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\centerline{Mathematics 241, section 1 -- Multivariable Calculus}
\centerline{Exam 1}
\centerline{October 1, 2010}
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\noindent
{\it Directions}
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Do all work in the blue exam booklet.  Do not place anything you want
considered for credit on this sheet.   {\it All answers should be given in 
exact form (e.g. as radicals, not decimal approximations), and all angles 
should be given in radian measure.} There are 100 possible points.
If you have time, you can also try for the 10 Extra Credit points.  
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\noindent
I.  In this question, $P = (0,0,1)$, $Q = (1,2,-4)$, and $R = (1,-1,0)$ in ${\bf R}^3$.
\item{A)}  (15) Find the equation of the plane containing $P,Q,R$.
\item{B)}  (10) Give a parametrization of the line segment from $R$ to $Q$
(in that direction), including the proper range of $t$-values.  
\item{C)}  (10) Which $t$-value gives the midpoint of the line segment 
from part B (the point equidistant from $R$
and $Q$) in your parametrization?
\item{D)}  (10)  Compute the angle between the vectors $\vec{PQ}$ and 
$\vec{PR}$.
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\noindent
II. Let ${\bf u},{\bf v},{\bf w}$ be vectors in ${\bf R}^3$ with tails at the origin.
\item{A)} (10)  Show that 
$${\bf w} \times ({\bf u} + {\bf v}) = {\bf w}\times {\bf u} + {\bf w} \times {\bf v}.$$ 
\item{B)} (5)  Show that if ${\bf w}$ is in the plane spanned by ${\bf u}$ and ${\bf v}$,
then $({\bf u} \times {\bf v}) \cdot {\bf w} = 0$.
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\noindent
III.  All parts of this question refer to $\alpha(t) = (\cos(3t)\cos(t),\cos(3t)\sin(t))$
called a {\it 3-leaved rose curve}:
\vglue 2.0truein
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\item{A)}  (15) Starting from $t = 0$, what is the first $t$ with
$\alpha(t) = (0,0)$?  At how many different $t$ is $\alpha(t) = (0,0)$
in the range $ 0 \le t < \pi$?
\item{B)}  (15) Find a parametrization of the tangent line to the rose curve 
at $t = \pi/3$. 
\item{C)}  (10) Give a parametrization of a circle surrounding
the ``leaf'' of $\alpha(t)$ in quadrants I and IV, {\it but not completely enclosing
either of the other leaves}.  (Any circle that does that is OK.)
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\noindent
{\bf Extra Credit}  (10) Let $\beta(t)$ be a parametric curve in 
${\bf R}^3$.  Suppose that $\beta(t)\cdot \beta(t) = 1$ (dot product)
for all $t$.  Show that $\beta'(t)$ is orthogonal
to $\beta(t)$ for all $t$.
\bye
