\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\def\R{{\Bbb R}}
\font\srm=cmr8
\magnification=\magstep1

\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 19\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 19} 
\bigskip

\item{1.}  A ship is at position $(1,0)$ on a nautical chart (with
north the positive $y$ direction sights a rock at position $(2,4)$.
The ship is pointing due north and traveling at a speed of 4 knots
relative to the water.  There is a current flowing due east at 1 knot;
The units on the chart are nautical mile; 1 knot = 1 nautical mile per
hour.

\medskip
\itemitem{a)}  If there were no current, what vector ${\bf u}$ would
represent the velocity of the ship relative to the sea bottom?

\medskip
\itemitem{b)}  If the ship were just drifting with the current, what
vector ${\bf v}$ would represent its velocity relative to the sea bottom?

\medskip
\itemitem{c)}  What vector ${\bf w}$ represents the total velocity of the
ship? 

\medskip
\itemitem{d)}  Where would the ship be after 1 hour?

\medskip
\itemitem{e)}  Should the captain change course?

\medskip
\itemitem{f)}  What if the rock were an iceberg?

\medskip 
\item{2.}  In 1618, Johann Kepler discovered that the square of the
time required by a planet for one revolution around the sun is
proportional to the cube of its mean distance from the sun.  In fact,
this law applies to all objects in orbit about a much larger object.
On April 24, 1990 NASA launched the Hubble Space Telescope which now
orbits the Earth 17 times a day at a distance of 3732 miles.

\medskip
\itemitem{a)}  How far away is the moon?

\medskip
\itemitem{b)}  A {\it geosynchronous} orbit is one in which the
sattelite remains over the same geographical location on the Earth.
What is the radius of a geosynchronous orbit?

\def\u{{\bf u}}
\def\v{{\bf v}}
\def\w{{\bf w}}
\eject
\item{3.}  Let $\u$ be the vector from the origin to $(u_1,u_2,u_3)$.
Let $\v$ be the vector from the origin to $(v_1,v_2,v_3)$.  Let $\w$
be the vector from the origin to $(w_1,w_2,w_3)$.   Let $a$ and $b$
denote real numbers.  

\medskip
\itemitem{a)}  For each of the following statements, say if they are
true always, sometimes, never, or are meaningless.  If always, never,
or meaningless, explain why.  If sometimes give examples of both
situations.

$$
\eqalign{
\pt {i} & (a\u)\cdot(b\v)=(ab)\u\cdot\v \cr 
\pt {ii} &  (a\u)\cdot(b\v)=(ab)\u\v \cr 
\pt {iii} & a(\u+\v)=a\u+a\v \cr 
\pt {iv} & a\cdot(\u+\v)=a\cdot\u+a\cdot\v \cr 
\pt {v} & \w(\u+\v)=\w\u+\w\v \cr 
\pt {vi} & \w\cdot(\u+\v)=\w\cdot\u+\w\cdot\v \cr 
\pt {vii} & \w\cdot(\u\cdot\v)=(\w\cdot\u)\cdot\v \cr 
\pt {viii} & (\u\cdot\v)\w=(\w\cdot\u)\v \cr 
\pt {ix} & \u\cdot\v=\v\cdot\u \cr}\hskip2.4in
$$

\itemitem{b)}  Verify that $(a+b)\u=a\u+b\u$.

\medskip
\item{4.}  Prove that if $\u$ is orthogonal to $\v$ and $\w$, then
$\u$ is orthogonal to $c\v+d\w$ for any scalars $c$ and $d$.  What
conclusions can you make?





\bye