\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 Feb. $19^\mathrm{th}$, 2021}

\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
$\displaystyle m = \int_C \delta\,ds \qquad \qquad \left(\bar{x},\bar{y},\bar{z}\right) = \frac{1}{m}\left( \int_C x\delta\,ds, \int_C y\delta\,ds, \int_C z\delta\,ds \right)$ \hfill 
$\tau = \dfrac{(\B r'(t) \times \B r''(t)) \bdot \B r'''(t)}{|\B r'(t) \times \B r''(t)|^2}$ \hfill
$\displaystyle{\kappa = \frac{|f''(x)|}{\left(1+(f'(x))^2\large\strut\right)^{\textstyle\frac32}}}$

\vspace{0.05in}

\noindent {\bf \large 1. (22 points)} Vector Basics: Let $\B u = \la 2,-2,1 \ra$, $\B v =\la -1,3,1 \ra$, and $\B w = \la 1,1,0 \ra$.
  
\begin{enumerate}[(a)]
   \item Find two unit vectors that are perpendicular to both $\B u$ and $\B v$.
   
\vfill

  \item Find the volume of the parallelepiped spanned by $\B u$, $\B v$, and $\B w$.

\vfill

   \item Find the angle between $\B u$ and $\B v$ (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 Match the statement on the left to the corresponding statement on the right\dots

\noindent \hspace{-0.15in}
   \begin{tabular}{lcl}
  \fbox{\varspace{0.2in}{0.2in}} $\B a \bdot \B b = 0$  & \myspace & {\bf\large A)}  $\B a$ and $\B b$ are parallel \\
  \fbox{\varspace{0.2in}{0.2in}} $\B a \times \B b = \B 0$   & \myspace &  {\bf\large B)}  $\B a$ and $\B b$ are orthogonal\\
  \fbox{\varspace{0.2in}{0.2in}} $(\B a \bdot \B b) \times (\B a \bdot \B b) = \B 0$ & \myspace & {\bf\large C)} is always true\\
  \fbox{\varspace{0.2in}{0.2in}} $(\B a \times \B b) \bdot \B b= 0$  & \myspace & {\bf\large D)}   is nonsense
  \end{tabular}
  
\vspace{0.05in}

   \item The vectors $\B a$ and $\B b$ are shown to the right. 
   
            They are based at the point $X$. Sketch the 
            
            vector $-\B a + \B b$ based at the point $P$ and sketch 
            
            the vector $2\B a - \B b$ 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. (8 points)} Let $\ell_1$ be parametrized by $\B r_1(t) = \la t, -t+1, 3t+2 \ra$ and let $\ell_2$ be the line which passes through the points $P=(-1,2,-1)$ and $Q=(2,1,0)$. Determine if $\ell_1$ and $\ell_2$ are\dots  (circle the correct answer)

\vspace{0.05in}

\force \hfill {\Large the same, \quad parallel} (but not the same){\Large, \quad intersecting, \quad } or {\Large \quad skew}. \hfill \force

\vspace{1in}

\vfill

\noindent {\bf \large 3. (12 points)} Plane old geometry.

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

\vfill

\item Find the area of the triangle $\Delta ABC$ where $A$, $B$, and $C$ are the same points as in part (a).

\vspace{1in}

\end{enumerate}

\noindent {\bf \large 4. (10 points)} A strange object is observed to have velocity function $\B v(t) = 3t^2\B i-6\B j+e^t\B k$. In addition, this object's initial position was known to be $\B r_0 = 5\B i + 100\B k$. \qquad {}[For what it's worth\dots{}measurements are made in meters and seconds.]


\vspace{0.15in}

\begin{enumerate}[(a)]

\item This object's acceleration function is $\B a(t)=$ \quad \underline{\hspace*{2in}}

\item Find this object's position function $\B r(t)$.

\vfill

Its initial speed was \underline{\hspace{0.5in}} meters per second.

\end{enumerate}

\newpage

\noindent {\bf \large 5. (10 points)} Parameterize and set up an integral that computes the arc length of the ellipse $\dfrac{(x-1)^2}{4}+\dfrac{(y+2)^2}{9}=1$.

\vfill

\noindent {\bf \large 6. (16 points)} Let $C$ be parameterized by $\B r(t) = \langle \sqrt{2} \sin(t), 2\cos(t), \sqrt{2}\sin(t) \rangle$.

\begin{enumerate}[(a)]
\item Find a parameterization, $\B \ell(t)$, for the line tangent to $C$ at $t=0$.

\vfill

\item Find the TNB-frame for $C$.

\vfill

\vfill

\item Compute the curvature of $C$.

\vspace{0.5in}

\end{enumerate}


\newpage

\noindent {\bf \large 7. (22 points)} Consider the curve $C$ parameterized by $\B r(t) = \la t^2, 3t, e^t \ra$, $-3 \leq t \leq 10$. 

\begin{enumerate}[(a)]
\item Compute the curvature of $\B r(t)$.

\vfill

\item Compute the torsion of $\B r(t)$.

\vfill

\item Set up the integral $\displaystyle\int_C (z+x\sin(y))\,ds$  \quad {}[Obviously we cannot hope to evaluate this by hand -- please don't try.]

\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}