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\begin{document}
 
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\title[Calculus II]
{Exam 3\\Math 2153 section 702, Spring 2005}
 
\author[Weiping Li]{Instructor: Weiping Li}
 
\maketitle

{\bf Print Name and Student $\#$}
\vspace{4ex}
\medskip

{\bf SHOW WORK FOR CREDIT !!! SHOW WORK FOR CREDIT !!!}
\medskip

\begin{enumerate}
\item (15pts) 
(a) Find a vector orthogonal to both ${\bf i}-{\bf j}$ and ${\bf j} +
2{\bf k}$.
\vspace{1in}

(b) Find the volume of the parallelpipe determined by vectors
$(1, -1, 0)$, $(-1, 1, 1)$ and $(0, 1, -1)$.
\vspace{1in}

(c) Find the line equation which passes $(1, 0, 6)$ and is
perpendicular to the plane $x+3y+z=5$.
\vspace{1in}

\item (10pts) Find the angle between these two planes 
$L_1: x-y+2z=1$ and $L_2: z=x-2y+1$.

\newpage

\item (10pts) (a) Find an equation of the plane through the point
$(0, 1, 1)$ and parallel to $2x-y+3z=1$.
\vspace{1.5in}

(b) Find the distance from the point $(1, -1, -1)$ to $x-y+z=2$.
\vspace{1.5in}

\item (15pts) (a) Find the domain of the function
$$f(x, y) = \frac{\sqrt{y-x^2}}{1-x^2},$$
\vspace{1in}

(b) and sketch the domain of the above function $f(x, y)$.
\vspace{1in}
\newpage

\item (10ts) (a) Write the equation $x^2+y^2-z^2=2z$ (i) in cylindrical coordinates and
(ii) in spherical coordinates.
\vspace{1.5in}

(b) Change the spherical coordinates $(2, \pi/2, \pi/3)$ to the rectangular
coordinates.
\vspace{1in}

\item (10pts) Find ${\bf r}(t)$ if ${\bf r}^{'}(t) = \sin t {\bf i} + \cos t 
{\bf j} - 3t^2 {\bf k}$ and ${\bf r}(0)= - {\bf i} + {\bf j} + {\bf k}$.
\vspace{2in}

\item (10pts) Find the length of arc of ${\bf r}(t) = \sin t {\bf i} + 
\sqrt{3} t {\bf j} - \cos t {\bf k}$ from $t=0$ to $t=2\pi$.
\newpage

\item (20pts) (a) Find the unit tangent vector ${\bf T}(t)$ and the unit
normal vector ${\bf N}(t)$ for the curve ${\bf r}(t) = (2 \sin t, 3t, 2\cos t)$.
\vspace{2.5in}

(b) Find the curvature of the curve.
\vspace{1.5in}

\item {\bf Bonus questions}: (10pts) Show that $\frac{d {\bf B}}{ds}$ is
perpendicular to ${\bf B}$ and ${\bf T}$.
\vspace{2in}

(5pts)  Prove that
$({\bf a}-{\bf b})\times ({\bf a} + {\bf b}) = 2 {\bf a} \times {\bf b}$.
\end{enumerate} 
\end{document}