\thispagestyle{fancy}
\fancyhead[C]{G1, G4}
\fancyhead[R]{Week One}
\renewcommand{\wknum}{1}

\iftoggle{questions}{
	\begin{center}{\Large \textbf{Math 2550 Worksheet - Week \wknum}}\\
	\end{center}
	
	\textbf{\large Learning Targets}
	\begin{itemize}
		\item \textbf{G1: Lines and Planes.} I can describe lines using the 
		vector equation of a line. I can describe planes using the general 
		equation of a plane. I can find the equations of planes using a point 
		and a normal vector. I can find the intersections of lines and planes.  
		I can describe the relationships of lines and planes to each other. I 
		can solve problems with lines and planes.
		
		\item \textbf{G4: Surfaces.} I can identify standard quadric surfaces 
		including: spheres, ellipsoids, elliptic paraboloids, hyperboloids, 
		cones, and hyperbolic paraboloids. I can match graphs of functions of 
		two variables to their equations and contour plots and determine their 
		domains and ranges. 
	\end{itemize}
	
	\textbf{\large Warmup}
	\begin{enumerate}
		\item If the statement is \textit{always} true, answer true.  If the 
		statement is \textit{ever} false, answer false.  Justify your answer.
		
		In each case, $\bf{u}$, $\bf{v}$, and $\bf{w}$ are vectors in 
		$\mathbb{R}^3$.
		\begin{enumerate}
			\item $\bf{u} \cdot \bf{v} = \bf{v} \cdot \bf{u}$
			\item $\bf{u}\cdot \bf{u} = |\bf{u}|^2$
			\item $(\bf{u} \times \bf{u}) \cdot \bf{u}=0$
			
		\end{enumerate} 
		
		\item Suppose ${\bf u}$, ${\bf v}$, and ${\bf w}$ are vectors in 
		$\R^3$.  Which of the following make sense, and which do not?
		For those that make sense, is the result a vector or a scalar?
		\begin{enumerate}
			\item $({\bf u} \times {\bf v}) \cdot {\bf w}$ 
			\item ${\bf u} \times ({\bf v} \cdot {\bf w})$
			\item ${\bf u} \times ({\bf v} \times {\bf w})$ 
			\item ${\bf u} \cdot ({\bf v} \cdot {\bf w})$
		\end{enumerate}
	\end{enumerate}
	\textbf{\large Focus Problems}
	\begin{enumerate}[resume=true]
		
		\item\textbf{[G1]} Find an equation for 
		\begin{enumerate}
			\item the line through point $P=(1,2,-1)$ and point $Q=(-1,0,1)$.
			
			\item the line through $(0,-7,0)$ perpendicular to the plane 
			$x+2y+2z=13$.
			
			\item the plane containing the point $(2,1,1)$ and orthogonal to 
			the vector $\langle 2,2,-2\rangle$.
		\end{enumerate}
		
		\item\textbf{[G1]} Find a vector in the direction of the line of 
		intersection $\ell$ 
		of the planes $2x+y-z=3$ and  $x+2y+z=2$.  Find a plane which goes	
		through $(2,1,-1)$ and is perpendicular to $\ell$ (and thus both 
		planes).  
		
		
		\item\textbf{[G4]} For the following, identify and describe (for 
		example, which way 
		is it oriented? what are the cross-sections?) the type of surface.
		
		\begin{enumerate}
			\item $x^2+4z^2=16$.
			
			\item $9x^2+z^2+y^2=9$.
			
			\item $y^2=3x^2+3z^2$.
			
			\item $z=x^2+y^2+4$.
			
			\item $x^2+y^2=16-z^2$.
			
		\end{enumerate}
	\end{enumerate}
	\textbf{\large Extra Problems}
	\begin{enumerate}[resume=true]
		
		\item\textbf{[G1]} Find 2 planes that are not parallel that both 
		contain the points 
		$P(1,-1,1)$, $Q(3,2,0)$, and $R(5,5,-1)$. When will 3 distinct points 
		NOT determine a unique plane? 
		
		\item\textbf{[G1]} Find the point where the line $\br(t)=\langle 2, 
		3+2t, 
		1+t\rangle$  intersects the plane
		$2x-y+3z=6$.
		
		\item\textbf{[G1]}  %12.5 68
		How can you tell when two planes $A_1x + B_1 y + C_1 z = D_1$ and  
		$A_2x + B_2 y + C_2 z = D_2$ are parallel?  Perpendicular?
		Justify your answer. 
		
		\item\textbf{[G1]} Find the point at which the lines $\ell_1(t)=\langle 
		2,3,1\rangle+\langle 1,-1,1\rangle t$ and $\ell_2(t)=\langle 
		2,1,-2\rangle 
		t+\langle 6,2,1\rangle$ intersect.
	\end{enumerate}
}{
	\begin{center}{\large \textbf{Math 2550 Worksheet Answers - Week \wknum}}\\
	\end{center}
	
	The Learning Targets for this week's worksheet are
	
	\begin{itemize}
		\item \textbf{G1: Lines and Planes.} I can describe lines using the 
		vector equation of a line. I can describe planes using the general 
		equation of a plane. I can find the equations of planes using a point 
		and a normal vector. I can find the intersections of lines and planes.  
		I can describe the relationships of lines and planes to each other. I 
		can solve problems with lines and planes.
		
		\item \textbf{G4: Surfaces.} I can identify standard quadric surfaces 
		including: spheres, ellipsoids, elliptic paraboloids, hyperboloids, 
		cones, and hyperbolic paraboloids. I can match graphs of functions of 
		two variables to their equations and contour plots and determine their 
		domains and ranges. 
	\end{itemize}
	
	\textbf{\large Warmup}
	\begin{enumerate}
		\item \begin{enumerate}
			\item True
			\item True 
			\item True
			
		\end{enumerate} 
		
		\item \begin{enumerate}
			\item Makes sense, scalar
			\item Does not make sense
			\item Makes sense, vector
			\item Does not make sense
		\end{enumerate}
	\end{enumerate}
	\textbf{\large Focus Problems}
	\begin{enumerate}[resume=true]
		
		\item\begin{enumerate}
			\item $\br(t)=\langle 1 -2t, 2-2t, -1+2t\rangle$
			
			\item $\br(t)=\langle t, -7+2t,2t\rangle$
			
			\item $2(x-2)+2(y-1)-2(z-1)=0$
		\end{enumerate}
		
		\item vector: $3\bi-3\bj+3\bk$.
		
		plane: $3(x-2)-3(y-1)+3(z+1)=0$
		
		\item 	
		\begin{enumerate}
			\item elliptical cylinder, oriented along the $y$-axis, 
			cross-sections are ellipses in the $y=k$ planes or 
			vertical/horizontal lines in the $x=k$ and $z=k$ planes.
			
			\item ellipsoid, centered at the origin, wider in the $z$ and $y$ 
			directions than the $x$ direction, cross-sections are circles in 
			the $x=k$ planes (if $k<1$), ellipses in the $z=k$ and $y=k$ planes 
			($k<3$)
			
			\item circular cone, oriented along the $y$-axis, cross sections 
			are circles in the $y=k$ planes and lines in the $x=k$ and $z=k$ 
			planes
			
			\item elliptical paraboloid, oriented in the positive $z$ 
			direction, shifted up $4$ units, cross-sections are circles in the 
			$z=k$ planes for $k\geq 4$ and parabolas in the $x=k$ or $y=k$ 
			planes
			
			\item sphere, centered at $(0,0,0)$, radius $4$, cross-sections are 
			circles in $x=k, y=k,$ and $z=k$ planes for $0\leq k\leq 4$.
			
		\end{enumerate}
	\end{enumerate}
	\textbf{\large Extra Problems}
	\begin{enumerate}[resume=true]		
		\item many correct solutions; pick two distinct points off of the line 
		$PQ$ which are not collinear with the given points.  Two such planes 
		are $-2(x-1)+3(y+1)+5(z-1)=0$ and $-3(x-1)+3(y+1)+3(z-1)=0$.
		
		\item $(2,7,3)$
		
		\item  Parallel: $\langle A_1, B_1, C_1\rangle = \lambda \langle A_2, 
		B_2, C_2\rangle$ for some $\lambda\neq 0$
		
		Perpendicular:  $\langle A_1, B_1, C_1\rangle \dotp \langle A_2, B_2, 
		C_2\rangle=0$
		
		\item $(4,1,3)$
		
	\end{enumerate}
}