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\begin{document}


{\bf Question}

\begin{description}
\item[(a)] For ${\bf c} = 5{\bf a} - {\bf b}$ and ${\bf d} = 3{\bf
a} + 2{\bf b}$ find ${\bf c} \times {\bf d}$ when
\begin{description}
\item[(i)] ${\bf a}$ and ${\bf b}$ are unit vectors at an angle
$\frac{\pi}{4}$
\item[(ii)]  ${\bf a}$ and ${\bf b}$ are perpendicular  with $|{\bf
b}| = 2|{\bf a}| = 2.$
\end{description}
\item[(b)] Evaluate $|{\bf c} \cdot {\bf d}|$ when ${\bf c} = {\bf
i} + 2{\bf j} + 3{\bf k}$ and ${\bf d} = 2{\bf i} - {\bf j} + {\bf
k}$
\end{description}

\vspace{.25in}

{\bf Answer}

\begin{description}
\item[(a)]${\bf c} = 5{\bf a} - {\bf b}$ and ${\bf d} = 3{\bf
a} + 2{\bf b}$

$\begin{array}{rcl} c\times d & = & (5{\bf a} - {\bf b}) \times
(3{\bf a} + 2{\bf b})\\ & = & 15{\bf a \times a} + 10{\bf a \times
b} - 3{\bf b \times a} - 2{\bf b \times b} \\ & = & 13{\bf a
\times b}\end{array}$

\begin{description}
\item[(i)] $13{\bf a \times b} = 13 |a| |b| \sin \frac{\pi}{4} \hat{n} = \frac{13}{\sqrt2}
\hat{n}$
\item[(ii)]  $13{\bf a \times b} = 13 |a| |b| \sin \frac{\pi}{2} \hat{n} =
26 \hat{n}$
\end{description}
\item[(b)] $| c \times d| = (1, 7, -5) \hspace{.2in} |c \times d| = 5 \sqrt 3$
\end{description}


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