\documentclass[12pt]{article}
\usepackage{physics,graphicx,multicol}
\pagenumbering{gobble}

\renewcommand{\vec}[1]{\ensuremath{\mathbf{#1}}}
\newcommand{\pts}[1]{(#1 points)}

\newif\ifkey
%\keytrue %comment to hide key

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	\begin{minipage}{0.875\textwidth}
	#1
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\begin{document}
\begin{large}
\noindent
\makebox[\textwidth]{
	\textbf{Quiz 1 | MA261 | June 16, 2017}
}
\makebox[\textwidth]{Christina Jamroz, Alden Bradford}

\ifkey\else
\noindent
All quizzes are scored out of 20 points. Show your work on all problems to receive full credit. Partial credit will be awarded where progress is shown. 20 points also earns you a sticker.
\fi

\begin{enumerate}
\ifkey\else
\itemsep2em
\fi

\item \pts{8} Let $\vec{u}=\langle 1,0,5\rangle$ and $\vec{v}=\langle -2,6,1\rangle$. Find:
	\begin{multicols}{4}
	\begin{enumerate}
		\item $2\vec{u}$
		\item $\vec{u}-\vec{v}$
		\item $\vec{u}\cdot\vec{v}$
		\item $\vec{u}\times\vec{v}$
	\end{enumerate}
	\end{multicols}

\key{
\begin{multicols}{2}
	\begin{enumerate}
		\item $\langle 2, 0, 10\rangle$
		\item $\langle 3, -6, 4\rangle$
		\item $3$
		\item $\langle -30, -11, 6\rangle$
	\end{enumerate}
\end{multicols}
}

\item \pts{6} Find a value for $t$ such that the vectors $18\vec{i}+t\vec{j}$ and $ 2\vec{i}-4t\vec{j}$ are perpendicular.

\key{
If the vectors are perpendicular, their dot product will be zero.
\begin{align*}
(18\vec{i}+t\vec{j})\cdot(2\vec{i}-4t\vec{j})&=0\\
36-4t^2&=0\\
(3+t)(3-t)&=0
\end{align*}
Hence, we must have $t=3$ or $t=-3$.
}



\item \pts{6} $\vec{a}$ and $\vec{b}$ form two sides of the equilateral triangle with side length 2 shown below. Find $|\vec{a}\times\vec{b}|$. Should $\vec{a}\times\vec{b}$ point into the page, or out of the page?

\ifkey
\begin{multicols}{2}
\raisebox{-5em}{\includegraphics[width=0.35\textwidth]{triangle}}
\fbox{
\begin{minipage}{0.6\textwidth}
	Using the formula $|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|\sin\theta$ with $\theta=\frac{\pi}{3}$ we find $|\vec{a}\times\vec{b}|=2\sqrt{3}$. The right-hand rule tells us the vector will point into the page.
\end{minipage}
}
\end{multicols}

\else
\makebox[0.9\textwidth]{\includegraphics[width=0.35\textwidth]{triangle}}
\fi


\end{enumerate}
\end{large}
\end{document}

UNUSED PROBLEMS
%%%%%%%%%%%%%%%
\item \pts{4} Find $(\vec{i}\times\vec{j})\times\vec{k}$. Show your work.

\item Find the center and radius of the sphere described by the equation $x^2+y^2+z^2=2x-4y-8z-5$.