\documentclass[minion]{homework}

\newcommand{\Reals}{\mathbb{R}}
\newcommand{\Quats}{\mathbb{H}}
\def\ip<#1,#2>{\left<#1,#2\right>}
\newcommand{\vi}{\mathbf{i}}
\newcommand{\vj}{\mathbf{j}}
\newcommand{\vk}{\mathbf{k}}
\renewcommand{\Re}{\mathbf{Re}}
\doclabel{Math F490: Homework 2}
\docdate{Due: January 28, 2019}

\begin{document}
\begin{problems}
\problem Andrew gave us the following definition.  A
space rotation is a linear map $R$ from $\Reals^3$ to itself
such that there exists perpendicular unit vectors $u$ and
$v$ and a real number $\theta$ such that
\begin{equation}
\begin{aligned}
R(u) &= u\\
R(v) &= \cos(\theta) v + \sin(\theta) w\\
R(w) &= -\sin(\theta) v + \cos(\theta) w
\end{aligned}
\end{equation}
where $w = u\times v$.
\begin{subproblems}
\item Show that for every axis $u$ and angle $\theta$,
there exists a corresponding rotation. Do this by
showing that conjugation by a particular unit quaternion 
suffices.  Show also that all
rotations with this axis and angle are the same maps.
That is, the role of $v$ in the definition is a bit immaterial.

\item Show that the only unit quaternions $t$ such that
\[
t^{-1} q t = q
\]
for all imaginary unit quaternions are $t=1$ and $t=-1$.
\item Conclude that for every axis $u$ and angle $\theta$,
there is a unique pair $t$ and $-t$ of unit quaternions
such that conjugation by $t$ yeilds the corresponding rotation.
\end{subproblems}

\problem Text: 1.4.2

\problem Text: 1.5.1, 1.5.2

\problem For $p,q \in \Quats$, we define $\ip<p,q>=\Re(p\overline q)$,
where $\Re q$ is the real part of $q$.
\begin{subproblems}
\item Show that if $p\in \Quats$, then $\ip<p,p>=|p|^2$.
\item Using the facts that $\overline{ab}=\overline b \overline a$ and $\Re(\overline a) = \Re(a)$
for all $a,b\in \Quats$, show that $\ip<p,q>=\ip<q,p>$.
\item For $x=(x_1,x_2,x_3,x_4)\in \Reals^4$, let $Q(x)=x_1 + x_2\vi+ x_3\vj+x_4\vk$
be its representation as a quaternion.
Show that for $x,y\in \Reals^4$, $x\cdot y = \ip<Q(x),Q(y)>$.
\item We say that $p$ and $q$ are orthogonal if $\ip<p,q>=0$. 
Show that $p\in\Quats$ is orthogonal to $1$ if and only if $p$ is pure imaginary.
\item Suppose $s\in \Quats$.  Show that for all $p,q\in\Quats$,
$\ip<ps,qs> = |s|^2\ip<p,q>$.  
\item Show that for all $p,q\in\Quats$, $\ip<\overline p, \overline q>=\ip<p,q>$.
\item Suppose $s\in \Quats$.  Show that for all $p,q\in\Quats$,
$\ip<sp,sq> = |s|^2\ip<p,q>$.  
\item Suppose $u\in\Quats$ and $u\neq 0$.  Show that conjugation by $u$ takes the real quaternions to themselves.
\item Conclude quickly that if $u\in\Quats$ and $u\neq 0$, then
conjugation by $u$ takes the imaginary quaternions to themselves.
\end{subproblems}

\end{problems}

\end{document}
