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\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{Math251-104}{Quiz 3}{Sep 12, 2025}

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\begin{enumerate}

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\item\points{3} Let $P(2,-1,-2)$, $Q(3,0,3)$ and $R(1,2,-1)$ be points in $\RR^3$.  Find the angle formed by the triangle $PQR$ at the vertex $Q$.  Leave your answer in terms of inverse trig functions.
\vfill

\item\ppoints{2}{3} Let $\mb{a}=\vect{1,-1,3}$ and $\mb{b}=\vect{5,1,-2}$.  
\begin{enumerate}
	\item Find $\mb{a}\times\mb{b}$.
	\vfill
	\item Find the area of the triangle formed by placing the tails of the vectors $\mb{a}$ and $\mb{b}$ at the origin.
	\vspace{2cm}
\end{enumerate}

\item\points{1} Let $\mb{a},\mb{b}\in\RR^n$ and let $T$ be the triangle with vertices $\mb{0}, \mb{a}, \mb{b}$.  The cross product is defined only in $3$-dimensions, so it cannot be used to compute the area of $T$.  Use the dot product theorem and the formula $\sin \theta = \sqrt{1-\cos^2 \theta}$ for $0\le \theta \le \pi$ to find a formula for the area of $T$ in terms of the magnitudes of $\mb{a}$ and $\mb{b}$ and the dot product $\mb{a}\cdot\mb{b}$.
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\end{enumerate}
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