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  MIDTERM 1 &\huge TQS 308  &\normalsize Fall 2009\\
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NAME:


\noindent{True/False: If the statement is \emph{always} true, give a brief explanation of why it is.  If the statement is false, give a counterexample.}


\normalsize
\begin{enumerate}
\item $[4]$  For every vector $\overrightarrow{u}$ and $\overrightarrow{v}$ in $\mathbb{R}^n$
 $$\overrightarrow{u}+ \overrightarrow{v} =\overrightarrow{v}+\overrightarrow{u}.$$
\vskip 2in
\item $[4]$ For every vector $\overrightarrow{u}$, $\overrightarrow{v}$, and $\overrightarrow{w}$ if $\overrightarrow{u}\cdot \overrightarrow{v}=\overrightarrow{u}\cdot \overrightarrow{w}$, then $\overrightarrow{v}=\overrightarrow{w}$.
\vskip 2in
\item $[4]$ For every vector $\overrightarrow{u}$, $\overrightarrow{v}$, and $\overrightarrow{w}$ $$(\overrightarrow{u}\cdot \overrightarrow{v})\cdot \overrightarrow{w}=\overrightarrow{u}\cdot (\overrightarrow{v}\cdot \overrightarrow{w}).$$
\vskip 2in

\newpage
\item $[4]$ For every vector $\overrightarrow{v}$ in $\mathbb{R}^n$ and scalar $c$, $\vert\vert c \overrightarrow{v}\vert\vert=c\vert\vert\overrightarrow{v}\vert\vert$.
\vskip 2in
\item $[4]$ $2*3+1= 1\mod 5$.
\vskip 2in
\item $[4]$ Every system of linear equations has a solution.
\vskip 2in



\newpage
\large
Free Responce: Show your work for the following problems.  The correct answer with 
no supporting work will receive NO credit.

\normalsize
\item Let the following vectors be in $\mathbb{R}^2$.
$$\overrightarrow{u}=[0,7]\hskip 1in \overrightarrow{v}=[4,-4]\hskip 1in \overrightarrow{w}=[8,-1]$$

\begin{enumerate}
\item $[3]$ Draw $\overrightarrow{u}-\overrightarrow{v}$ on the axes below.

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\item $[3]$ Find the tail if the vector $\overrightarrow{w}$ (in standard position) is translated so it's head is at the point $(-4,-8)$
\vskip .6in
\item $[6]$ Find the exact angle between $\overrightarrow{u}$ and $\overrightarrow{v}$.
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\item $[5]$ Is $\overrightarrow{u}$ a linear combination of $\overrightarrow{v}$ and $\overrightarrow{w}$, if so write $\overrightarrow{u}$ as a linear combination of $\overrightarrow{v}$ and $\overrightarrow{w}$.
\vfill
\item $[1]$ Verify your answer to (d) graphically.
\end{enumerate}




\newpage
\item Let the following vectors be in $\mathbb{R}^3$.

$$\overrightarrow{u}=[4,3,-1]\hskip 1in \overrightarrow{v}=[-1,1,0]$$

\begin{enumerate}
\item $[2]$ Write the vector form for a line that is parallel to $\overrightarrow{u}$ and passes through $(1,2,3)$.
\vskip 1in
\item $[5]$ Let $\mathcal{P}$ be the plane defined by $2x-2y+2z=5$.  Determine if $\mathcal{P}$ is parallel, perpendicular, or neither to $\overrightarrow{u}$. 
\vskip 2.5in
\item $[5]$ Let $\mathcal{L}$ be the line that passes through the point $A=(3,1,1)$ and has a direction vector $ \overrightarrow{v}$. Find the distance from the point $B=(1,0,2)$ to the line $\mathcal{L}$. 
\vskip 2in
\end{enumerate}


\newpage
\item $[8]$ Use Gaussian or Gauss-Jordan elimination to find solution(s) if they exist, to the following system of linear equations:
$\left\{ \begin{matrix}
x_1-3x_2-2x_3=0\\
-x_1+2x_2+x_3=0\\
2x_1+4x_2+6x_3=0\\
\end{matrix}\right\}$

\vskip 4in

\item $[3]$ Find the reduced row echelon (or row echelon) form for $A$ where  $A= \left[  \begin{matrix}
   				   	1 & -3 & -2 & 0 \\
      					-1 & 2 & 1 & 0\\
					2 & 4 & 6 & 0\\
   			\end{matrix}\right]$
\vskip 1in

\begin{enumerate}
\item $[2]$ What is the rank of $A$?
\end{enumerate}



\newpage
\item $[8]$ Find all vectors $\overrightarrow{u}$ and $\overrightarrow{v}$ in $\mathbb{R}^3$ that exhibit the property:
$$\vert\vert \overrightarrow{u}+\overrightarrow{v}\vert\vert=\vert\vert\overrightarrow{u}\vert\vert+\vert\vert\overrightarrow{v}\vert\vert.$$
\noindent Be sure to show (prove) you've found \emph{all} the vectors that work.




\end{enumerate}

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