9.3 The Intersection of Two Planes
Chapter
Chapter 9
Section
9.3
Lectures 4 Videos
Solutions 21 Videos

A system of two equations in three unknowns has been manipulated, and, after correctly using elementary operations, a student arrives at the following equivalent system of equations:

• x - y + z = 1
• 0x + 0y + 0z = 3

a. Explain what this equivalent system means.

b. Give an example of a system of equations that might lead to this solution.

0.57mins
Q1

A system of two equations in three unknowns has been manipulated, and, after correctly using elementary operations, a student arrives at the following equivalent system of equations:

1. 2x - y + 2z = 1

2. 0x + 0y + 0z = 0

a. Write a solution to this system of equations, and explain what your solution means.

b. Give an example of a system of equations that leads to your solution in part a.

1.07mins
Q2

A system of two equations in three unknowns has been manipulated, and, after correctly using elementary operations, a student arrives at the following equivalent system of equations:

1. x - y + z = -1

2. 0x + 0y +2z = -4

a. Write a solution to this system of equations, and explain what your solution means.

b. Give an example of a system of equations that leads to your solution in part a.

1.15mins
Q3

Consider the following system of equations:

1. 2x + y + 6z = p

2. x + my + 3z = q

a) Determine values of m, p, and q such that the two planes are coincident. Are these values unique? Explain.

1.01mins
Q4a

Consider the following system of equations:

1. 2x + y + 6z = p

2. x + my + 3z = q

b) Determine values of m, p, and q such that the two planes are parallel and not coincident. Are these values unique? Explain.

1.02mins
Q4b

Consider the following system of equations:

1. 2x + y + 6z = p

2. x + my + 3z = q

• A value of m such that the two planes intersect at right angles. Is this value unique? Explain.

• Determine values of m, p, and q such that the two planes intersect at right angles. Are these values unique? Explain.

0.43mins
Q4cd

Consider the following system of equations:

1. x + 2y - 3z = 0

2. y + 3z = 0

a. Solve this system of equations by letting z = s.

0.48mins
Q5a

Consider the following system of equations:

1. x + 2y - 3z = 0

2. y + 3z = 0

b. Solve this system of equations by letting y = t.

1.02mins
Q5b

Consider the following system of equations:

1. x + 2y - 3z = 0

2. y + 3z = 0

Show that the solution you found in part a. is the same as the solution you found in part b.

0.36mins
Q5c

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. x + y + z = 1

2. 2x + 2y + 2z = 2

0.31mins
Q6a

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. 2x - y +z +1 = 0

2. 2x - y + z +2 =0

0.23mins
Q6b

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. x - y + 2z = 2

2. x + y + 2z = -2

1.21mins
Q6c

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. x + y + 2z = 4

2. x - y = 6

1.07mins
Q6d

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. 2x - y + 2 = 2

2. -x + 2y + z = 1

2.13mins
Q6e

The following systems of equations involve two planes. State whether the planes intersect, and, if they do intersect, specify if their intersection is a line or a plane.

1. x - y + 2z = 0

2. z = 4

0.34mins
Q6f

A system of equations is given as follows:

1. x + y + 2z = 1

2. kx + 2y + 4z = k

a. For what value of k does the system have an infinite number of solutions? Determine the solution to the system for this value of k.

b. Is there any value of k for which the system does not have a solution? Explain.

2.23mins
Q8

Determine the vector equation of the line that passes through A(-2,3,6) and is parallel to the line of intersection of the planes \pi_1: 2x - y + z = 0 and \pi_2: y + 4z =0.

1.26mins
Q9

For the panes 2x - y + 2z = 0 and 2x + y + 6z = 4, show that their line of intersection lies on the plane with equation 5x + 3y + 16z - 11 = 0.

4.48mins
Q10

The line of intersection of the planes \pi_1: 2x + y - 3z = 3 and \pi_2: x - 2y + z = -1 is L.

a) Determine parametric equations for L.

2.25mins
Q11a

The line of intersection of the planes \pi_1: 2x + y - 3z = 3 and \pi_2: x - 2y + z = -1 is L.

b) If L meets the xy-plane at point A and the z-axis at point B, determine the length of line segment AB.

Determine the Cartesian equation of the plane that is parallel to the line with equation x = - 2y = 3z and that contains the line of intersection of the planes with equations x - y + z = 1 and 2y - z = 0.