9.3 The Intersection of Two Planes
Chapter
Chapter 9
Section
9.3
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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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

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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

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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

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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

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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

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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.

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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.

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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.

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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.

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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.

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2.46mins
Q11b

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.

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2.55mins
Q12