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\centerline{Mathematics 241, section 1 -- Multivariable Calculus}
\centerline{Discussion 1 -- The Cartesian Coordinate System in ${\bf R}^3$}
\centerline{September 7, 2010}
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{\it Background}
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Over the last couple of days, we introduced Cartesian coordinates in ${\bf R}^2$ and ${\bf R}^3$
 and we saw how various geometric objects in the plane or space could be described using
these coordinates.  Today, in our first small-group discussion day,
we want to practice some more using these ideas.  Each group should
write up {\it one set of solutions to these questions to hand in}.
If you are done by the end of the period, you can hand them in at
that time; if not, arrange to meet outside of class to finish up.
The write-ups are due no later than the start of class on Wednesday,
September 8.
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I.  Use set notation to give a symbolic description of each of
the subsets of ${\bf R}^2$ shown below:
\vglue 3.5truein
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II.  Describe each of the following subsets of ${\bf R}^3$ in words.  If
the set is defined by more than one equation or inequality, say what you 
get with each one separately, and how that relates to the set you are looking at.
\item{A)}  $S_1 = \{(x,y,z) \in {\bf R}^3 : x = 3 {\rm\ and\ } z = 3\}$
\item{B)}  $S_2 = \{(x,y,z) \in {\bf R}^3 : x^2 + y^2 - 6y + z^2 = 0\}$ (complete the square in $y$)
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III.  We will study the set ${\cal C} = \{(x,y,z) \in {\bf R}^3 : x^2 + y^2 - z^2 = 0\}$
by the ``method of slices'' in this question.  
(Note:  This looks similar to an example we saw in class, but the right-hand
side of the equation is {\it zero}, not {\it one}.  This makes a difference!)
\item{A)}  What are the slices of ${\cal C}$ by planes $z = c$?  How is the case $c = 0$
different from the others?
\item{B)}  What are the slices of of ${\cal C}$ by planes $x = c$?  Again, the case $c = 0$
is different, what is the slice in that case?
\item{C)}  Draw a reasonably accurate picture of ${\cal C}$.  What is the usual name
of this surface?
\item{D)}  The equation $z = 2x + 1$ defines a slanted plane in ${\bf R}^3$ passing
through the point $(0,0,1)$ on the $z$-axis.  What happens if you substitute for $z$
in the equation of ${\cal C}$ -- what kind of curve in the $xy$-plane is defined
by the equation you get?  
\item{E)}  The equation $z = y + 1$ defines another slanted plane in ${\bf R}^3$ passing
through the point $(0,0,1)$ on the $z$-axis.  What happens if you substitute for $z$
in the equation of ${\cal C}$ -- what kind of curve in the $xy$-plane is defined
by the equation you get? 
\item{F)}  Why are circles, ellipses, hyperbolas, and parabolas called {\it conic sections}? 
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IV.  In class we introduced the vector sum $v + w$ in geometrical and coordinate form.
There is also a scalar multiplication operation $c\cdot v$ which is defined like this:
If $v = (v_1,v_2)$ (respectively $(v_1,v_2,v_3)$), then
$c\cdot (v_1,v_2) = (c v_1,c v_2)$ (resp. $(cv_1,cv_2,cv_3)$).  
\item{A)}  Draw $v = (1,2)$ together with ${1\over 2}\cdot (1,2), 2\cdot (1,2), (-1)\cdot (1,2)$
in the plane.  On the basis of this, how are $c\cdot v$ and $v$ related in general?  (Say how
the magnitude and direction of $c\cdot v$ relate to the magnitude and direction of $v$.) 
\item{B)}  Now draw the vectors
$$(-1,3) + (1,2), (-1,3) + {1\over 2}\cdot (1,2), (-1,3) + 2\cdot (1,2), (-1,3) + (-1)\cdot (1,2)$$
as arrows, all with tail at the origin in ${\bf R}^2$.  
\item{C)}  If you formed the vectors $(-1,3) + t\cdot (1,2)$ for all $t \in {\bf R}$, 
what kind of geometric figure would be formed by the set of heads of these vectors?    
\bye
