In-class work for 7 June 2012
1.) Find the distance from the point $(4,-4,3)$ to the plane $2x-2y+5z+8=0$.
2.) Find the distance between the two parallel planes $x-5y-z=1$ and $5x-25y-5z=-3$.
3.) Consider the lines $\vec{r} = <1,2,3> + t<2,3,4>$ and $\vec{r} = <6,-7,4> + s<-1,4,1>$.
a.) Find the point of intersection of the two lines.
b.) Find an equation of the plane that contains both of these lines.
4.) Find a vector equation, parametric equation, and symmetric equation for the line passing through the point $(2,0,1)$ parallel to the vector $<1,-1,6>$.
5.) Find a vector equation of the line in which the planes $2x-5y+3z=12$ and $3x+4y-3z=6$ meet.
6.) Determine whether the lines
$$L_1 : \frac{x-4}{2} = \frac{y+5}{4} = \frac{z-1}{3},$$
and
$$L_2 : \frac{x-2}{1} = \frac{y+1}{3} = \frac{z}{2}$$
are parallel, skew, or intersecting. If they intersect, find the point of intersection.
7.) Find the distance from $(0,2,1)$ to the line $<1,1,1>+t<-2,1,3>$.
8.) Find the distance between the point $(1,2,3)$ and the plane $-3x+y-3z=4$.