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\LARGE 

{\hspace{50mm}Exam 1
\hspace{50mm}Name:\hspace{20mm}}

%\includegraphics{model}  This is a line that will not run with the commented out percent sign... however this is how to include a picture.
\large

Directions:  Answer each question to the best of your ability.  The number of points the question is worth follows the question. Calculators are allowed, but partial credit can only be given if proper work is shown.
\small

\begin{enumerate}
\item Given that $\vec{v}=i-6j-2k$ and $\vec{w}=4i+2j-4k$ on the given set of coordinate axes, find the following: (5 pts each)
\begin{enumerate}
\item $3\vec{v}-2\vec{w}$
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\item $\vec{v}*\vec{w}$
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\item $\vec{w}\,\mathtt{ x }\,\vec{v}$
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\item The magnitude of $\vec{w}$, or $||\vec{w}||$
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\item The unit vector parallel to $\vec{w}$ pointing in the opposite direction.
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\item The angle between $\vec{v}$ and $\vec{w}$.
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\end{enumerate}
\newpage
\item Consider each of the following statements.  Write T if the statement is always true, and F if the statement is not always true. $\vec{a}$, $\vec{b}$, and $\vec{c}$ are non-zero vectors.  (4 pts each)
\begin{enumerate}
\item $(\vec{a}*\vec{b})*\vec{c}=\vec{a}*(\vec{b}*\vec{c})$
\vspace{10mm}
\item $(\vec{a}\,\mathtt{x}\,\vec{b})*\vec{c}=-(\vec{b}\,\mathtt{x}\,\vec{a})*\vec{c}$
\vspace{10mm}
\item If $\vec{a}$, $\vec{b}$, and $\vec{c}$ are parallel, then $(\vec{a}\,\mathtt{x}\,\vec{b})*\vec{c}=0$
\vspace{10mm}
\item If $\vec{a}\,\mathtt{x}\,\vec{b}=\vec{0}$, then $\vec{a}\perp\vec{b}$
\vspace{10mm}
\item $(\vec{a}-\vec{b})*(\vec{a}+\vec{b})=||\vec{a}||^2-||\vec{b}||^2$
\vspace{10mm}
\end{enumerate}
\item Given the points $P$=(1,2,4), $Q$=(2,1,1), and $R$=(-1,0,2), answer the following: (5 pts each)
\begin{enumerate}
\item What is the vector $\vec{n}=\vec{PQ}\,\mathtt{x}\,\vec{PR}$?
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\item What is the defining characteristic of $\vec{n}$?
\vspace{10mm}
\item What is the equation of the plane containing $P$, $Q$, and $R$?
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\end{enumerate}
\newpage
\item  For what value(s) of $\lambda$ are the vectors $\vec{v}=\lambda i-3j+3k$ and $\vec{w}=\lambda i+\lambda j-6k$ perpendicular? (5 pts)
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\item The displacement (in meters) of an object is represented by the vector $\vec{d}=3i-j-2k$, while the constant force (in Newtons) acting on the object is represented by the vector $\vec{F}=2i-3j+k$.  Find the work done on the object. (5 pts)
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\item Find the projection of vector $\vec{v}=2i-3j+k $ onto the vector $\vec{w}=8i-9j-12k$.  (5 pts)
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\newpage
\item An airplane flies southwest with an airspeed of 400 $mph$.  While in the air, the plane encounters a wind that blows due south at a speed of 50 $mph$.  Assume a coordinate system such that the positive $x$ and $y$ are aligned with the East and North directions respectively. (5 pts each)
\begin{enumerate}
\item On the axes below, sketch the graphs of the vectors that represent the speed and direction of both the plane and the wind.
\begin{center}
\includegraphics[scale=1.25]{examq7.pdf} 
\end{center}
\item Express in component form the vector that represents the velocity of the airplane relative to the ground.
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\item What is the speed of the airplane relative to the ground?
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\item In which direction is the plane traveling relative to the ground?  (express your answer in term of degrees relative to West)
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\end{enumerate}
\end{enumerate}

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