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\usepackage{enumerate}
\usepackage{ifthen}
\usepackage{ulem} % for strikethrough \sout{...} command
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\usetikzlibrary{arrows, positioning, calc, intersections, decorations.pathreplacing}

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

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    (#1) -- (#2) node [midway,xshift=23pt] {\Huge $\displaystyle #3$};
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%\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 February $14^\mathrm{th}$, 2018}

\vspace{0.15in}

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

\vspace{0.1in}

\noindent
$\displaystyle{\mbox{proj}_\B v(\B u) = \frac{\B u \bdot \B v}{\,|\B v|^2}\B v }$ \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
$\displaystyle m = \int_C \rho\,ds \qquad \qquad \left(\bar{x},\bar{y},\bar{z}\right) = \frac{1}{m}\left( \int_C x\rho\,ds, \int_C y\rho\,ds, \int_C z\rho\,ds \right)$ \hfill \hfill
$\displaystyle{\kappa = \frac{|f''(x)|}{\left(1+(f'(x))^2\large\strut\right)^{\textstyle\frac32}}}$

\vspace{-0.1in}

\noindent {\bf \large 1. (24 points)} Vector Basics: Let $\B u = \la -1,-1,1 \ra$, $\B v =\la 1,0,3 \ra$, and $\B w = \la 2,-1,2 \ra$.
  
\begin{enumerate}[(a)]
   \item Compute the area of the parallelogram spanned by $\B v$ and $\B w$.

\vfill

\vfill

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

\vfill

   \item Find two unit vectors which are parallel to $\B w$. 
   
\vfill

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

\vfill
      
      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}} $(\B a \times \B b) \bdot \B b= 0$  & \myspace & {\bf\large A)}  cannot ever be true / is nonsense \\
  \fbox{\varspace{0.2in}{0.2in}} $\B a \bdot \B b = \B a \times \B b$   & \myspace &  {\bf\large B)} is always true\\
  \fbox{\varspace{0.2in}{0.2in}} $\B a \times \B b = \B 0$ & \myspace & {\bf\large C)} $\B a$ and $\B b$ are orthogonal\\
  \fbox{\varspace{0.2in}{0.2in}} $\B a \bdot \B b = 0$  & \myspace & {\bf\large D)}   $\B a$ and $\B b$ are parallel
  \end{tabular}
 
%\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 $\B a + \B b$ 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}$X$}};

% 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. (10 points)} Let $\ell_1$ be parametrized by $\B r_1(t) = \la 1,1,3 \ra + \la 2,1,-1 \ra t$ and let $\ell_2$ be the line which passes through the points $P=(2,0,2)$ and $Q=(3,4,2)$. 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

\vfill

\vfill

\noindent {\bf \large 3. (14 points)} Plane old geometry.

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

\vfill

%\item Find the area of the triangle with vertices $A$, $B$, and $C$ (from part (a)).

%\vfill

\item The planes: $x+2y+3z=4$ and $5x-y-z=1$ are\dots [Circle {\bf ALL} that apply.]

\vspace{0.05in}

\force \hfill {\Large parallel \qquad perpendicular \qquad intersecting \qquad the same} \hfill \force

\vspace{1.25in}

\end{enumerate}


%\noindent {\bf \large 4. (8 points)} Is the curvature of $y=x^3+3x^2+1$ ever zero? \qquad {\Large Yes \quad / \quad No}

%\vspace{1.25in}


\newpage

\noindent {\bf \large {\color{red}\sout{5}\;}4. (14 points)} Parameterization, arc length, and a line integral.

Let $C$ be the {\bf upper-half} of the circle $x^2+y^2=4$. Parameterize $C$, compute its arc length (with an integral -- not just using geometry), and then set up $\displaystyle \int_C y\,e^{x^2+y^2}\,ds$.

\vfill

\vfill

\vfill


\noindent {\bf \large {\color{red}\sout{6}\;}5. (13 points)} Let $C$ be parameterized by $\B r(t) = \la t^2, -t^2, t^3 \ra$.

\begin{enumerate}[(a)]

\item Find the curvature of $\B r(t)$.

\vfill

\vfill

\item Find the tangential and normal components of acceleration for $\B r(t)$ (i.e. $a_{\bf T}$ and $a_{\bf N}$).

\vfill

\end{enumerate}

\newpage

\noindent {\bf \large {\color{red}\sout{7}\;}6. (13 points)} Find the TNB-frame for the curve $C$ parameterized by $\B r(t) = \la 3\sin(t), 5\cos(t), 4\sin(t) \ra$. 

\vfill

\vfill

Find the curvature of $C$

\vspace{1in}

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

\vspace{0.5in}

\noindent {\bf \large {\color{red}\sout{8}\;}7. (12 points)} No numbers here. Choose {\bf ONE} of the following:
  \begin{enumerate}[I.]
   \item Suppose that $\B v$ and $\B w$ have the same length. Show that $\B v+\B w$ and $\B v - \B w$ are orthogonal. Draw a picture to go with your proof.
   \item Given vectors $\B a$ and $\B b$, is it always true that $|\B a \times \B b| = |\B a| \, |\B b|$? Either explain why this always holds or explain what would make it hold or fail.
   \end{enumerate}

\vfill


\end{document}