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\begin{document} \huge

\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
{\bf Vectors 1} (Classwork January 7, 2013) \\
\end{center}
\setcounter{equation}{0}

\rightline{Group \#~~~~~~~~}

\leftline{Participating students (print):} \vskip 60pt

\centerline{------------------------------------------------------}

\bigskip

In the following $i=1,\dots,n$, $j=1,\dots,n$.

\begin{enumerate}
\item Let $\hat{x}_i$ and $\hat{x}_j$ be Cartesian unit
vectors. It holds the relation
\begin{equation}
  \hat{x}_i\cdot\hat{x}_j\ =~~~~~~~~~~\
\end{equation} % \bigskip
Name the r.h.s.\ quantity: \bigskip
\item Write down $n$-dimensional ($nD$ henceforth) column vectors.
\vskip 20pt
\begin{equation}
  \vec{a}\ =~~~~~~~~~~~~~~~~~~~~~~
  \vec{b}\ =~~~~~~~~~~~~~~~~~~~~~~
\end{equation} 
\vskip 20pt

\eject

\item Write down the scalar product.
\bigskip \bigskip
\begin{equation} \label{sab}
  \vec{a}\cdot\vec{b}\ =\ \sum_{i=~~}
  ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{equation} \bigskip
\item Write down the scalar products
\bigskip \bigskip
\begin{equation}
  \vec{a}\cdot\vec{a}\ =~~~~~~~~~~~~~~~~~
  \vec{b}\cdot\vec{b}\ =~~~~~~~~~~~~~~~~~
\end{equation} \bigskip
\item Give the definition of the magnitude of $\vec{a}$.
\bigskip \bigskip
\begin{equation}
  |\vec{a}|\ =~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{equation} \bigskip
\item Express the unit vectors $\hat{a}$ and $\hat{b}$
through previously defined quantities.
\bigskip \bigskip
\begin{equation}
  \hat{a}\ =~~~~~~~~~~~~~~~~~~~~~
  \hat{b}\ =~~~~~~~~~~~~~~~~~~~~~
\end{equation} \bigskip
\eject
\item Use $\vec{b}$ to find a unit vector that is perpendicular
to $\hat{a}$. \bigskip
\begin{equation}
  \vec{a}_{\perp}\ =\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{equation} \smallskip

Which condition has $\vec{b}$ to fulfill, so that this is possible?  
\vskip 80pt

Show $\vec{a}_{\perp}\cdot\vec{a}=0$. \vskip 80pt

\item Expand the vectors $\vec{a}$ and $\vec{b}$ in
terms of the unit vectors $\hat{x}_i$.
\bigskip \bigskip
\begin{equation}
  \vec{a}\ =\ \sum_{i=1}^n~~~~~~~~~~~~~~~~
  \vec{b}\ =\ \sum_{j=1}^n~~~~~~~~~~~~~~~~
\end{equation} \bigskip
\eject
\item Calculate the scalar products for the r.h.\ sides of the previous 
equation and show that the results agrees with Eq.~(\ref{sab}). \bigskip
\begin{eqnarray} \nonumber
  \vec{a}\cdot\vec{b}\ =\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\end{eqnarray} \bigskip

\end{enumerate}

\end{document}

