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)
b). line: r = (-2, 1, 8) + t(6, -3, -2)
plane: 2x + 2y + 3z - 11 = 0
Answer: no intersection
C). Line: r = (0, -5, -9) + t(4, 1, 3)
Plane: x + 8y - 4z = 0
Answer: line lies on plane
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)
Answer: lines are coincident
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
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)
D) line: r = (-3, 1, 7) + s(1, 2, -5)
line: r = (3, 9, 8) + t(5, 6, -3)
Answer: lines are skew
3. Determine the point at which the line r = (-3, 8, 4) + t (1, -2, 3) intersects the xz-plane
Answer : (1, 0, 16)
4. Determine whether each of the following lines has a y-intercept
a) r = (10, 7, -15) + t(2, 4, -3)
Answer: yes (0, -27, 0)
B). r = (-7, 8, 3) + t(9, -1, -8)
Answer: no
Chapter 9.2 - Systems of Equations (and Appendix work with Matrices)
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 )
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
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)
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.
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
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
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
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
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
answer : (3/4, -2, -3)
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)
Answer: 112 sqrt(65) / 65
b) Line 1: 3x - 8y + 27 = 0
Line 2: 3x - 8y - 9 = 0
Answer: 36 sqrt(73) / 73
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
Answer: 21 sqrt(130) / 26
b) P(7, 6); L: r = (7, 6) + t (8, -1)
Answer: 23 sqrt(5) / 4
18. Determine the distance from the point (-7, -1, 5) to the line
r = (-8, 2, -1) + t(11, -4, 1)
Answer: approximately 6.3
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)
Answer: approximately 12.3
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)
Answer: (30, 9, -7)
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'
Answer: (2, 0, 0)
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
Answer: 142 sqrt(77) / 77
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
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)
Answer: 2 sqrt(10)
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)
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.
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
Answer: (18, 15, -8)
29.
Point (-2, -8, 6) is reflected in the plane with equation 3x+7y-z-50=0. Determine the coordinates of the image point.
Answer:(10, 20, 2)