\documentclass[12pt]{article}
\usepackage{amsmath}
%%%--------- Set up page size and formatting information ----------%%%
\def\newpage{\par\vfill\eject}	%new page macro
\def\tab{\ \ \ \ \ }		%tab in five spaces
\def\dtab{{\tab}{\tab}}         %double tab

\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.0in}
\setlength{\topmargin}{-0.51in}
\setlength{\oddsidemargin}{0.0in}
\parskip 2ex
\parindent=0pt
%%%----------------------------------------------------------------------%%%
\begin{document}

\centerline{\bf ME 115(a):  Homework $\#2$}
\vskip 0.15 true in
\centerline{(Due Wednesday, February 6, 2008)}
\vskip 0.5 true in

\vskip 0.2 true in
{\bf Problem 1:} (5 points) Consider the following rotation matrix:
  \[ \begin{bmatrix} 0.866025  & -0.353553 &  0.353553 \\
	  	    0.353553  &  0.933013 & 0.0669873 \\
		 -0.353553 & 0.0669873 &  0.933013 \end{bmatrix} \]

Find the axis of rotation and angle of rotation associated with this rotation.

\vskip 0.2 true in
{\bf Problem 2:} (10 points) Can every orthogonal matrix be represented by a the
exponential of a real matrix?  That is, if $A\ \in \ {\cal O}(n)$, can
$A$ be represented by
  \[ A\ =\ e^{C} \]
for some {\em real} matrix $C$?  (Hint: the determinant of $e^{C}$ can
be expressed as an exponential.)

{\bf Problem 3:} (15 points) Let Z-X-Y Euler angles be denoted by $\psi$, $\phi$, and
$\gamma$.  That is, successfully rotate a body about its body fixed $z$, $x$,
and $y$ axes by the angles $\psi$, $\phi$, and $\gamma$.  

\begin{itemize}
\item{}{\bf Part (a):} Develop an expression for the rotation matrix that
describes the Z-X-Y rotation as a function of the angles $\psi$, $\phi$, and
$\gamma$.

\item{}{\bf Part (b):} Given a rotation matrix of the form:
  \[ R = \begin{bmatrix} r_{11} & r_{12} & r_{13} \\
		       r_{21} & r_{22} & r_{23} \\
		       r_{31} & r_{32} & r_{33} \end{bmatrix} \]
compute the angles $\psi$, $\phi$, and $\gamma$ as a function of the $r_{ij}$.
\end{itemize}


\vskip 0.2 true in
{\bf Problem 4:} (10 points) Do Problem 4(a,b) in Chapter 2 of MLS. 

\vskip 0.15 true in
{\bf Problem 5:} (10 points) Do Problem 5 in Chapter 2 of MLS.

\vskip 0.15 true in 
{\bf Problem 6:} (10 points) Do Problem 10 (a,b) in Chapter 2 of MLS.  Do not worry about
the question of surjectivity in 10(b).

\end{document}

