\documentclass [12pt] {article}

\usepackage{graphicx}
\usepackage{amssymb}
\usepackage{epstopdf}
\usepackage{amsmath}
\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}

\newenvironment{packed_itemize}{
\begin{itemize}
  \setlength{\itemsep}{1pt}
  \setlength{\parskip}{0pt}
  \setlength{\parsep}{0pt}
}{\end{itemize}}

\topmargin 0pt
\advance \topmargin by -\headheight
\advance \topmargin by -\headsep
\textheight 8.9in
\oddsidemargin 0pt
\evensidemargin \oddsidemargin
\marginparwidth 0.5in
\textwidth 6.5in

\begin{document}
\begin{center}

\Huge{Dot Products}
\end{center}
\vskip .1 in

Some Properties: If $\overrightarrow{a}$, $\overrightarrow{b}$ and $\overrightarrow{c}$ are vectors in a vectors space $V$, and $c$ is a scalar, then 
\begin{itemize}
\item $\overrightarrow{a} \cdot \overrightarrow{a}=\vert \vert \overrightarrow{a} \vert \vert ^2$
\item $\overrightarrow{a} \cdot \overrightarrow{b}=\overrightarrow{b} \cdot \overrightarrow{a}$
\item $\overrightarrow{a} \cdot (\overrightarrow{b}+\overrightarrow{c})=\overrightarrow{a} \cdot \overrightarrow{b}+\overrightarrow{a} \cdot \overrightarrow{c}$
\item $(c\overrightarrow{a}) \cdot \overrightarrow{b}=c(\overrightarrow{a} \cdot \overrightarrow{b})=\overrightarrow{a} \cdot (c \overrightarrow{b})$
\item $\overrightarrow{0} \cdot \overrightarrow{a}=\overrightarrow{0}$
\end{itemize}

While working in a group make sure you:
\begin{packed_itemize}
\item Expect to make mistakes but be sure to reflect/learn from them!
\item Are civil and are aware of your impact on others.
\item Assume and engage with the strongest argument while assuming best intent.
\end{packed_itemize}

\begin{enumerate}
\item Let $\overrightarrow{v}=2\overrightarrow{i}+2\overrightarrow{j}-\overrightarrow{k}$ and $\overrightarrow{w}=\langle 3, -2\rangle$. Find (if possible):
\begin{enumerate}
\item $(\overrightarrow{v} \cdot \overrightarrow{j})\cdot \overrightarrow{k}$
\vspace{.3in}
\item $\vert\vert \overrightarrow{v}\vert \vert$
\vspace{.3in}
\item $\overrightarrow{v} \cdot \overrightarrow{w}$
\vspace{.3in}
\end{enumerate}

\item For each $\overrightarrow{v}$ and $\overrightarrow{w}$, determine if the two vectors are parallel, perpendicular, or neither. \\
Note that ``orthogonal" and ``normal" are alternatives to the word "perpendicular".
\begin{itemize}
\item $\overrightarrow{v}=\langle 2,2,-1\rangle$ and $\overrightarrow{w}=\langle 5,-4,2\rangle$ 
\vspace{1in}

\item $\overrightarrow{v}=\langle 3,7,-\frac{1}{2}\rangle$ and $\overrightarrow{w}=\langle -1,-\frac{7}{3},\frac{1}{6}\rangle$ 
\vspace{1.5in}
\end{itemize}


\newpage

\begin{center}
\Huge{Cross Products}
\end{center}
\vskip .1 in

\item Find the following
\begin{itemize}
\item $\overrightarrow{j} \times \overrightarrow{i}$
\vspace{1.5in}

\item $\overrightarrow{3i} \times \overrightarrow{j}$
\vspace{1.5in}
\end{itemize}


\item Find a unit vector orthogonal to both $\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{k}$ and $2\overrightarrow{i}+\overrightarrow{k}$.
\vfill

Some nice properties of cross products: If $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are vectors and $c$ is a scalar, then   
\begin{itemize}
\item $\overrightarrow{a}\times \overrightarrow{b}=-\overrightarrow{b}\times \overrightarrow{a}$
\item $(c\overrightarrow{a})\times \overrightarrow{b}=c(\overrightarrow{a}\times \overrightarrow{b})=\overrightarrow{a}\times (c\overrightarrow{b})$
\item $\overrightarrow{a}\times (\overrightarrow{b}+\overrightarrow{c})=\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{c}$
\item $(\overrightarrow{a} + \overrightarrow{b})\times \overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{c}+\overrightarrow{b}\times \overrightarrow{c}$
\item  $\overrightarrow{a} \cdot ( \overrightarrow{b}\times \overrightarrow{c})=(\overrightarrow{a}\times \overrightarrow{b})\cdot \overrightarrow{c}$
\item $\overrightarrow{a} \times ( \overrightarrow{b}\times \overrightarrow{c})=(\overrightarrow{a}\cdot \overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}\cdot \overrightarrow{b})\overrightarrow{c}$
\end{itemize}


\end{enumerate}





\end{document}
