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{\bf Question}

Given ${\bf u} = (4,2,0),$  ${\bf v} =(-3,1,1),$  ${\bf w} =
(5,1,5),$  ${\bf s} = (1,2,1) $ find: (a) ${\bf u}\cdot{\bf v}$,
(b) $({\bf u} \cdot {\bf s}){\bf \hat v}$

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{\bf Answer}

$${\bf u} = (4,2,0) \hspace{.1in} {\bf v} =(-3,1,1) \hspace{.1in}
{\bf w} = (5,1,5)\hspace{.1in} {\bf s} = (1,2,1) $$
\begin{description}
\item[(a)]
${\bf u} \cdot {\bf v} = 4 \times -3 + 2 \times 1 + 0 \times 1 =
-12 + 2 = -10$
\item[(b)]
\begin{eqnarray*}
({\bf u} \cdot {\bf s}){\bf \hat v} & = & [4 \times 1 + 2 \times 2
+ 0 \times 1] \times \frac{(-3,1,1)}{\sqrt{(-3)^2 + 1^2 + 1^2}} \\
& = & \frac{8}{\sqrt{11}}(-3,1,1) \\ & = & \left(
\frac{-24}{\sqrt{11}}, \frac{8}{\sqrt{11}}, \frac{8}{\sqrt{11}}
\right) \end{eqnarray*}
\end{description}


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