Determine the coordinates of the point of intersection of the line defined by the parametric equations and the plane defined by the scalar equation.

\displaystyle \ell:\left\{\begin{array}{l} x=4+t \\ y=2-2 t \\ z=6+3 t \end{array}\right.

\displaystyle \pi: x+5 y+z-8=0

In each case, verify that the plane and line are parallel, and then determine if they are distinct or coincident.

\displaystyle 3 x+5 y+z-5=0

\displaystyle [x, y, z]=[1,2,-8]+t[2,-1,-1]

In each case, verify that the plane and line are parallel, and then determine if they are distinct or coincident.

\displaystyle 4 x-y+6 z-12=0

\displaystyle [x, y, z]=\mid 4,3,10]+t \mid 7,-14,-7]

In each case, determine if the plane and the line intersect. If so, state the solution.

\displaystyle 3 x-y+4 z-8=0

\displaystyle [ x, y, z]=|3,0,5|+t \mid 7,-11,-8]

In each case, determine if the plane and the line intersect. If so, state the solution.

\displaystyle -2 x+6 y+4 z-4=0

\displaystyle [x, y, z] = [5,-1,4]+t[1,-2,3 ]

Use direction vectors to determine if each line intersects the plane \displaystyle 3 x-2 y+4 z=5

\displaystyle \vec{r}=[-3,2,7]+t[3,6,2 \mid

Use direction vectors to determine if each line intersects the plane \displaystyle 3 x-2 y+4 z=5

\displaystyle \vec{r}=[-3,-5,1]+t[-2,1,2]

Use direction vectors to determine if each line intersects the plane \displaystyle 3 x-2 y+4 z=5

\displaystyle \vec{r}=[0,1,2]+t[4,4,1]

Does each line intersect the plane

\displaystyle [x, y, z]=[4,-15,-8]+s[1,-3,1]+t[2,3,1] ?

If so, how many solutions are there?

\displaystyle [x, y, z]=[5,-9,3]+k[1,-12,2]

Find the distance between the parallel line and plane.

\displaystyle \ell: \vec{r}=[2,0,1]+t[1,4,1]

\displaystyle \pi: 2 x-y+2 z=4

Find the distance between the planes.

\displaystyle \begin{aligned} \pi_{1}: 2 x-y-z-1=0 \\ \pi_{2}: 2 x-y-z-4=0 \end{aligned}

Determine the distance between each point and the given plane.

\displaystyle \begin{array}{l} \begin{array}{l} P(2,1,6) \\ 3 x+9 y-z-1=0 \end{array}\end{array}

Write an equation of the plane that contains the points \displaystyle \mathrm{P}(2,-3,6) and \displaystyle \mathrm{Q}(4,1,-2) and is parallel to the line \displaystyle [x, y, z]=[3,3,-2]+t[1,2,-3] .

Does the line through \displaystyle \mathrm{A}(2,3,2) and \displaystyle \mathrm{B}(4,0,2) intersect the plane \displaystyle 2 x+y-3 z+4=0 ? Explain.

Consider these lines.

a) Determine if lines \displaystyle \ell_{1} and \displaystyle \ell_{2} are skew.

b) Write the equations of parallel planes that each contain one of \displaystyle \ell_{1} and \displaystyle \ell_{2} .

\displaystyle \ell_{1}:\left\{\begin{array}{l}x=3 s \\ y=2+s \\ z=1+s\end{array}\right.

and

\displaystyle \ell_{2}:\left\{\begin{array}{l}x=1+2 t \\ y=-3-t \\ z=t\end{array}\right.

Consider these lines.

`$\displaystyle \begin{array}{l} \ell{1}:[x, y, z]=[1,-2,4]+s[1,1,-3] \text { and } \ \ell{2}:[x, y, z]=[4,-2, k]+t[2,3,1] \end{array}

$ Determine an equation of the plane that contains$\displaystyle \ell{1} $and is parallel to$\displaystyle \ell{2} $`

In each case, determine the distance between point \displaystyle \mathrm{P} and the line.

\displaystyle \begin{array}{l} \mathrm{P}(1,3,-4) \\ {[x, y, z]=[4,1,4]+t[2,4,-3]} \end{array}

Math Contest Determine the value of \displaystyle k if \displaystyle [3,7,-2] \times[a, 3, b]=[-1, k,-5]