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?