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\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
{Levi-Cevita Tensor Homework 1} \\ (January 18, 2013) \\
\end{center}
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\begin{enumerate}

\item Use the $3D$ identity
\begin{eqnarray} \label{epsdel}
  \sum_{i=1}^3 \epsilon_{ijk}\epsilon_{ilm} =
  \delta_{jl}\delta_{km}-\delta_{jm}\delta_{kl}
\end{eqnarray} 
to calculate
\begin{eqnarray} 
  (\hat{a}\times\hat{b})\cdot(\hat{a}\times\hat{b})
\end{eqnarray} 
where $\hat{a}$ and $\hat{b}$ are unit vectors. Eliminate all 
$\hat{a}\cdot\hat{b}$ in favor of $\cos\theta$. Use the notation
\begin{eqnarray} \label{hataxhatb}
  \hat{a}=\left(\matrix{\tilde{a}_1\cr\tilde{a}_2\cr\tilde{a}_3}\right)
  ~~{\rm and}~~ \hat{b}=
  \left(\matrix{\tilde{b}_1\cr\tilde{b}_2\cr\tilde{b}_3}\right)\,.
\end{eqnarray} 

\item Use the previous result to show $\vec{a}\times\vec{b}
=a\,b\,\sin\theta$ (book p.21).

\end{enumerate}
\end{document}
