\documentclass[12pt]{article}
\usepackage{fullpage,amsmath,amsfonts}

\usepackage[homework]{quizmachine}

\newcommand{\x}{\boldsymbol{x}}
\renewcommand{\a}{\boldsymbol{a}}
\renewcommand{\b}{\boldsymbol{b}}
\renewcommand{\c}{\boldsymbol{c}}
\newcommand{\p}{\boldsymbol{p}}
\newcommand{\w}{\boldsymbol{w}}
\renewcommand{\v}{\boldsymbol{v}}
\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}
\renewcommand{\i}{\mathrm{i}}


\begin{document}

\title{Engineering and Science Mathematics 2B}
\author{Homework 1}
\date{due February 13, 2013, before 12:00}

\quizleft{ESM 2B}
\quizcenter{Quiz 1}
\quizright{February 21, 2013}

\hwpreamble{\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.}

\defproblem{Show that $\a \cdot (\b \times \c) = \b \cdot (\c \times \a)$.}

\defproblem{
\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}}

\defproblem{
\begin{itemize}
\item[(E)] Find the distance between the point $\p=(1,2,3)$ and the
line 
\[
  \x =
  \begin{pmatrix}
  -1 \\ 1 \\ 6
  \end{pmatrix}
  + \lambda \,
  \begin{pmatrix}
  1 \\ -1 \\ 0
  \end{pmatrix} \,.
\]

\item[(A)] Show that, as an alternative to the formula given in class,
the distance between a point $\p$ and the line $\x=\a + \lambda \v$ is
given by
\[
  d = | \w - \w \cdot \hat \v \, \hat \v| \,,
\]
where $\w = \a-\p$, and $\hat \v$ is the unit vector in the direction
of $\v$.
\end{itemize}}

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

\defproblem{
\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 that
\[
  (\a \times \b) \times \c = (\a \cdot \c) \, \b - (\b \cdot \c) \, \a 
  \,.
\]
\emph{Hint:} You can write out everything in components.
Alternatively, write $\a$, $\b$, and $\c$ in terms of an orthonormal
basis (i.e., three perpendicular basis vectors of unit length) where
the first basis vector is the unit vector in the direction of $\a$ and
the second lies in the plane spanned by $\a$ and $\b$.
\end{itemize}}



\defproblem{Let $z=3+4\i$ 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.)}

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


\quizzes{100}





\end{document}
