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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 10/19/98}\\
{\it Closed book \hfill Van Dommelen \hfill 11:00-11:50am}
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Show all reasoning and intermediate results leading to your answer.
One book of mathematical tables, such as Schaum's Mathematical Handbook,
may be used, as well as a standard size handwritten formula sheet.

\begin{enumerate}

\item What is the equation for the unknowns $a$, $b$, $c$, and $d$
in order that the point P, (1,-1,5) is in the plane
\begin{displaymath}
    a x + b y + c z = d
\end{displaymath}
What are the equations that need to be satisfied in order that the
points Q, (-1,1,7), R, (-1,-1,11), S, (-1,-2,13), and T, (-1,3,3), are
also all in this same plane?  Solve all equations to find the plane,
if it exists.
\begin{rawhtml}
<A HREF="figures/1.gif">Solution.</A>
\end{rawhtml}

\item We want to push a drill in between two intersecting planes of a
part to drill a hole through the line where they meet.  The drill will
only fit if the angle between the planes is at least 30 degrees.  One
of the two planes passes through the three points P, (1,2,3), Q,
(1,3,2), and R, (2,1,3).  The other plane is normal to the vector
$\vec n_2 = (-1,-2,-3)$.  Determine the (smaller) angle
between these two planes to see whether it is more than 30 degrees.
\begin{rawhtml}
<A HREF="figures/2.gif">Solution.</A>
\end{rawhtml}

\item  The potential energy near a position of equilibrium (0,0,0)
is given by
\begin{displaymath}
    V = x^2 + y^2 + z^2 + 2xy - 4xz +2yz
\end{displaymath}
The equilibrium is stable if the potential energy is always positive
away from the origin (0,0,0).  Determine whether the corresponding
matrix is positive definite to see whether it is stable.
\begin{rawhtml}
<A HREF="figures/3.gif">Solution.</A>
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\end{enumerate}

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