8.6 Intersections of Planes
Chapter
Chapter 8
Section
8.6
Intersection Line between Two Planes Lecture 1 Videos
Intersection of 3D Planes Lectures 5 Videos Possible Outcomes of 3 Planes Three Planes intersecting at a Point Three Planes Intersecting in a Line Triangular Prism formation
Solutions 57 Videos

Find the vector equation of the line of intersection for each pair of planes.

\displaystyle \begin{array}{llllllll} &x + 5y - 3z - 8 = 0 \\ &y + 2z - 4 = 0 \end{array} 

1.11mins
Q1a

Find the vector equation of the line of intersection for each pair of planes.

\displaystyle \begin{array}{llllllll} &5x - y +z - 22= 0\\ &x + 3y - 9=0 \end{array} 

1.10mins
Q1b

Find the vector equation of the line of intersection for each pair of planes.

\displaystyle \begin{array}{llllllll} &2x - y+z-22=0\\ &x -11y+2z -8=0 \end{array} 

2.27mins
Q1c

Find the vector equation of the line of intersection for each pair of planes.

\displaystyle \begin{array}{llllllll} &3x + y - 5z - 7=0\\ &2x - y -21z + 33 = 0 \end{array} 

Q1d

Which pairs of planes are parallel and distinct and which are coincident?

\displaystyle \begin{array}{llllllll} &2x +3y - 7z -2= 0\\ &4x + 6y -14z -8=0 \end{array} 

Q2a

Which pairs of planes are parallel and distinct and which are coincident?

\displaystyle \begin{array}{llllllll} &3x + 9y - 6z -24 = 0\\ &4x + 12y -8z - 32= 0 \end{array} 

Q2b

Which pairs of planes are parallel and distinct and which are coincident?

\displaystyle \begin{array}{llllllll} &4x-12y-16z-52 =0\\ &-6x +18y+24z+78=0 \end{array} 

Q2c

Which pairs of planes are parallel and distinct and which are coincident?

\displaystyle \begin{array}{llllllll} &x - 2y+2.5z -1=0\\ &3x -6y+7.5z -3= 0 \end{array} 

Q2d

Show that the line (x,y,z)=[10, 5, 16]+t[3, 1,5] is contained in each of these planes.

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

1.00mins
Q3a

Show that the line (x,y,z)=[10, 5, 16]+t[3, 1,5] is contained in each of these planes.

\displaystyle 9x -2y-5z= 0 

Q3b

For each system of equations, determine the point of intersection.

\displaystyle \begin{array}{llllllll} &x + y +z-7=0\\ &2x + y+3z-17=0\\ &2x -y-2z+5=0 \end{array} 

Q4a

For each system of equations, determine the point of intersection.

\displaystyle \begin{array}{llllllll} &2x + y + 4z = 15\\ &2x + 3y + z = -6\\ &2x -y + 2z = 12 \end{array} 

Q4b

For each system of equations, determine the point of intersection.

\displaystyle \begin{array}{llllllll} &5x -2y - 7z -19=0\\ &x - y + z -8=0\\ &3x + 4y+ z - 1=0 \end{array} 

Q4c

For each system of equations, determine the point of intersection.

\displaystyle \begin{array}{llllllll} &2x-5y-z = 9\\ &x + 2y + 2z = -13\\ &2x + 8y + 3z = - 19 \end{array} 

Q4d

Determine the line of intersection of each system of equations.

\displaystyle \begin{array}{llllllll} &2x+y+z= 7\\ &4x +3y-3z=13\\ &4x+2y+2z=14 \end{array} 

1.47mins
Q5a

Determine the line of intersection of each system of equations.

\displaystyle \begin{array}{llllllll} &x + 3y-z=4\\ &3x+8y-4z= 4\\ &x +2y-2z= -4 \end{array} 

Q5b

Determine the line of intersection of each system of equations.

\displaystyle \begin{array}{llllllll} &x + 9y+3z= 23\\ &x + 15y + 3z= 29\\ &4x -13y + 12z = 43 \end{array} 

Q5c

Determine the line of intersection of each system of equations.

\displaystyle \begin{array}{llllllll} &x-6y+ z= -1\\ &x - y = 5\\ &2x-12y+ 2z= -2 \end{array} 

Q5d

In the system, at least one pair of planes are parallel. Describe each system.

\displaystyle \begin{array}{llllllll} &3x+15y-9z=12\\ &6x+30y-18z=24\\ &5x+25y -15z =10 \end{array} 

Q6a

In the system, at least one pair of planes are parallel. Describe each system.

\displaystyle \begin{array}{llllllll} &2x-y+4z= 5\\ &6x-3y+12z =15\\ &4x -2y + 8z =10 \end{array} 

0.20mins
Q6b

In the system, at least one pair of planes are parallel. Describe each system.

\displaystyle \begin{array}{llllllll} &3x+2y-z=8\\ &12x + 8y-4z =20\\ &18x+12y -6z = -3 \end{array} 

Q6c

In the system, at least one pair of planes are parallel. Describe each system.

\displaystyle \begin{array}{llllllll} &8x+4y+6z= 7\\ &12x + 6y + 9z = 1\\ &3x+ 2y +4z = 1 \end{array} 

Q6d

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &3x+y-2z= 18\\ &6x -4y + 10z = -10\\ &3x -5y+10z =10 \end{array} 

2.55mins
Q7a

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &2x+ 5y-3z= 12\\ &3x-2y+3z=5\\ &4x + 10y-6z= -10 \end{array} 

Q7b

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &2x-3y+2z= 10\\ &5x-15y + 5z = 25\\ &-4x + 6y -4z = -4 \end{array} 

Q7c

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &3x-5y-5z = 1\\ &-6x+ 10y + 10z = -2\\ &15x-25y -25z = 20 \end{array} 

Q7d

Planes that are perpendicular to one another are said to be orthogonal . Determine which of the systems from question 4 contain orthogonal planes.

Q9

For each system of planes below, find the triple scalar product of the normal vectors. Compare the answers. What does this say about the normal vectors of planes that intersect in a line?

a) \displaystyle 2 x+y+z=7

\displaystyle 4 x+3 y-3 z=13

\displaystyle 4 x+2 y+2 z=14

b) \displaystyle x+3 y-z=4

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

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

c) \displaystyle x+9 y+3 z=23

\displaystyle x+15 y+3 z=29

\displaystyle 4 x-13 y+12 z=43

d) \displaystyle x-6 y+z=-1

\displaystyle x-y=5

\displaystyle 2 x-12 y+2 z=-2

Q10

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &3x+2y-z=-2\\ &2x+y-2z= 7\\ &2x-3y + 4z= -3 \end{array} 

Q11a

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &3x-4y+2z= 1\\ &6x-8y+4z= 10\\ &15x-20y+10z=-3 \end{array} 

Q11b

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &x + 4y + 8z= 1\\ &2y - 4z = 10\\ &0=12 \end{array} 

0.19mins
Q11c

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &x-y+z=20\\ &x + y + 3z = -4\\ &2x -5y-z= -6 \end{array} 

Q11d

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &4x+8y+ 4z= 7\\ &5x+ 10y + 5z = -10\\ &3x-y -4z= 6 \end{array} 

Q11e

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &2x+ 2y + z = 10\\ &5x + 4y-4z = 13\\ &3x+5y-2z= 6 \end{array} 

Q11f

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &2x+5y =1\\ &4y+ z= 1\\ &7x-4z = 1 \end{array} 

3.51mins
Q11g

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &3x-2y-z=0\\ &x -z= 0\\ &3x+2y-5z=0 \end{array} 

Q11h

Determine if the system of planes is consistent or inconsistent. If possible, solve the system.

\displaystyle \begin{array}{llllllll} &y + 2z= 11\\ &4y -4z = -16\\ &3y-3z= -12 \end{array} 

Q11i

A student solved three systems of planes and obtained these algebraic solutions. Interpret how the planes intersect as exactly as you can in each case.

\displaystyle \begin{array}{llllllll} &x = 5\\ &y = 10\\ &z = -3 \end{array} 

0.29mins
Q12a

A student solved three systems of planes and obtained these algebraic solutions. Interpret how the planes intersect as exactly as you can in each case.

\displaystyle \begin{array}{llllllll} &x -y+2z=4 &y-3z=8 &0=0 \end{array} 

0.20mins
Q12b

In each case, describe all the ways in which three planes could intersect.

a) The normals are not parallel.

b) Two of three three normals are parallel.

c) All three of the normals are parallel.

Q13

Show that the planes in each set are mutually perpendicular and have a unique solution.

\displaystyle \begin{array}{llllllll} &5x + y+4z= 18\\ &x+3y-2z= -16\\ &x -y-z=0 \end{array} 

Q14a

Show that the planes in the set are mutually perpendicular and have a unique solution.

\displaystyle \begin{array}{llllllll} &2x-6y+z= -16\\ &7x+3y+4z = 41\\ &27x + y - 48 z = -11 \end{array} 

Q14b

You are given the following two planes:

\displaystyle \begin{array}{llllllll} &x+4y-3z-12= 0\\ &x + 6y-2z-22= 0 \end{array} 

a) Determine if the planes are parallel.

b) Find the line of intersection of the two planes.

c) Use 3-D graphing technology to check your answer to part a).

d) Use the two original equations to determine two other equations that have the same solution as the original two.

f) Find a third equation that will have a unique solution with the original two equations.

Q15

A dependent system of equations is one whose solution requires a parameter to express it. Change one of the coefficients in the following system of planes so that the solution is consistent and dependent.

• a) \displaystyle x-3 z=-3 , \displaystyle y+5 z=20 , \displaystyle 3 x+5 z=3

• b) \displaystyle 2 x+8 y=-6  , \displaystyle 2 x-z=4  , \displaystyle 3 x+12 y+6 z=-9

Q16

Determine the equations of three planes that

are all parallel but distinct.

Q17a

Determine the equations of three planes that

intersect at the point P(3, 1, -9).

Q17b

Determine the equations of three planes that

intersect in a sheaf of planes

Q17c

Determine the equations of three planes that

intersect in pairs

Q17d

Determine the equations of three planes that

intersect in the line (x, y, z)= (1, ,3, -4)+ t(4, 1, 9).

Q17e

Determine the equations of three planes that

intersect in pairs and are all parallel to the y-axis

Q17f

Determine the equations of three planes that

intersect at the point (-2, 4, -4) and are all perpendicular to each other

Q17g

Determine the equations of three planes that

intersect along the z-axis.

Q17h

Find the volume of the figure bounded by the following planes.

\displaystyle \begin{array}{llllllll} &x + z = -3\\ &10x -3z = 22\\ &4x - 9z =-38\\ &y = -4\\ &y = 10 \end{array} 

Q18

Solve the following system of equations.

\displaystyle \begin{array}{llllllll} &2w + x+y+2z = - 9\\ &w-x + y + 2z= 1\\ &2w +x+ 2y - 3z = 18\\ &3w+2x + 3y + z = 0 \end{array} 

Q19

Create a four—dimensional system that has a solution (3, 1, -4, 6).

The parallelepiped formed by the vectors \vec{a} = [1,2,-3], \vec{b} = [1, k , -3], and \vec{c} = [2, k, 1] has volume 147 unit^3. Determine the possible values of k.