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{\bf Worksheet \#6; date: 02/06/2018}
{\bf MATH 53 Multivariable Calculus}
\begin{enumerate}
\item {\bf Correction about scalar projections: It is SIGNED.}
\item {\em (Stewart 12.5.5)} Find a vector equation and parametric equations for the line through the point $(1, 0, 6)$ and perpendicular to the plane $x + 3y + z = 5$.
\item {\em (Stewart 12.5.11)} Find a parametric equation and a symmetric equation for the line through $(-6, 2, 3)$ and parallel to the line
\[
\frac{1}{2}x = \frac{1}{3}y = z + 1.
\]
How does the symmetric equation of the new line compare to the old one?
\item {\em (Stewart 12.5.19)} Determine whether the lines $L_1$ and $L_2$ are parallel, skew, or intersecting. If they intersect, find the point of intersection.
\begin{align*}
L_1: & ~~~~ x = 3 + 2t, ~~~~ y = 4 - t, ~~~~ z = 1 + 3t \\
L_2: & ~~~~ x = 1 + 4s, ~~~~ y = 3 - 2s, ~~~~ z = 4 + 5s
\end{align*}
\item {\em (Stewart 12.5.35)} Find an equation of the plane that passes through the point $(3, 5, -1)$ and contains the line
\[
x = 4 - t, ~~~~ y = 2t - 1, ~~~~ z = -3t.
\]
\item {\em (Stewart 12.5.55)} Determine whether the planes are parallel, perpendicular, or neither. If neither, find the angle between them. (Leave as inverse trigonometric functions where necessary.)
\[
2x - 3y = z, ~~~~ 4x = 3 + 6y + 2z.
\]
\item {\em (Stewart 12.5.73)} Find the distance between the given parallel planes
\[
2x - 3y + z = 4, ~~~~ 4x - 6y + 2z = 3.
\]
\item {\em (Stewart 12.5.77)} Show that the lines with symmetric equations $x = y = z$ and $x + 1 = y/2 = z/3$ are skew, and find the distance between these lines.
\item {\em (Stewart 12.6.11)} Use traces to sketch and identify the surface
\[
x = y^2 + 4z^2.
\]
\item {\em (Stewart 12.6.37)} Reduce the equation to one of the standard forms, classify the surface, and sketch it.
\[
x^2 - y^2 + z^2 - 4x - 2z = 0.
\]
\end{enumerate}
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