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\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Homework 1}
\date{due February 7, 2003}
\maketitle

\noindent
Normal questions and advanced questions (A) are worth $5$ points; easy
questions (E) are worth $4$ points.  Complete either the easy, or the
advanced version, not both.

\begin{enumerate}
\item Find an equation for the plane that contains the point $(2,4,6)$
and the line 
\[
  \x =
  \begin{pmatrix}
  7 \\ 3 \\ 5
  \end{pmatrix}
  + \lambda \,
  \begin{pmatrix}
  -3 \\ 4 \\ 2
  \end{pmatrix} \,.
\]

\item
\begin{itemize}
\item[(E)] Find the angle between the vectors $(3,-4,0)$ and
$(-2,1,0)$, and find a vector that is perpendicular to both.
\item[(A)] Prove, by writing out in component form or by following the 
suggestion in Edwards \& Penney, p.\ 733), that
\[
  (\a \times \b) \times \c = (\a \cdot \c) \, \b - (\b \cdot \c) \, \a 
  \,.
\]
\end{itemize}

\item
\begin{itemize}
\item[(E)] Find the distance between the point $(5,12,-13)$ and
the plane with equation $3x+4y+5z=12$.
\item[(A)] Show that the distance of the point $(x_0,y_0,z_0)$ to the
plane $ax+by+cz=d$ is given by
\[
  D=\frac{|ax_0+by_0+cz_0-d|}{\sqrt{a^2+b^2+c^2}} \,.
\]
\end{itemize}

\item Let $z=3+4i$ and $w=-5$.  Sketch the following quantities in the
complex plane: $z^*$, $z+w$, $z-w$, $zw$, $z/w$.  (This is sometimes
called an Argand diagram plot.)

\item Simplify the following expressions:
\begin{enumerate}
\item $\displaystyle \Re \frac{1+i}{1-i}$
\item $\displaystyle \Im (\exp 2iz)$
\item $\displaystyle \ln i$
\end{enumerate}


\item
\begin{itemize}
\item[(E)] Find the (complex) roots of the equation $z^2 - 2z + 2=0$. 
\item[(A)] Find, in terms of $x$ and $y$, those points $z=x+iy$ that
satisfy the equation
\[
  \frac{\Im z^2}{z^2} = -i  \,.
\]
\end{itemize}

\end{enumerate}



\end{document}
