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\centerline{\bf ME 115(a):  Homework $\#3$}
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\centerline{(Due Friday Feb. 14, 2014)}
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{\bf Problem 1:} (20 points) Do Problem 6(a,b,d,e) in Chapter 2 of the MLS text.

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{\bf Problem 2:} (15 points) Do Problem 11(a,b,d) in Chapter 2 of the MLS text.

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{\bf Problem 3:}  (15 points) Consider $2\times 2$ complex matrices of the form:

   \[ M=\begin{bmatrix} z & w \\ -w^{*} & z^{*} \end{bmatrix}
      =\begin{bmatrix} (a+ib) & (c+id) \cr -(c-id) & (a-ib) \end{bmatrix} \]
where: 
	\[ det(M) = z z^{*} + w w^{*} = 1 \]
and $z,w\in {\cmplx}$, and $*$ denotes complex conjugation.  Such
matrices form a matrix group termed the ``special unitary matrices''
of dimension 2, $SU(2)$.  

\begin{itemize}
\item{} {\bf Part (a):} Show that matrices:
\[ \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \ \ \ \ \ 
   \begin{bmatrix} i & 0\\ 0 & -i \end{bmatrix} \ \ \ \ \ 
   \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix} \ \ \ \ \ 
   \begin{bmatrix} 0 & i\\ i & 0 \end{bmatrix} \]
form a basis for $SU(2)$.  The element $i$ is $\sqrt{-1}$.  
I.e., all elements of $SU(2)$ can be 
expressed as some combination of these elements.  Next show that
elements of $SU(2)$ are
isomorphic to the unit quaternions.  That is, there is a one-to-one
correspondence between each element of $SU(2)$ and a unit quaternion.  

\item{} {\bf Part (b):} Show that the special unitary representation
of a rotation in terms of z-y-x Euler Angles can be computed as :
\def\halfpsi{ {\psi \over 2} }
\def\halfphi{ {\phi \over 2} }
\def\halfgamma{ {\gamma \over 2} }
  \[ \begin{bmatrix} \cos\halfpsi & i\sin\halfpsi \\
		  i\sin\halfpsi & \cos\halfpsi \end{bmatrix}
    \begin{bmatrix}\cos\halfphi & \sin\halfphi \\
		  -\sin\halfphi & \cos\halfphi \end{bmatrix}
    \begin{bmatrix} e^{i\halfgamma} & 0 \\ 0 & e^{-i\halfgamma}\end{bmatrix} \]
where $\psi$, $\phi$, and $\gamma$ are respectively the rotations about the z,
y, and x axes.

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\item{} {\bf Part (c):} Suppose a rotation is represented by a
special unitary matrix of the form:
  \[ \begin{bmatrix} z & w\\ -w^{*} & z^{*} \end{bmatrix} \]
determine the angle of rotation, $\phi$, and the axis of rotation, $\hat{s}$.

\end{itemize}

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{\bf Problem 4:} (10 points)  Find the three z-y-x Euler angles from a given rotaion matrix

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