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\title{ESP Workshop, Worksheet \#10}
\date{\today}
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\begin{center}
{\bf \Large ESP Workshop, Worksheet \#10}

{\bf \large Thursday \today}

{\bf \large AI: Eric Katerman}
\end{center}
\vspace{.15in}
\begin{enumerate}
\item 
\begin{enumerate}
\item Compute the following determinants:
\begin{displaymath}
\textrm{(a)}\ \left | \begin{array}{cc}
2 & 1 \\ 1 & 1 \end{array} \right |
\quad \textrm{(b)}\ \left | \begin{array}{ccc}
2 & 1 & -1 \\ 3 & 1&0 \\ 4 & -5 & 2 \end{array} \right |
\quad \textrm{(c)}\ \left | \begin{array}{ccc}
\bf{i} & \bf{j} & \bf{k} \\ 3 & 1 & 0 \\ 4 & -5 & 2 \end{array} \right |
\end{displaymath}

\item Can you find two vectors $\mathbf{a}$ and $\mathbf{b}$ such that
  $\mathbf{a} \times \mathbf{b}$ is the answer to 1(c)?
  
\item What is the unit vector pointing in the same direction as
  $\mathbf{a} \times \mathbf{b}$?

\item Find two unit vectors that are orthogonal to both $\langle 1,
  -1, 1 \rangle$ and $\langle 0,3,3 \rangle$.

\end{enumerate}

\item Let $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ be vectors and
  let $a$ and $b$ be real numbers. Determine which of the following
  are always true, and which are not always true.  If possible, give
  an example where the statement is not true.
\begin{enumerate}
\item $(a\mathbf{u})\cdot (b\mathbf{v}) = (ab)\mathbf{u}\cdot \mathbf{v}$
\item $a(\mathbf{u}+\mathbf{v}) = a\mathbf{u}+a\mathbf{v}$
\item $\mathbf{w}\cdot (\mathbf{u}+\mathbf{v}) = \mathbf{w}\cdot \mathbf{u} + \mathbf{w}\cdot \mathbf{v}$
\item $(\mathbf{u}\cdot \mathbf{v}) \mathbf{w} = (\mathbf{w}\cdot \mathbf{u}) \mathbf{v}$
\item $\mathbf{u}\cdot \mathbf{v} = \Vert \mathbf{u} \Vert \Vert \mathbf{v} \Vert$
\item $(\mathbf{u}-\mathbf{v}) - \mathbf{w} = \mathbf{u} - (\mathbf{v}-\mathbf{w})$
\item $\mathbf{u}\cdot \mathbf{v} = \mathbf{v}\cdot \mathbf{u}$
\item $\Vert \mathbf{u} \times \mathbf{v} \Vert = \Vert \mathbf{v}\times \mathbf{u} \Vert$
\item $\mathbf{u} \times \mathbf{v} = \mathbf{v}\times \mathbf{u}$
\item $a\mathbf{u} \times b\mathbf{v} = ab\mathbf{u}\times \mathbf{v}$

\end{enumerate}  

\item Prove that $\mathbf{a} \times \mathbf{b}$ is orthogonal to
  $\mathbf{a}$.

\item (A Chemistry problem!) A molecule of methane, $\textrm{CH}_4$,
  is structured with the four hydrogen atoms at the vertices of a
  regular tetrahedron and the carbon atom at the centroid. The
  \textit{bond angle} is the angle formed by the H--C--H combination;
  it is the angle between the lines that join the carbon atom to two
  of the hydrogen atoms. Show that the bond angle is about 109.5
  degrees. [\textit{Hint:} Take the vertices of the tetrahedron to be
  the points $Q =(1,0,0), R = (0,1,0), P =(0,0,1),$ and $S=(1,1,1)$ as
  in Figure 1 (on the back).  Then the centroid is (1/2, 1/2, 1/2).]

\newpage
\begin{figure}
\begin{center}
\psfrag{u}{\bf{u}}
\psfrag{v}{\bf{v}}
\psfrag{s}{$S$}
\psfrag{q}{$Q$}
\psfrag{p}{$P$}
\psfrag{r}{$R$}
\includegraphics{worksheet10}
\caption{The vectors on the left are for question 5, and the tetrahedron on the right is for questions 4 and 6.}
\end{center}
\end{figure}

\item Suppose that $\Vert \mathbf{u} \Vert = 6$ and $\Vert \mathbf{v}
  \Vert = 3$. Is $\mathbf{u} \times \mathbf{v}$ a vector pointing into
  the page or out of the page? Also, what is $\Vert \mathbf{u} \times
  \mathbf{v} \Vert$?
  
\item (The geometry of a tetrahedron.) A tetrahedron is a
  3-dimensional solid with four vertices, $P, Q, R,$ and $S$, and four
  triangular faces as shown in the figure.
\begin{enumerate}
\item Let $\mathbf{v_1}, \mathbf{v_2}, \mathbf{v_3},$ and $\mathbf{v_4}$ be vectors with lengths equal to the areas of the faces opposite the vertices $P, Q, R,$ and $S$ respectively, and directions perpendicular to the respective faces pointing outward. Show that
\begin{displaymath}
\mathbf{v_1} + \mathbf{v_2} + \mathbf{v_3} + \mathbf{v_4} = 0
\end{displaymath}

\item The volume $V$ of a tetrahedron is one-third the distance from a vertex to the opposite face, times the area of that face.
\begin{enumerate}
\item Find a formula for the volume of a tetrahedron in terms of the coordinates of its vertices, $P, Q, R,$ and $S$.
\item Find the volume of the tetrahedron whose vertices are $P=
  (1,1,1), Q=(1,2,3), R=(1,1,2),$ and $S=(3,-1,2)$.
\end{enumerate}
\item Suppose that the tetrahedron in the figure has a trirectangular
  vertex $S$. (This means that the three angles at $S$ are all right
  angles.) Let $A,B,$ and $C$ be the areas of the three faces that
  meet at $S$, and let $D$ be the area of the opposite face $PQR$.
  Using the result of Problem 1, or otherwise, show that
\begin{displaymath}
D^2 = A^2 + B^2 + C^2
\end{displaymath}
This is a three-dimensional version of the Pythagorean Theorem. Can
you think of a three-dimensional version of Fermat's Last Theorem? Can
you find any integer solutions to such an equation??
\end{enumerate}


\end{enumerate}

\end{document}