8.5 Intersection of Lines and Planes
Chapter
Chapter 8
Section
8.5
Solutions 18 Videos

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

1.23mins
Q1

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]

1.22mins
Q2a

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]

1.12mins
Q2b

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]

0.52mins
Q3a

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 ]

1.52mins
Q3b

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

0.36mins
Q4a

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]

1.04mins
Q4b

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]

Q4c

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]

Q5a

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

1.26mins
Q6a

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} 

Q7a

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} 

Q8a

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] .

Q9

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.

Q10

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.

4.25mins
Q14

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$\displaystyle \ell{2}$

4.03mins
Q17a

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]