie Groups: Fall-2022 HW 1 : Due 06 Mon 12 Dec
Quaternionic rotations
S0 4487S
Identify imaginary quaternions \(v=xi+yj+zk\in\mathrm{Im}\mathbb{H}\) with \((x,y,z)\in\mathbb{R}^3\) and let \(q=\cos\frac\alpha2+u\sin\frac\alpha2\), with \(u\in\mathrm{Im}\mathbb{H}\) of unit norm, that is, \(u\bar{u}=1\). Consider the transformation
\[
T_q(v) = v\longmapsto qvq^{-1}
\]
Show that the only direction fixed by \(T_q\) is \(u\), that is, show that \(T_q(w)=w\) for \(w\in\mathrm{Im}\mathbb{H}\) if and only if \(w\) is a (real) multiple of \(u\).
If \(u\in\mathrm{Im}\mathbb{H}\) is perpendicular to \(v\), show that \(T_q(v)\) with \(\alpha=\frac\pi2\) is perpendicular to both \(u\) and \(v\).
A reasonable conclusion (which you do not need to verify) is that \(T_q\) represents rotation through an angle \(\alpha\) about the \(u\) direction. (Every rigid rotation in \(\mathbb{R}^3\) can in fact be accomplished as a single rotation about some axis.) Using this interpretation, determine the effective angle of a rotation about the \(z\) axis by \(\beta\) followed by a rotation about the \(u\) axis by \(\alpha\).
You do not need to determine the effective axis of rotation.
For the special case where \(u=j\) (the \(y\) axis) and \(\alpha=\frac\pi2=\beta\), determine both the angle of rotation and its axis. You are encouraged to check your result by actually performing these rotations on a physical object!