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~
\vspace{2cm}

\noindent
\textbf{Directions:}  
\begin{enumerate}
	\item Section: Math251 007
	\item Write your name with one character in each box below.
	\item Show all work.  No credit for answers without work.
\end{enumerate}

\newpage
\headers{Math251}{Test 1}{Sep 24, 2025}

~
\vspace{1.5cm}

\begin{enumerate}

\newcommand{\proj}{\mathrm{proj}}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Page 1: linear algebra, 12.1
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item\points{10} Use matrices and row operations to give a simple description for the set of  solutions to the following system.
\[  
	\kgmarray{7}{
	 x_1   &-&   2x_2   &-&  9x_3   &=&  5\\
	 3x_1  &+&   x_2   &-& 13x_3   &=&  8
    }
\]
\vfill

\item Let $S$ be the sphere $x^2 + 3x + y^2 - 2y + z^2 + 4z = 13$.
\begin{enumerate}
	\item\points{10} Find the center and radius of $S$.
	\vfill
	
	\item\points{6} Give a full geometric description of the curve obtained at the intersection of $S$ and the plane $z=0$.
	\vspace{4cm}
\end{enumerate}

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%% Page 2: 12.2, 12.3
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\newpage
~
\vspace{1.5cm}

\newcommand{\myspc}{\vspace{2cm}\\~}
\item\ppoints{4}{4} Let $\mb{a} = \vect{1,-5,2}$ and $\mb{b} = \vect{1,-1,-2}$.  Compute the following.
\begin{multicols}{2}
\begin{enumerate}
\item $3\mb{a} - 2\mb{b}$
\myspc

\item $|\mb{a}|$
\myspc

\columnbreak
\item $\mb{a}\cdot \mb{b}$
\myspc

\item $\proj_{\mb{a}} \mb{b}$
\myspc
\end{enumerate}
\end{multicols}

\item\points{6} Find numbers $h$ and $k$ such that $\vect{3,2,-5}$ and $\vect{h,-1,k}$ are parallel.
\vfill

\item Let $\mb{a}=\vect{2,1,0}$ and $\mb{b} = \vect{-1,3,5}$.  
\begin{enumerate}
	\item\points{8} Find the angle between $\mb{a}$ and $\mb{b}$.  Leave your answer in terms of inverse trigonometric functions.
	\vfill
	\item\points{2} Is the angle between $\mb{a}$ and $\mb{b}$ acute, obtuse, or right?  Explain.
	\vspace{1cm}
\end{enumerate}


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%% Page 3: 12.4, 12.5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
~
\vspace{1.5cm}

%\item\points{8} Compute $\vect{3,-1,4}\times \vect{2,-3,-7}$.
%\vfill

\item Let $T$ be the triangle with vertices $(2,1,0)$,  $(3,1,4)$ and $(1,5,1)$.
\begin{enumerate}
	\item\points{10} Find a simple equation for the plane containing $T$.
	\vfill
	\item\points{6} Find the area of $T$.
	\vfill
\end{enumerate}
  
\item\points{10} If the following lines intersect, then find the point of intersection.  If the lines do not intersect, determine if they are parallel or skew.
\begin{align*}
\mb{r}_1 &= \vect{9,3,-16} + t\vect{1,1,4}  &
\mb{r}_2 &= \vect{5,14,-5} + t\vect{-2,3,1}
\end{align*}
\vfill

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Page 4: 12.6
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
~
\vspace{1.5cm}
\item Consider the surface $\frac{x^2}{4} + y + \frac{z^2}{9} = 1$.
\begin{enumerate}
	\item\points{6} Describe the axis-aligned traces.
	\vspace{5cm}
	
	\item\points{10} Classify the surface and sketch it, labeling any significant points.
	\vfill
\end{enumerate}


\newpage
~

\end{enumerate}
\end{document}
