# Ch. 9 - Rel'nships b/w Points, Lines and Planes incl. Matrices

Chapter 9.1 - The Intersection of a Line with a Plane and the Intersection of Two Lines

1.  For each of the following, determine whether the line intersects the plane, and if so how.

a). line:     r = (1,5,1) + t(3, -9, 1)

plane: 2x + y + 15z + 4 = 0

Answer: intersect at the point (2, 2, -2/3)

Video solution to 1a)

b). line:      r = (-2, 1, 8) + t(6, -3, -2)

plane:   2x + 2y + 3z - 11 = 0

Video solution to 1 b)

C). Line:        r = (0, -5, -9) + t(4, 1, 3)

Plane:      x + 8y - 4z = 0

Video solution to 1c)

2. Determine whether each of the following pairs of lines intersects and if so, how they intersect.

a) line: r = (2, -5, -7) + s(1, -9, 3)

line: r = (0, 13, 1) + t(-2, 18, -6)

Video solution to 2a)

b) line: r = (-3, 9, -4) + s(6, 3, 5)

line: r = (-2, -8, 3) + t(18, 9, 15)

Answer: lines are parallel, non coincident

Video solution to 2b)

c) line: r = (2, 3, -9) + s(3, 4, -1)

line: r = (-11, 11, 22) + t(1, -5, -7)

Answer: lines intersect at only one point (-7, -9, -6)

Video solution to 2c)

D) line: r = (-3, 1, 7) + s(1, 2, -5)

line: r = (3, 9, 8) + t(5, 6, -3)

Video solution to 2d)

3.  Determine the point at which the line r = (-3, 8, 4) + t (1, -2, 3) intersects the xz-plane

Video solution to #3

4.  Determine whether each of the following lines has a y-intercept

a) r = (10, 7, -15) + t(2, 4, -3)

Video solution to 4a)

B). r = (-7, 8, 3) + t(9, -1, -8)

Video solution to 4b)

Chapter 9.2 - Systems of Equations (and Appendix work with Matrices)

5. Write a solution to each of the following using parameters

a) 2x + 3y = 7
Answer: x = -3/2 t + 7/2
y = t
Video solution to 5a)

b) 9x - 4y + 2z = 11
y = t
z = -9/2 s + 2t + 11
Video solution to 5b)

C)     4x + 3y - 9z = 12
7x - y + 5z = -8
y = -83/6 t - 2
z = -25/6 t - 2
Video solution to 5 c)

6. Determine the values of k so that the following system of equations

(k - 3)^2    x         +          4 y     =      84
4   x         +             y     =      3k

a)  an infinite number of solutions
b)  zero solutions
c)  one solution

b) k = -1
c) k does not equal 7 or -1, and k is a real #

Video solution # 6

7.

Solve each of the following systems of equations using matrices and interpret the results:

a)

2x - 2y + 3z = 14

x + 4y - 2z = 11

4x - 4y + 6z = 28

Answer:  No solution, inconsistent system.      Video solution to 7a)

b)

3x + y - 4z = -5

2x + 9y - 8z = 21

4x - 2y + 5z = -41

Answer:  intersect at the point (-6, 1, -3).           Video solution to 7b)

c)

2x - y + z = 3

4x - 2y + 2z = 6

x + y + z = 2

Answer:  intersect at a line.                                 Video solution to 7c)

d)

3x + 2y + z = 12

6xd + 4y + 2z = 24

9x + 6y + 3z = 36

Answer:  3 coincident planes.                               Video solution to 7d)

8.   Solve the following system of equations

1/x    +    1/y     -    3/z    =    -16

1/x    +    3/y    +    1/z    =     -8

2/x    +    1/y    +    1/z    =     -9

Ans: ( -1/5 , -1/2 , 1/3 )

Video solution to # 8

9.

For what value(s) of k will the following system of equations have

a)  an infinite number of solutions

b)  one solution

c)  zero solutions

2x + y + 4z = 3

20x - 5y - 128z = 10

5x + ky - z = 5

Answer:  a) k = 5/8;  b) impossible;  c) k does not equal 5/8, kER.                 Video solution to # 9

Chapter 9.3 - The Intersection of Two Planes

10.

Determine whether each of the following system of equations is consistent or inconsistent.  If the system is consistent, then state the equation of the line of intersection or show that the planes are coincident

a)

2x + 3y - 4z = 7

6x + 9y - 12z = 22

Answer: inconsistent system; two parallel, non-coincident planes.             Video solution to 10a

b)

3x + y + 8z = 23

6x + 2y + 16z = 46

|Answer: consistent system; two coincident planes.                           Video solution to 10b

c)

x - 4y + 3z = 4

2x + y - z = 6

Answer:  line of intersection r = (22/7, 0, 2/7) + t(1,7,9), tER.                       Video solution to 10c

11.  Show that the line r = (3, -1, 2) + t(4, -1, 3)  lies on the plane 5x-28y+16z-75=0

Video solution to #11

12. State the equation of the line 3x = 2y + 10 = -4z in vector form using only integers.

Answer: r = (0, -5, 0) + t (4, 6, -3)

Video solution to #12

Chapter 9.4 - The Intersection of Three Planes

VIDEO  LESSON

When dealing with three planes,  there are 8 possible scenarios.  Here is a video lesson in two parts.

Part 1.                     Part 2.

13.

Determine whether the following system of equations is consistent or inconsistent.  If the system is consistent, then state the single point of intersection or the equation of the line of intersection or show that the planes are coincident.

a)

3x - y + 4z + 8 = 0

2x + y + z + 7 = 0

2x + 5y - 2z + 9 = 0

Answer:  consistent; single point of intersection (-5,1,2).     Video solution to 13a

b)

x + y + z = 2

4x - y - z = -7

2x - y - z = -5

Answer:  consistent; line of intersection r = (-1,3,0) + t (0,1,-1), t ER.        Video solution to 13b

c)

x + y = 3

y + 2z = 7

8x + 3y - 10z = 62

Answer: inconsistent system; triangular prism.             Video solution to 13 c

d)

x + 4y - 5z = 7

2x + 8y -10z = 17

3x + 2y - 6z = 1

Answer:  inconsistent: 2 parallel non-coincident planes and a non-parallel plane intersecting both

Video solution to 13 d

e)

x + y - z = 4

2x - y + 5z = 11

6x - 3y + 15z = 33

Answer: consistent; line of intersection is r = (5, -1, 0) + t (4, -7, -3), t ER

Video solution to 13 e

f)

x - 2y + z = 4

3x - 6y + 3z = 12

x/2 - y + z/2 = 2

Answer:  consistent; 3 coincident planes.                              Video solution to 13 f

g)

4x - 3y + z = 10

8x - 6y + 2z = 20

x - 3/4 y + 1/4 z = 3

Answer:   inconsistent; 2 coincident planes and a third parallel, non-coincident plane

Video solution to 13 g

h)

4x + 2y - z = 11

8x + 4y - 2z = 23

12x + 6y - 3z = 34

Answer: inconsistent; 3 parallel, non coincident planes.                 Video solution to 13 h

14.

Determine a and b such that the following three planes

3x + 2y + 5z = 10

6x + y + 8z = a

bx + 3y + 6z = 2

a) intersect along a line

b) do not intersect

c)  intersect at just one point

Answer: a) a = 158/3, b = 36/11;  b) b = 36/11, a does not equal 158/3, a ER;

c)  b does not equal 36/11, a ER, b ER

Video solution to 14

15.   Solve the following system of equations

4/x    -    2/y    +    10/z    =    3

3/x    +    4/y    -     3/z     =    3

2/x    -     6/y    +    2/z     =    5

Video solution to 15

Chapter 9.5 - The Distance from a Point to a Line in R2 and R3

16. Calculate the distance between the following pairs of lines

a)   Line 1: r = (-9, 1) + t (4, -7)

Line 2: r = (3, 8) + s (4, -7)

Video solution to 16a

b)  Line 1: 3x - 8y + 27 = 0

Line 2: 3x - 8y - 9 = 0

Video solution to 16b

17.  Determine the distance from the point P to the line L in each of the following situations

a)  P(5, -1); L: 9x - 7y + 53 = 0

Video solution to 17a

b)  P(7, 6); L: r = (7, 6) + t (8, -1)

Video solution to 17b

18.  Determine the distance from the point (-7, -1, 5) to the line

r = (-8, 2, -1) + t(11, -4, 1)

Video solution to 18

19.  Determine the distance between the lines

line 1: r = (3, -8, 2) + t(1, 7, 6)

line 2: r = (9, 7, 0) + s(1, 7, 6)

Video solution to 19

20. Blank

21.  Determine the coordinates of the point on the line r = (48, -18, 14) + t(6, -9, 7) that produces the shortest distance between the line and a point with coordinates (-6, -8, 2)

Video solution to 21

22.  The point P(5, -3, 6) is reflected in the line with equation

r = (2, -1, 4) + t(3, -1, -2) to give the point P'. Determine the coordinates of P'

Video solution to 22

Chapter 9.6 - The Distance from a Point to a Plane

23.  Determine the distance from the point P(-7, 4, -5) to the plane

2x - 3y + 8z - 76 = 0

Video solution to 23

24. The distance from the point (1, -2, 1) to the plane 2x + y + az - 4 = 0 is 2 units. Solve for a.

Answer: a = -2/3 or a = -2

Video solution to 24

25. Determine the distance between the lines

line 1: r = (-3, 4, 0) + t(1, -6, 3)      and      line 2: r = (2, -1, -5) + s(3, 8, 9)

Video solution to 25

26. Determine the coordinates of points on line 1 and line 2 that produce the minimal distance between the lines

line 1: r = (1, 2, -1) + t(1, 1, 2)      and      line 2: r = (-4, 4, -5) + s(1, -1, 1)

Answer: A (3/14, 17/14, -18/7)   and   B (-6/7, 6/7, -13/7)

Video solution to 26

Chapter 9 Review

27.

Show that the points A(5, 10, -40), B(0, 60, 0), C(-30,6,4) and D(-15, 15, -10) all lie on the same plane.

Video solution to 27

28.

A perpendicular line is drawn from the point A(26, -1, -2) to the plane 4x-8y+3z+72=0 and meets the plane at point B.  Determine the coordinates of point B