Exercise 8.4 - mrsk.ca

Part A

1. Explain why two planes can never intersect in a single point.

2. Do the following pairs of planes intersect in a straight line?

a. 6x12y9z90 and 4x8y6z90 b. 2xy2z30 and 6x3y6z90 c. r៬(6, 0, 1)p(1, 1, 2)q(4, 2, 3)

and r៬(1, 1,9)s(5, 3, 5)t(3, 1, 1) d. r៬(1, 1, 1)p(0, 0, 1)q(0, 1, 0)

and r៬(0, 0, 0)s(0, 0, 1)t(1, 0, 0)

3. Determine which of the following pairs of planes are parallel. For each pair that is not parallel, find the parametric equations of the line of intersection.

Use algebraic elimination.

a. xy3z4 and x2yz1

b. 5x2y2z10 and 5x2y2z30 c. x3yz30 and 2x4yz50 d. xyz1 and x0

e. x3yz40 and 2x6y2z80 Part B

4. Write the augmented matrix for each of the following linear systems.

a. 3x7yz12 b. 4x3y2z4

xy2z 3 2y5z5

c. x4z16 d. 5y2z6x4

y8z 2 3z5y2x 4

5. Write the system of equations that corresponds to each of these matrices.

a. 冤 冥 ^b. 冤 冥

c. 冤 冥 ^d. 冤 ⁰0冥

4 9 0 1 1 0 8

6 10

4 0 3 5 0

6 9 3 6 2 6 8 2 9

4 4 6 0 1 1 0

Knowledge/

Understanding Communication

Exercise 8.4

C H A P T E R 8 300

6. Use Gauss-Jordan elimination to find the vector equation of the line of intersection of each pair of planes.

a. x2y7z4 b. x4y3z5

x3y3z1 2xy6z0

c. 2x8y2z7 d. 4x8y3z6

x4yz3 3x6yz 2

e. 3x2y6z5 f. 6x8y3z9

2x3y9z 10 10x2y5z15

Part C

7. What is the geometrical interpretation of the system of equations that corresponds to these matrices?

a.

冤 冥

冤 冥

冤 冥

8. a. Let A₁xB₁yC₁zD₁0 and A₂xB₂yC₂zD₂0 be two non-parallel planes in space. Show that for any fixed k,

(A₁xB₁yC₁zD₁)k(A₂xB₂yC₂zD₂)0

is the equation of the plane through the intersection of the two planes.

As kvaries, this equation generates the family of all such planes (except the second plane itself).

b. Find the scalar equation of the plane that passes through the origin and the line of intersection of the planes 3x4y7z20 and

2x3y40.

c. Find the scalar equation of the plane that is parallel to the line x2y3z and passes through the line of intersection of the planes

4x3y5z100 and 4xy3z150.

9. Find the scalar equation of the plane that is perpendicular to the plane r៬(2, 1, 3)s(5,2,2)t(1, 0, 1) and intersects it along the line r៬(9,1,5)p(2,2, 2).

Application Thinking/Inquiry/

Problem Solving

10 22 15 9 4

4 1 5 3 6 7 2 6 3 8 5 5

2 2 8 3 2 1

4 3 5

5 3 2

1 4 6 5 3 2 6 2

Communication Knowledge/

Understanding

8 . 4 T H E I N T E R S E C T I O N O F T W O P L A N E S 301