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{\bf Question}

Given ${\bf p} = (2,1,0), \hspace{.1in} {\bf q} = (-1,-1,-1),
\hspace{.1in} {\bf r} = (1,2,1),$ find: (a) $({\bf p} \times {\bf
r})\cdot {\bf q}$, (b) $({\bf p} \times {\bf r})\times {\
bf q} $


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{\bf Answer}

$${\bf p} = (2,1,0) \hspace{.1in} {\bf q} = (-1,-1,-1)
\hspace{.1in} {\bf r} = (1,2,1)$$

Then ${\bf p} \times {\bf r}$ is given by\ \ 
$\left|\begin{array}{ccc} {\bf i}&{\bf j}&{\bf k}\\ 2&1&0\\1&2&1
\end{array}\right|$

Thus ${\bf p} \times {\bf r} = (1\times 1 - 2 \times0, 0 \times1 -
1 \times 2, 2 \times 2 - 1 \times 1) = (1,-2,3)$

\begin{description}
\item[(a)] \begin{eqnarray*}({\bf p} \times {\bf r})\cdot {\bf q} & = &
 (1,-2,3) \cdot(-1,-1,-1) \\ & = & 1 \times -1 + -2 \times -1 + 3 \times -1 \\ & = &
 -2\end{eqnarray*}
\item[(b)] \begin{eqnarray*} ({\bf p} \times {\bf r})\times {\bf q} & = & (1,-2,3)
\times(-1,-1,-1) \\  & = & (-2 \times  1 - (-1 \times 3), 3 \times
-1 - (-1 \times 1), 1 \times 1 - (-1 \times -2)) \\ & = &
(5,-2,-3)
\end{eqnarray*}
\end{description}


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