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\begin{document}
\begin{center}

\Huge{
Three-Dimensions}
\end{center}
\vskip .1 in



\begin{enumerate}
\item For each of the following set of axis below, identify the positive $z$-axis:

\vspace{-3.2in}\includegraphics[height=8in]{IDZaxis}

\vspace{-3.7in} \item Use the coordinate axis provided \\
on the right for the following questions:
\begin{enumerate}
\item Plot the points \\
$A=(0,0,0)$,\\
$B=(1,2,3)$, \&\\
$C=(0,-2,1)$.

 \vspace{-4in}\hspace{1.2in}\includegraphics[height=8.5in]{3Daxis}

\vspace{-4.5in}Note that the points plotted in \\
part (a) cast `shadows' on the \\
$xy$-plane.  That is, if we drop a \\
perpendicular from a point \\
$P=(a,b,c)$ to the $xy$-plane, the \\
point $Q=(a,b,0)$ is the \emph{projection} of\\
 $P$ to the $xy$-plane.\\
\item Find the $xy$-plane projections for each of the three points you plotted in part (a).  Graph the projections and explain why I described these projections as `shadows'.
\vspace{.5in}
\item Find the $yz$-plane projections of the three points you plotted in part (a).
\vspace{.5in}
\item Find the distance between the point $(0,0,0)$ and $(0,-2,1)$.
\vfill
\item Find the distance between the point $(0,0,0)$ and $(1,2,3)$.
\vfill 
\item Consider the point $P=(a,b,c)$, find the distance between $(0,0,0)$ and $P$. Justify yourself.
\end{enumerate}
\end{enumerate}

\newpage
\begin{center}

\Huge{Things to do with vectors...}
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\vskip .1 in

\begin{enumerate}
\item Let $A(0,0,0)$, $B(1,2,3)$, \& $C(0,-2,1)$.

\begin{enumerate}
\item Plot the vectors\\
$\overrightarrow{AB}$, $\overrightarrow{AC}$, \& $\overrightarrow{AB}+\overrightarrow{AC}$.\\

 \vspace{-4in}\hspace{1.2in}\includegraphics[height=8.5in]{3Daxis}

\vspace{-4.5in}
Notation: vectors $\overrightarrow{v}$ who's initial \\
point is at the origin with their \\
 terminal point $(a, b, c)$ have what \\
 Stewart calls components,   $\langle a, b, c \rangle$.

\item Write the components of $\overrightarrow{AB}$ \& $\overrightarrow{AC}$.
\vskip .5in
\item If a vector doesn't have an initial point at the origin, we can move (translate) the vector so that its initial point is at the origin. \\ Find the components of $\overrightarrow{BA}$ and $\overrightarrow{BC}$.
\vskip .5in
\item In general, write down the components of the vector $\overrightarrow{XY}$ if $X(x_1,y_1,z_1)$ and $Y(x_2,y_2,z_2)$.
\vskip .5in
\end{enumerate}

\hspace{2.5in}\includegraphics[height=2in]{blank_5}

\vspace{-2.2in}
\item Let the components of $\overrightarrow{v}$ and $\overrightarrow{w}$ \\
be $\langle 1,2\rangle$ and $\langle 2, -1\rangle$ respectively.  
\begin{enumerate}
\item Draw $\overrightarrow{v}$, $\overrightarrow{w}$, and $\overrightarrow{v}+\overrightarrow{w}$.
\item What are the components of \\
$\overrightarrow{v}+\overrightarrow{w}$?
\vskip .5in
\item In general, if the components \\of $\overrightarrow{a}$ and $\overrightarrow{b}$ are $\langle a_1, a_2\rangle$ and $\langle b_1, b_2\rangle$ respectively, write down the components of $\overrightarrow{a}+\overrightarrow{b}$.
\vfill
\item  In general, if the components of $\overrightarrow{a}$ are $\langle a_1, a_2\rangle$, find $\vert \overrightarrow{a}\vert$.
\vfill
\end{enumerate}

\end{enumerate}



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