\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\def\R{{\Bbb R}}
\font\srm=cmr8
\magnification=\magstep1
\def\A{{\bf A}}
\def\B{{\bf B}}
\def\C{{\bf C}}
\def\U{{\bf U}}
\def\V{{\bf V}}
\def\F{{\bf F}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\D{{\bf D}}
\def\0{{\bf 0}}

\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 24\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 24} 
\bigskip


\item{1.}  Find the line in the plane of $(0,0,0)$, $(2,2,0)$, and
$(0,1,-2)$ that passes through the origin perpendicular to the line
$$
{x\over3}={y\over2}={2z}.
$$

\item{2.}  Two surface ships on manuevers are trying to determine a
submarine's course and speed to prepare for an aircraft intercept.
Ship $A$ is located at $(4,0,0)$ while ship $B$ is located at
$(0,5,0)$.  All coordinates are given in thousands of feet.  Ship $A$
locates the submarine in the direction of the vector
$2\i+3\j-(1/3)\k$, and ship $B$ locates it in the direction of the
vector $18\i-6\j-\k$.  Four minutes ago, the ship was located at
$(2,-1,-1/3)$.  The aircraft is due in 20 minutes.  Assuming the
submarine moves in a straight line at a constant speed, to what
position should the surface ships direct the aircraft?

\medskip
\item{3.}  Two helicopters, $H_1$ and $H_2$, are traveling together.
At time $t=0$ hours, they separate and follow different straight line
paths given by
$$
\eqalign{
H_1:\quad x&=6+40t,\quad y=-3+10t,\quad z=-3+2t\cr
H_2:\quad x&=6+110t,\quad y=-3+4t,\quad z=-3+t\cr}
$$
all coordinates measured in miles.  Due to system malfunctions, $H_2$
stops its flight at $(446,13,1)$ and, in a negligible amount of time,
lands at $(446,13,0)$.  Two hours later, $H_1$ is advised of this fact
and heads toward $H_2$ at 150 mph.  How long will it take $H_1$ to
reach $H_2$?

\medskip
\item{4.}  Prove that four points $A$, $B$, $C$, and $D$ are coplanar
(lie in a common plane) if and only if
$\vec{AD}\cdot(\vec{AB}\times\vec{BC})=0$. 

\medskip
\item{5.} If the four vectors $\A$, $\B$, $\C$, and $\D$ are coplanar,
show that $(\A\times\B)\times(\C\times\D)=\0.$

\medskip
\itemitem{6.\hskip12pt a)}  Let $\F\colon\R\to\R^3$.  
Describe the domain and range of this function.  What uses might such
a function have?

\medskip
\itemitem{b)}  If $\F(t)=(x(t),y(t),z(t))$, then the derivative
$d\F/dt$ is defined by 
$$
{d\F\over dt}=\left({dx\over dt},{dy\over dt},{dz\over dt}\right).
$$
Find $d\F/ds$ as a function of $s$ if $t=s^2-1$ and
$\F(t)=(1,\sin(t+1),\sqrt t)$. 

\medskip
\itemitem{c)}  Prove the following differentiation rules for two
vector functions $\U$ and $\V$, 
$$
\eqalign{
{d\over dt}(\U\cdot\V)&={d\U\over dt}\cdot\V+U\cdot{d\V\over dt}\cr\cr
{d\over dt}(\U\times\V)&={d\U\over dt}\times\V+U\times{d\V\over dt}.\cr}
$$







\bye