\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\input epsf
\def\R{{\Bbb R}}
\font\srm=cmr8
\magnification=\magstep1
\def\u{{\bf u}}
\def\v{{\bf v}}
\def\w{{\bf w}}
\def\F{{\bf F}}
\def\d{{\bf d}}

\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 20\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 20} 
\bigskip


\item{1.}  Recall that if $\u$ and $\v$ are vectors in $\R^n$, we
define their {\it dot product} to be
$$
\u\cdot\v=||\u||\,||\v||\cos\theta
$$
where $\theta$ measures the smaller angle made by $\u$ and $\v$ when
their initial points coincide.  For vectors $\u=(u_1,u_2,u_3)$ and
$\v=(v_1,v_2,v_3)$ in $\R^3$, prove the formula
$$
\u\cdot\v=u_1v_1+u_2v_2+u_3v_3.
$$
\hbox{\vbox{\hsize=3in
\itemitem{Hint:}  Consider the triangle formed by the three vectors
$\u$, $\v$, and $\v-\u$.  Write down the lengths of these three
vectors and then apply the law of cosines.}
\hskip.3in
\epsfxsize=1.5in\epsfbox{/home/oehrtman/m210f96/vector_addition.eps}}

\medskip 
\item{2.}  You may be familiar with the equation $W=fd$ from physics.
It says that the amount of work $W$ done in moving an object a
distance $d$ by applying a constant force $f$ in the direction of
motion is the product of $f$ and $d$.

\medskip
\centerline{
\epsfxsize=3in\epsfbox{/home/oehrtman/m210f96/force1.eps}}

\itemitem{a)}  Suppose that the force applied to the object is not
constant but varies as the object moves.  How could you modify this
formula to compute the work done?  Justify your answer.  What
assumptions did you make?

\medskip
\itemitem{b)}  Suppose a constant force is applied to the object but
now in a direction different from the direction of motion.  For
example, a rope may pull a heavy block at an upward angle but the
block slides horizontally.  Represent the force by a vector $\F$ which
points in the direction of the force and whose magnitude is the amount
of force being applied.  Represent the distance with a vector $\d$
pointing in the direction of motion and with magnitude equal to the
distance traveled.  Find a formula for the work done in this case in
terms of the vectors $\F$ and $\d$.

\medskip
\centerline{
\epsfxsize=3in\epsfbox{/home/oehrtman/m210f96/force2.eps}}

\eject
\itemitem{c)}  How much work does it take to slide a crate 20 m along
a loading dock by pulling on it with a constant force of 200 Newtons
at an angle of $30^\circ$ from the horizontal?

\medskip
\itemitem{d)}  What general approach would you follow to come up with
a formula for the work done if the force vector was continuously
changing and the path of motion was not a straight line.

\centerline{
\epsfxsize=3.5in\epsfbox{/home/oehrtman/m210f96/force3.eps}}

\item{3.}  Find the angles between the following two curves at each
intersection point:
$$
y=x^3\qquad\hbox{ and }\qquad y=\sqrt x.
$$

\bigskip
\bigskip
\item{4.}  What does it mean for vectors to be orthogonal?  Explain
how you can determine if two vectors are orthogonal, and why that
method works.  What does it mean for vectors to be parallel?  Explain
how you can determine if two vectors are parallel, and why it is that
your method works.

\bigskip
\bigskip
\item{5.}  Find conditions on the vectors $\u$ and $\v$ which ensure
that $\u+\v$ is orthogonal to $\u-\v$.

\bigskip
\bigskip
\item{6.}   A 1-kilogram (1-kg) mass located at the origin is
suspended by ropes attached to the points $(1,1,1)$ and $(-1,-1,1)$.
If the force of gravity is pointing in the direction of the vector
$(0,0,-1)$, what is the vector describing the force along each rope?

\medskip
\item{}  [Hint:  Use the symmetry of the problem.  A 1-kg mass weighs
9.8 Newtons (N).] 

\bye