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\newcommand{\ra}{\rangle}

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\begin{document}

\pagestyle{empty}

\noindent
\parbox{2in}{\bf Math 2130}
\hfill {\Large \bf Test \#1} \hfill
\parbox{2in}{\bf \hfill July $11^\mathrm{th}$, 2016}

\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{0.05in}

\noindent {\bf \large 1. (20 points)} Vector Basics: Let $\B v = \la 3,-1,2 \ra$ and $\B w =\la 2,1,-1 \ra$.
  
\begin{enumerate}[(a)]
  \item Find a {\bf unit} vector perpendicular to both $\B v$ and $\B w$.

\vfill

   \item Compute $\mathrm{proj}_{\B v}(\B w)$
   
\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 Fill in the blanks ($\B a$, $\B b$, and $\B c$ are vectors)\dots

\begin{enumerate}[(i)]
\item  $|\B a \times \B b|$ computes the \underline{\hspace*{1.5in}} of the \underline{\hspace*{1.75in}} spanned by $\B a$ and $\B b$.

\vspace{0.1in}

\item $|\B a \bdot (\B b \times \B c)|$ computes the \underline{\hspace*{1.5in}} of the \underline{\hspace*{1.75in}} spanned by $\B a$, $\B b$, and $\B c$.
\end{enumerate}

\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+1.8*13/15,3.6) circle (0.1);
\node[scale=1] (ptP) at (-2+1.8*13/15 +0.1,3.6 -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 $\B r_1(t) = \la 2t+1, -t+3, t \ra$ and let $\ell_2$ be the line which passes through the points $P=(2,1,-1)$ and $Q=(-2,3,-3)$. 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. (13 points)} Plane old geometry.

\begin{enumerate}[(a)]
\item Find a (scalar) equation for the plane that containing the line $\B r(t) = \la 1+2t, -1-t, -2t \ra$ and the point $P=(1,0,-2)$.

\vfill

\item Consider the two planes: $x+2y+2z+4=0$ and $2x-y-2z+5=0$. If these are parallel planes explain why they are parallel. If these planes intersect, find the angle between these planes (don't worry about evaluating inverse trig. functions).

\vfill

\end{enumerate}

\noindent {\bf \large 4. (8 points)} Find the area of the triangle with vertices $A=(1,0,2)$, $B=(2,2,5)$, and $C=(1,3,3)$.

\vfill

\newpage

\noindent {\bf \large 5. (8 points)} Compute the curvature of $\B r(t) = \la t^2, \sin(t), e^{2t} \ra$

\vfill

\noindent {\bf \large 6. (13 points)} Consider the circle $C$: $(x-a)^2+(y-b)^2=R^2$ (where $R>0$).

\begin{enumerate}[(a)]

\item Parameterize $C$ and compute its arc length using your parameterization.

\vfill

\item Compute the curvature of $C$ using your parameterization.

\vfill


\end{enumerate}

\newpage

\noindent {\bf \large 7. (16 points)} Consider the curve parameterized by $\B r(t) = \la 3\sin(t), 4t, -3\cos(t) \ra$ where $0 \leq t \leq 2\pi$. 

\begin{enumerate}[(a)]
\item Compute the centroid of this curve.

\vfill

\item Find the TNB-frame for $\B r(t)$. 

\vfill
\vfill

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

\end{enumerate}

\vspace{0.5in}


\noindent {\bf \large 8. (12 points)}  Choose {\bf ONE} of the following: [In both cases, drawing a good explanatory picture will earn you some partial credit -- but for full credit you need more.]
  \begin{enumerate}[I.]
   \item Let $\B a$ and $\B b$ be vectors. Simplify $(2\B a-\B b) \bdot (\B a \times \B b)$. Explain your answer geometrically.
   \item Let $C$ be the curve in the plane defined by $y=f(x)$ (where $f$ is a function with at least two continuous derivatives). Use the special formula for curvature to explain why $\kappa(x)=0$ exactly when $C$ is a line (or part of a line).
   \end{enumerate}

\vfill


\end{document}