\documentclass{article}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{enumerate}
\usepackage{ifthen}
\usepackage{ulem} % for strikethrough \sout{...} command
\usepackage{color}
\setlength{\oddsidemargin}{-0.5in}
\setlength{\evensidemargin}{-0.5in}
\setlength{\topmargin}{-0.5in}
\setlength{\headheight}{0.0in}
\setlength{\headsep}{0.0in}
\setlength{\topskip}{0.0in}
\setlength{\textheight}{10in}
\setlength{\textwidth}{7.5in}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Adding Tikz %%%
\usepackage{tikz}
\usetikzlibrary{arrows, positioning, calc, intersections, decorations.pathreplacing}

\newcommand\tikzmark[1]{%
  \tikz[overlay,remember picture,baseline] \coordinate [anchor=base] (#1);}

\newcommand\DrawBrace[3]{%
  \draw [ultra thick, decorate,decoration={brace,amplitude=5pt,mirror,raise=2pt}]
    (#1) -- (#2) node [midway,xshift=23pt] {\Huge $\displaystyle #3$};
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%\newcommand{\bdot}{\, {\bf \cdot} \,}
\newcommand{\bdot}{\, {\scriptstyle{\bullet}} \,}
\newcommand{\B}[1]{{\bf {#1}}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}

\newcommand{\force}{{$\mbox{\;}$}}

\newcommand{\myspace}{\mbox{\parbox[c][0.25in]{0.25in}{\hfill}}}  % height then width
\newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\hfill}}}

\begin{document}

\pagestyle{empty}

\noindent
\parbox{2in}{\bf Math 2130-101}
\hfill {\Large \bf Test \#1} \hfill
\parbox{2in}{\bf \hfill June $5^\mathrm{th}$, 2019}

\vspace{0.15in}

\noindent {\large\bf Name:} \underline{\hskip 3.0 truein}
\hfill {\bf Be sure to show your work!}

\vspace{0.1in}

\noindent
$\displaystyle{\mbox{proj}_\B w(\B v) = \frac{\B v \bdot \B w}{\,|\B w|^2}\B w }$ \hfill 
$\displaystyle{\mathbf{r}''(t) = \left(\frac{\mathbf{r}'(t) \bdot \mathbf{r}''(t)}{|\mathbf{r}'(t)|} \right) \mathbf{T}(t) + 
                            \left(\frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|} \right) \mathbf{N}(t)}$ \hfill
$\displaystyle{\kappa = \left|\frac{d\mathbf{T}}{ds}\right| = \frac{|\mathbf{T}'(t)|}{|\mathbf{r}'(t)|} = \frac{|\mathbf{r}'(t) \times \mathbf{r}''(t)|}{|\mathbf{r}'(t)|^3}}$

\vspace{0.1in}

\noindent
$\displaystyle m = \int_C \delta(x,y,z)\,ds \qquad \left(\bar{x},\bar{y},\bar{z}\right) = \frac{1}{m}\left( \int_C x\delta(x,y,z)\,ds, \int_C y\delta(x,y,z)\,ds, \int_C z\delta(x,y,z)\,ds \right)$ \hfill \hfill
$\displaystyle \tau = \dfrac{(\B r'(t) \times \B r''(t)) \bdot \B r'''(t)}{|\B r'(t) \times \B r''(t)|^2}$

\vspace{0.1in}

\noindent {\bf \large 1. (24 points)} Vector Basics: Let $\B v = \la -2,0,1 \ra$ and $\B w = \la 1,2,-2 \ra$.
  
\begin{enumerate}[(a)]
  \item Compute $\mathrm{proj}_{\B w}(\B v)$.

\vfill

   \item Find two {\it unit} vectors which are {\it orthogonal} (i.e., perpendicular) to both $\B v$ and $\B w$.
   
\vfill

\vfill

   \item Find the angle between $\B v$ and $\B w$ (don't worry about evaluating inverse trig. functions). 

\vspace{0.75in}
      
      Is this angle... {\bf \quad right, \quad acute, \quad} or {\bf \;\; obtuse} \;\;? (Circle your answer.)

\vspace{0.05in}

   \item Let $\B a$, $\B b$, and $\B c$ be vectors in $\mathbb{R}^3$. Fill in the blanks:

\vspace{0.15in}

If $\B a \times \B b = \B 0$, then $\B a$ and $\B b$ are \underline{\hspace*{2in}}.

\vspace{0.15in}

If $\B a \bdot \B b = 0$, then $\B a$ and $\B b$ are \underline{\hspace*{2in}}.

\vspace{0.15in}

If $(\B a \times \B b) \bdot \B c = 0$, then $\B a$, $\B b$, and $\B c$ are \underline{\hspace*{2in}}.

\vspace{0.1in}

   \item The vectors $\B a$ and $\B b$ are shown to the right. 
   
            They are based at the point $X$. Sketch the 
            
            vector $2\B a - \B b$ based at the point $P$ and sketch 
            
            the vector $-\B a + \B b$ based at the point $Q$.
   
\vspace{-0.85in}

\force \hfill
\begin{tikzpicture}[scale=0.9]

\colorlet{grayLines}{gray!50}

% parallel to b
\foreach \i in {1,...,7}
{
    \draw[ultra thick,color=grayLines] (-6,0.9*\i)--(6,0.9*\i);
}

% parallel to a
\draw[ultra thick,color=grayLines] (-6,5)--(-4.7,6.5);   % 3.4 horizontal step to next diagonal, slope of diagonals = 15/13
\draw[ultra thick,color=grayLines] (-6,6.5-15/13*4.7)--(-1.3,6.5);
\draw[ultra thick,color=grayLines] (2.1-6.1*13/15,0.4)--(2.1,6.5);
\draw[ultra thick,color=grayLines] (5.5-6.1*13/15,0.4)--(5.5,6.5);
\draw[ultra thick,color=grayLines] (8.9-6.1*13/15,0.4)--(6,3.153846154);

% vector a
\draw[->,ultra thick] (-5.4,1.8)--(-5.4+13/15*0.9,2.7);
\node[scale=1.5] (vecA) at (-5.4,2.3) {{$\B a$}};

% vector b
\draw[->,ultra thick] (-5.4,1.8)--(-2,1.8);
\node[scale=1.5] (vecB) at (-3.9,1.4) {{$\B b$}};

% point X (in red)
\filldraw[color=red] (-5.4,1.8) circle (0.1);
\node[scale=1.25] (ptX) at (-5.3,1.45) {{\color{red}$X$}};

% over by 0.9*13/15 and up by 0.9

% point P
\filldraw (-2+0.9*13/15,2.7) circle (0.1); 
\node[scale=1] (ptP) at (-2+0.9*13/15 +0.1,2.7 -0.35) {$P$};

% point Q
\filldraw (-2+2.7*13/15,4.5) circle (0.1);
\node[scale=1] (ptP) at (-2+2.7*13/15 +0.1,4.5 -0.35) {$Q$};

\end{tikzpicture}

\end{enumerate}

\newpage

\noindent {\bf \large 2. (14 points)} Give vector valued functions which parameterize the following curves.\\ \force \hfill {\bf Don't forget} to specify a domain for your parameter.

\begin{enumerate}[(a)]

\item The line segment from $A = (1,2,0)$ to $B = (4,0,1)$.

\vspace{0.6in}

\item The circle $(x+1)^2+(y-3)^2=25$.

\vspace{0.6in}

\end{enumerate}

\noindent {\bf \large 3. (14 points)} Lines and Planes

\begin{enumerate}[(a)]
\item Find an equation for the plane which passes through the points $A=(1,0,-1)$, $B=(2,3,0)$, and $C=(3,-1,1)$.

\vfill

\vfill

\item Consider the line parameterized by $\B r(t) = \la 2t,-4t+1,6t-2\ra$  and the plane $-x+2y-3z=10$. 

 The line and plane are (circle {\bf all} that apply)\dots \hfill {\Large parallel \qquad perpendicular \qquad intersecting}.

\vfill

\end{enumerate}

\noindent {\bf \large 4. (12 points)} Suppose that a particle's velocity is given by $\B v(t) = t^2\B i + 5e^t\B j$. In addition, we have that this particle's initial position is $\B r_0 = \B i +2\B j$. \hfill [For what it's worth, measurements are made in meters and seconds.]

\begin{enumerate}[(a)]

\item The particle's initial speed is \underline{\hspace{0.5in}} meters per second.

\vspace{0.2in}

\item Find the particle's acceleration $\B a(t)$.

\vspace{0.5in}

\item Find the particle's position function $\B r(t)$. 


\vfill

\end{enumerate}

\newpage

\noindent {\bf \large 5. (18 points)} Let $C$ be the curve parameterized by $\B r(t) = \la 3t, 4\sin(t), 4\cos(t) \ra$, $-\pi \leq t \leq 3\pi$.

\begin{enumerate}[(a)]

\item Compute the $\B T \B N \B B$-frame of $C$.

\vfill

\vfill

\vfill

\item Compute the curvature of $C$.

\vfill

\item Compute the arc length of $C$.

\vfill

\item Set up (but {\bf do not evaluate}) the line integral $\int_C x\,e^{y^2+z^2}\,ds$.

\vfill
\end{enumerate}

\newpage

\noindent {\bf \large 6. (18 points)} Consider the curve $C$ parameterized by $\B r(t) = \la t, t^2, t^3 \ra$. 

\begin{enumerate}[(a)]
\item Find a parameterization for the line tangent to $C$ at $t=2$.

\vfill

\item Compute the curvature of $C$. 

\vfill

\item Compute the torsion of $C$.

\vfill

\item Compute the tangential and normal components of acceleration of $\B r(t)$.

\vfill

\item Does this curve lie in a plane? Why or why not?

\vspace{0.5in}

\end{enumerate}


\end{document}