In-class work for 6 June 2012
1.) Find a vector equation in variable $t$ for the line through the point $(2,1,-3)$ parallel to the vector $<1,2,3>$.

2.) Find a vector equation in variable $t$ for the line through the point $(-1,3,2)$ and perpendicular to the plane $3x-2y+4z=5$.

3.) Find the parametric equation of the line between the point $P=(1,2,3)$ and the point $Q=(-2,3,1)$.

4.) Let $L_1$ be the line given by the parametric equations $$L_1 : \left\{ \begin{array}{ll} x &= 1+2t \\ y &= 3t \\ z &= 2-t, \end{array} \right.$$ and let $L_2$ be the line given by the vector equation $$\vec{r} = <-1,4,1> + s<1,1,3>.$$ Are $L_1$ and $L_2$ parallel, skew, or intersecting? If intersecting, find the point of intersection.

5.) Find an equation of the plane through the point $(1,2,3)$ with normal vector $< 3,-5, 1 >$.

6.) At which point does the line $$\left\{ \begin{array}{ll} x &= 1+2t \\ y &= 2-3t \\ z &= -5 + t \end{array} \right.$$ intersect the plane $2x+5y-3z=6$?

7.) Are the planes $x+y+z=1$ and $x-y+z=1$ parallel, perpendicular, or neither? How do you know?

8.) Let $\vec{x} = <2,0>$ and $\vec{y} = <1,1>$. Find the vector projection of $\vec{x}$ onto $\vec{y}$.

9.) Are the vectors $<0,1>$ and $<1,0>$ orthogonal? Why or why not?