\nopagenumbers
\def\pt#1{\hbox{#1) }}
\input amssym.def
\input amssym.tex
\def\R{{\Bbb R}}
\font\srm=cmr8
\magnification=\magstep1
\def\u{{\bf u}}
\def\v{{\bf v}}
\def\n{{\bf n}}
\def\B{{\bf B}}
\def\a{{\bf a}}
\def\b{{\bf b}}
\def\i{{\bf i}}
\def\j{{\bf j}}
\def\k{{\bf k}}
\def\c{{\bf c}}


\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 22\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 22} 
\bigskip


\item{1.}    Show that the point $({1\over\sqrt2},{-1\over2},{1\over2})$
lies on the unit sphere.  What is the equation of the plane tangent to
the unit sphere at this point?

\bigskip 
\item{2.}  {\bf ``The sexiest binary operation on vectors in
three-space:''} 
\item{}  Recall the {\it cross product} of two vectors
$\u=(u_1,u_2,u_3)$ and $\v=(v_1,v_2,v_3)$ in $R^3$ is the determinant
of the matrix 
$$
\left(\matrix{\i&\j&\k\cr u_1&u_2&u_3\cr v_1&v_2&v_3\cr}\right){\atop.}
$$
List the properties of the cross product.  Which of these can you justify?

\bigskip
\item{3.}    Let $\u$ be a unit vector, and $\B$ be any vector which is
not parallel to $\u$.  Describe what happens to the sequence of
vectors 
$$
\u\times\B,\quad\u\times(\u\times\B),\quad\u\times(\u\times(\u\times\B)),
\quad\ldots
$$

\bigskip
\item{4.}  How many planes contain at least three of the points
$(1,0,2)$, $(-1,3,1)$, $(4,4,4)$, and $(0,0,0)$?  Find the distance
from each point to any of the planes not containing that point.

\bigskip
\item{5.}  A fluid flows across a plane surface with uniform vector
velocity $\v$.  Let $\n$ be a unit normal to the plane surface.  How
much fluid passes through a unit area of the plane in unit time?

\bigskip
\itemitem{6.\hskip12pt a)}  Suppose that $\a\cdot\b=\a'\cdot\b$ for all
$\b$.  Show that $\a=\a'$.

\medskip
\itemitem{b)}  If $\a\times\b=\a'\times\b$ for all $\b$, is it
necessary that $\a=\a'$. 

\bigskip
\item{7.} Consider the parallelepiped spanned by three vectors
$\a$,$\b$ and $\c$ in $\R^3$.

\medskip
\itemitem{a)}  Prove that the volume of the parallelepiped is the
{\it scalar triple product}
$$
|\a\cdot(\b\times\c)|.  
$$

\itemitem{b)}  Find the volume of the parallelepiped spanned by the
vectors 
$$
(1,0,1),\qquad (1,1,1),\qquad\hbox{and}\qquad (-3,2,0).
$$




\bye