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\title{General Mathematics and CPS II}
\author{Exercise 7}
\date{February 25, 2015}
\maketitle

\begin{enumerate}

\item (Ivanov, p.\ 36, Problem~14.)  Show that
  \begin{enumerate}
  \item $R_{\ell_1} \, R_{\ell_2} = R_{\ell_2} \, R_{\ell_1}$ if and
  only if $\ell_1$ and $\ell_2$ are perpendicular;
  \item  $R_{\ell} \, H_A = H_A \, R_{\ell}$ if and
  only if $A \in \ell$.
  \end{enumerate}

\item Use the matrix form of the equation for a reflection (see
handout) to show that the composition of reflections about parallel
lines is a translation $\Pi_{\boldsymbol v}$.  Find an expression for the
translation vector $\boldsymbol v$.

% \item A \emph{Euclidean transformation} $F$ or, in short, a
% \emph{motion} of the plane is a map which preserves the Euclidean
% distance between points, i.e.
% \[
%   \lvert F(a) - F(b) \rvert = \lvert a - b \rvert
% \]
% where $\lvert a-b \rvert$ denotes the distance between two arbitrary
% points $a$ and $b$.

% Show that every motion of the plane is bijective.

% \emph{Hint:} To prove that it is surjective, notice that the
% transformation must map lines onto lines and circles onto circles.

\end{enumerate}
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