Chapter 10
1. Compute a + b: a = 8, 2 , b = 7, –1
A)
56, –2
B)
1, 3
C)
8, 2, 7, –1
D)
15,1
Ans: D
Difficulty: Easy
2. Compute –3a – 4b: a = –6,15 , b = –13,13
A)
–19, 28
B)
70, –97
C)
–34, 7
D)
70, 28
Ans: B
Difficulty: Easy
3. Compute ||–4a + 5b||: a = –6i – 5 j, b = 3i + 6 j
A)
4021
B)
2101
C)
61 + 3 5
D)
2245
Ans: A Difficulty: Moderate
Page 455
4. Illustrate the sum a + b graphically.
a = –2i – j; b = –3i + 2j
A)
B)
C)
Page 456
D)
Ans: C
Difficulty: Easy
Page 457
5. Illustrate the difference a – b graphically.
a = –i + 4j; b = –3i + 3j
A)
B)
C)
Page 458
D)
Ans: C
Difficulty: Easy
6. Determine whether the vectors a and b are parallel.
a = 4, –2 , b = –6,16
A) parallel
B) not parallel
Ans: A Difficulty: Easy
7. Determine whether the vectors a and b are parallel.
a = 6, –2 , b = 2, –2
A) parallel
B) not parallel
Ans: B Difficulty: Easy
Page 459
8. Find the vector with initial point A and terminal point B.
A = (–1, 6), B = (3, 9)
JJJG
A)
AB = –4, –3
JJJG
B)
AB = 2,15
JJJG
C)
AB = 3,9
JJJG
D)
AB = 4,3
Ans: D
Difficulty: Easy
9. Find a unit vector in the same direction as the given vector.
3i + 11j
3
11
A)
i+ j
14 14
1
1
B)
i+
j
130
130
3
11
C)
i+
j
130
130
3
11
D)
i+
j
3
11
Ans: C Difficulty: Easy
10. Find a vector with the given magnitude in the same direction as the given vector.
magnitude 6, v = i – 2j
A) 6i – 12j
6
12
i–
j
B)
5
5
1
2
C)
i–
j
5
5
D) 6i
Ans: B Difficulty: Easy
11. Suppose that there are two forces acting on a barge being towed along a river. One force
is exerted due south by the river current and has a magnitude of 3 units (the exact nature
of the force unit is unimportant). The other force is exerted by a tugboat and has a
magnitude of 7 units towards the north and 3 units towards the east. What is the net force
acting on the barge?
A) 7 units towards the north, 0 units towards the east
B) 10 units towards the north, 3 units towards the east
C) 4 units towards the north, 3 units towards the east
D) 3 units towards the north, 4 units towards the east
Ans: C Difficulty: Easy
Page 460
12. The motor of a small boat produces a speed of 4 mph in still water. The boat is travelling
in flowing water with a current velocity given by –3, –3 . In what direction should the
boat head to travel due east? (Represent the direction with a unit vector.)
A) i
1
1
B)
i+
j
2
2
1
3
C)
7i + j
4
4
D) The boat cannot travel due east against this current.
Ans: C Difficulty: Moderate
13. Find the distance between the given points.
(–2, –7, 5), (6, 0, 7)
3 13
A)
B) 117
C)
209
D) 209
Ans: A Difficulty: Easy
14. Compute –2a – b.
a = 6, 0, –7 , b = –2, 7, –9
A)
–10, –7, 23
B)
8, –7, 2
C)
4, 7, –16
D)
–14, 7,5
Ans: A
Difficulty: Easy
15. Compute ||–3a + 4b||: a = 4i + 4 j – 3k , b = –2i + j + 6k
A)
1553
B)
5 41
41 + 41
C)
D)
2 113
Ans: A Difficulty: Moderate
Page 461
16. Find two unit vectors parallel to the given vector.
–5, –1, –2
A)
B)
5
1
2
– ,– ,–
2 10 5
5 1 2
, ,
2 10 5
–
5
1
2
,–
,–
26
26
26
5
1
2
,
,
26 26 26
C)
–
5
1
2
,–
,–
30
30
30
5
1
2
,
,
30 30 30
D)
–
–
Ans: C
5
1
2
,–
,–
30
30
30
1
5
2
,–
,–
30
30
30
Difficulty: Moderate
Page 462
17. Find two unit vectors parallel to the given vector.
From 1, –7,14 to 4, 0, 7
3 1 1
, ,
49 3 3
3
1 1
– ,– ,–
49 3 3
A)
–
B)
3
7
7
,–
,
58
58 58
3
7
7
,
,–
58 58
58
C)
–
3
7
7
,–
,
107
107 107
3
7
7
,
,–
107 107
107
D)
–
–
Ans: C
3
7
7
,–
,
107
107 107
7
3
7
,–
,
107
107 107
Difficulty: Moderate
18. Write the given vector as the product of its magnitude and a unit vector. (Write the unit
vector so that it is in the same direction as the given vector. This means that the signs of
the components of the unit vector should be the same as the signs of the corresponding
components of the given vector.)
–1,3, –2
A)
6
1 3 2
, ,
6 2 3
B)
10 –
1
3
2
,
,–
10 10
10
C)
14 –
1
3
2
,
,–
14 14
14
D)
1
3
2
,
,–
14 14
14
Difficulty: Moderate
14 –
Ans: C
Page 463
19. Write the given vector as the product of its magnitude and a unit vector. (Write the unit
vector so that it is in the same direction as the given vector. This means that the signs of
the components of the unit vector should be the same as the signs of the corresponding
components of the given vector.)
From 4, 9, –5 to 0, 2, –8
A)
84
4 7 3
, ,
21 12 28
B)
65
4
7
3
,
,
65 65 65
C)
74
4
7
3
,
,
74 74 74
D)
74
Ans: C
4
7
3
,
,
74 74 74
Difficulty: Moderate
20. Find a vector with the given magnitude and in the same direction as the given vector.
Magnitude 13, v = 3, –2, –7
A)
B)
39
26
91
,–
,–
62
62
62
39, –26, –91
3
2
7
,–
,–
62
62
62
C)
D)
Ans: A
–
39 26 91
,
,
62 62 62
Difficulty: Moderate
21. Find a vector with the given magnitude and in the same direction as the given vector.
Magnitude 8, v = 5i – j + k
40
8
8
A)
i–
j+
k
3 3 3 3
3 3
B)
40i – 8 j + 8k
5
1
1
C)
i–
j+
k
3 3 3 3
3 3
40
8
8
–
D)
i+
j–
k
3 3 3 3
3 3
Ans: A Difficulty: Moderate
Page 464
22. Find an equation of the sphere with radius r and center (a, b, c).
r = 4, (a, b, c) = (–8, 2, 5)
A)
B)
C)
D)
( x + 8 )2 + ( y – 2 )2 + ( z – 5 )2 = 4
( x + 8)2 + ( y – 2 )2 + ( z – 5)2 = 16
( x – 8)2 + ( y + 2 )2 + ( z + 5)2 = 16
( x – 8 )2 + ( y + 2 )2 + ( z + 5 )2 = 4
Ans: B
Difficulty: Easy
23. Identify the plane as parallel to the xy-plane, xz-plane, or yz-plane.
x=4
A) yz-plane
B) xz-plane
C) xy-plane
D) xz-plane and xy-plane
Ans: A Difficulty: Easy
JJJG
JJJG
24. Find the displacement vectors PQ and QR and determine whether the points P = (–8, –
7, 0), Q = (–1, 0, 7), and R = (–3, –2, 5) are collinear (on the same line).
A) collinear
B) not collinear
Ans: A Difficulty: Easy
25. Use vectors to determine if the points (–7, –18, –8), (2, –3, –3), (7, –2, –15), and (–2, –
17, –20) form a square.
A) yes
B) no
Ans: A Difficulty: Moderate
26. A small communications satellite is being manuevered by two on-board thrusters. One
thruster generates a force with magnitude 15 newtons in a direction –3, –4,1 . In
response to the applied thrusts, the satellite moves under a total force of –10, –13,13
(the components are in units of newtons). Find the thrust vector for the second thruster.
45
60
15
A)
–10 +
, –13 +
,13 –
26
26
26
B)
C)
45
60
15
, –13 –
,13 +
26
26
26
35, 47, –2
–10 –
D) cannot be calculated
Ans: A Difficulty: Moderate
Page 465
27. Compute ||4a + 3b||: a = 5, –7, 7, –6 , b = 6,3, –1,5
9 31
A)
B)
3183
C)
159 + 71
D)
1498
Ans: A Difficulty: Moderate
28. Compute a ⋅b .
a = 5, –4,9 , b = –2,9, 6
A)
B)
C)
–105, –48,37
8
–10, –36,54
D) 4312
Ans: B Difficulty: Easy
29. Compute a ⋅b .
a = –9i + 2 j + 4k , b = 7i + 2 j + k
A)
–6, 37, –32
B)
C)
–55
–63, 4, 4
D) 4001
Ans: B Difficulty: Easy
30. Compute the angle between the vectors.
a = 5, –4, 4 , b = 2, 7,9
A)
B)
C)
 3 
cos −1 
 ≈ 1.57
 1273 
 6 
cos −1 
 ≈ 1.37
 10 
cos −1 ( 0 ) = 0
 18 
cos −1 
 ≈ 1.36
 7638 
Ans: D Difficulty: Moderate
D)
Page 466
31. Determine if the vectors are orthogonal.
a = 5i – 4 j + 13k , b = –5i – 3 j + k
A) orthogonal
B) not orthogonal
Ans: A Difficulty: Easy
32. Determine if the vectors are orthogonal.
a = –7i + 6 j – 5k , b = 6i + 3j – 8k
A) orthogonal
B) not orthogonal
Ans: B Difficulty: Easy
33. Find a vector perpendicular to the given vector. Show all your work.
0,8, –3
Ans: Answer may vary. Find a non-zero vector x, y, z such that 0,8, –3 ⋅ x, y, z = 0 .
For example, 1,1,
8
.
3
Difficulty: Moderate
34. Find a vector perpendicular to the given vector. Show all your work.
–7i – 8 j – 9k
Ans: Answer may vary. Find a non-zero vector x, y, z such that
–7, –8, –9 ⋅ x, y, z = 0 . For example, 1,1, –
Difficulty: Moderate
35. Find compb a for the given vectors.
a = 7, –2 , b = 9, 7
49
130
49
B)
53
49
C)
9, 7
130
49
D)
9, 7
53
Ans: A Difficulty: Moderate
A)
Page 467
5
.
3
36. Find compb a for the given vectors.
a = 3,5, –3 , b = –4, –9, –7
36
146
36
B)
43
18
C)
–4, –9, –7
73
36
D)
–4, –9, –7
43
Ans: A Difficulty: Moderate
A)
37. Find projb a for the given vectors.
a = –7, –8 , b = 5, –8
29
89
29
B)
113
29
C)
5, –8
89
29
D)
5, –8
113
Ans: C Difficulty: Moderate
A)
38. Find projb a for the given vectors.
a = 7,1, –9 , b = –7, –3,9
133
139
133
B)
131
133
C)
–7, –3,9
139
133
D)
–7, –3,9
131
Ans: C Difficulty: Moderate
A)
Page 468
39. A constant force of 15, –35 pounds moves an object in a straight line from the point (0,
0) to the point (23, –19). Compute the work done.
A) –962
B) 0
C) –320
D) 1010
Ans: D Difficulty: Moderate
40. A constant force of 1,8 pounds moves an object in a straight line a distance of 15 feet,
and the work done is 95 ft-lb. If the motion of the object started at the point (0, 0), find
the coordinates of the final position of the object. [Assume that the final position is
somewhere in the first quadrant of the coordinate system.]
 32 14 + 19 –4 14 + 152 
,
A)


13
13


B)
16, 23
C)
15, 0
D) The final position cannot be determined.
Ans: A Difficulty: Difficult
41. Who is doing more work: a weight lifter who is holding a 450-pound barbell motionless
over his head, or senior citizen sitting on a park bench? Explain.
Ans: The two individuals are doing the same amount of work: zero! The weight lifter is
exerting a much greater force, but in both cases, the displacement d is zero.
Difficulty: Easy
42. Prove that projc(a + b) = projc a + projc b for any non-zero vectors a, b, and c.
Ans:
(a + b ) ⋅ c c
projc ( a + b ) =
2
c
=
a ⋅c + b ⋅c
c
2
c
 a ⋅c b ⋅c 
c
=
+
 c 2 c 2


a ⋅c
b ⋅c
=
c+ 2 c
2
c
c
= projca + projcb
Difficulty: Easy
Page 469
43. Use the Cauchy-Schwartz Inequality in n dimensions to show that
1/ 2
1/ 2
 n
 n
2/m 
2( m−1) / m 
≤
a
a
a




∑ k ∑ k  ∑ k

k =1
 k =1
  k =1

where m is an odd natural number.
Ans: Cauchy-Schwartz Inequality: b ⋅ c ≤ b c
n
Let bk = ak1/ n and ck = ak (
. Then,
1/ 2
 n
2
=  ∑ ak1/ n 


 k =1

1/ 2
2
 n
n −1 / n 
=  ∑ ak ( )



 k =1

 n 2
b =  ∑ bk 


 k =1

 n

c =  ∑ ck 2 


 k =1

Also,
b ⋅c =
n −1) / n
(
)
(
,
1/ 2
 n
2/n 
=  ∑ ak



 k =1

1/ 2
)
1/ 2
1/ 2
 n
2 n −1 / n 
=  ∑ ak ( ) 


 k =1

n
∑ bk ck
k =1
Substituting into the Cauchy-Schwartz Inequality:
1/ 2
1/ 2
 n
 n
2/n 
2( n −1) / n 
b
c
≤
a
a



 .
∑ k k ∑ k  ∑ k

k =1
 k =1
  k =1

Further, the Cauchy-Schwartz Inequality is true even if b and c happen to be
n
parallel (in which case b ⋅ c is at a maximum and is equal to
n
∑ bk ck
k =1
1/ 2
1/ 2
 n
 n
2/n 
2( n −1) / n 
b
c
≤
b
c
≤
a
a



 .
∑ k k ∑ k k ∑ k  ∑ k

k =1
k =1
 k =1
  k =1

Dropping the leftmost expression and substituting for bk and ck:
n
n
n
1/ n
∑ ( ak )
k =1

( ak )( n−1) / n ≤  ∑ ak
n

 k =1
1/ 2
 n
2/n 
∑ ak ≤  ∑ ak 
k =1
 k =1

Difficulty: Difficult
n
1/ 2 n

2/n 



 ∑ ak

 k =1
1/ 2
 n
2( n −1) / n 
 ∑ ak



 k =1

Page 470
1/ 2
2( n −1) / n 



). That is,
44. Compute the given determinant.
8 –2 7
6 –6 –7
3 1 –2
A)
B)
C)
D)
Ans:
–338
8
338
–8
C Difficulty: Moderate
45. Compute the cross product a × b.
a = 2, –4, –4 , b = 2, 6,8
A)
–8, 24, 20
B)
–56, 8, 4
C)
–8, 24, –20
D)
–8, –24, 20
Ans: D
Difficulty: Moderate
46. Compute the cross product a × b.
a = –4i – 9 j, b = 9i – 7k
A)
63i – 28 j + 81k
B) 109k
C)
63i + 28 j + 81k
D)
53k
Ans: A Difficulty: Moderate
47. Find two unit vectors orthogonal to the two given vectors.
a = –2, 2,5 , b = 2,8, 4
A)
±
–16
1
–10
,
,
357 357 357
B)
±
–16
9
–10
,
,
437 437 437
C)
±
–8 9 –5
,
,
2 2 2 2
D)
±
Ans: B
–16
9
–10
,
,
613 613 613
Difficulty: Moderate
Page 471
48. Find two unit vectors orthogonal to the two given vectors.
a = –2i – 5 j – k , b = 4i – 9 j + 4k

±


B)
±


C)
±


D)
±

Ans: B
A)
–29
12
38

i–
j+
k
2429
2429
2429 
–29
4
38

i+
j+
k
2301
2301
2301 
–29
4
38 
i+
j+
k
33
33
33 
–29
4
38

i+
j+
k
269
269
269 
Difficulty: Moderate
49. Use the cross product to determine the angle betweeen the vectors, assuming that
0 ≤θ ≤
A)
B)
C)
D)
Ans:
π
. Round to the nearest thousandth.
2
a = 9,9, –2 , b = 7, –8, –7
1.571
1.528
1.540
0.030
C Difficulty: Moderate
50. Use the cross product to determine the angle betweeen the vectors, assuming that
0 ≤θ ≤
π
. Round to the nearest thousandth.
2
a = 4i – 6 j + 3k , b = –8i + 2 j + 9k
A) 1.571
B) 1.318
C) 1.392
D) 0.179
Ans: C Difficulty: Moderate
51. Find the distance from the point Q to the given line. Round to the nearest thousandth.
Q = ( –6, 0, –3) , line through ( –8, –5, 0 ) and (1, –8,9 )
A)
B)
C)
D)
Ans:
6.164
0.375
5.140
5.885
D Difficulty: Moderate
Page 472
52. If you apply a force of magnitude 38 pounds at the end of an 12-inch wrench at an angle
π
to the wrench, find the magnitude of the torque applied to the bolt. Round to the
3
nearest tenth of an inch-pound.
A) 394.9 in-lb
B) 228.0 in-lb
C) 456.0 in-lb
D) 789.8 in-lb
Ans: A Difficulty: Easy
of
53. Assume that the ball is moving into the page (and away from you) with the indicated
spin. Determine the direction of the Magnus force.
A)
B)
C)
D)
E)
Ans:
to the right
to the left
up
down
The Magnus force is zero.
A Difficulty: Easy
Page 473
54. Assume that the ball is moving into the page (and away from you) with the indicated
spin. Determine the direction of the Magnus force.
A)
B)
C)
D)
E)
Ans:
to the right
to the left
up
down
The Magnus force is zero.
E Difficulty: Easy
55. Which of the following statements are true?
i. a × b = b × a
ii. a ⋅ b = b ⋅ a
iii. a ⋅ ( b × c ) = ( a ⋅ b ) × ( a ⋅ c )
A)
B)
C)
D)
E)
Ans:
i only
ii only
i and ii only
i and iii only
iii only
B Difficulty: Easy
56. Find the indicated area.
Area of the parallelogram with two adjacent sides formed by –5, –3 and –4,9
–57
2
B) –33
C) –57
–33
D)
2
Ans: C Difficulty: Easy
A)
Page 474
57. Find the indicated area.
Area of the triangle with vertices ( 0, 0, 0 ) , ( 5, –2, 7 ) , and ( 4, 4, –1)
1
2
11
B)
2
1
C)
2
5
D)
2
Ans: A
A)
2549
13
65
Difficulty: Moderate
58. Find the indicated volume.
Volume of the parallelpiped with three adjacent edges formed by ( 9, –1, –4 ) ,
( 3, 4,9 ) , and ( 5,9,3)
A) 595
B)
595
C)
685
D) 685
Ans: D Difficulty: Moderate
59. Use geometry to identify the cross product.
i × ( j× k )
A)
B)
C)
D)
Ans:
0
j
k
–i
A
Difficulty: Easy
60. Use geometry to identify the cross product.
j × ( 7k )
A)
B)
C)
D)
Ans:
–7j
0
7i
7k
C Difficulty: Easy
Page 475
61. Use the parallelpiped volume formula to determine if the vectors are coplanar.
–1, 0, –3 , 5,9, –7 , –7, –8,8
A) coplanar
B) not coplanar
Ans: B Difficulty: Moderate
62. Use the parallelpiped volume formula to determine if the vectors are coplanar.
7, –6,87 , 1, –7,37 , –8,5, –92
A) coplanar
B) not coplanar
Ans: A Difficulty: Moderate
Page 476
63. Show that a × ( b × c ) = ( a ⋅ c ) b − ( a ⋅ b ) c . Show all of your work.
Ans: Let a = a1, a2 , a3 , b = b1, b2 , b3 , and c = c1, c2 , c3 .
i j k
a × ( b × c ) = a × b1 b2 b3
c1 c2 c3
= a × b2c3 − b3c2 , b3c1 − b1c3 , b1c2 − b2c1
i
a1
=
j
a2
k
a3
b2c3 − b3c2 b3c1 − b1c3 b1c2 − b2c1
(
)
(
) (
)
(
)
= a2 b1c2 − b2c1 − a3 b3c1 − b1c3 , a3 b2c3 − b3c2 − a1 b1c2 − b2c1 ,
(
)
(
a1 b3c1 − b1c3 − a2 b2c3 − b3c2
)
( a2c2 + a3c3 ) b1 − ( a2b2 + a3b3 ) c1, ( a1c1 + a3c3 ) b2 − ( a1b1 + a3b3 ) c2 ,
( a1c1 + a2c2 ) b3 − ( a1b1 + a2b2 ) c3
= ( a1c1 + a2c2 + a3c3 ) b1 − ( a1b1 + a2b2 + a3b3 ) c1,
( a1c1 + a2c2 + a3c3 ) b2 − ( a1b1 + a2b2 + a3b3 ) c2 ,
( a1c1 + a2c2 + a3c3 ) b3 − ( a1b1 + a2b2 + a3b3 ) c3
= ( a1c1 + a2c2 + a3c3 ) b1, b2 , b3 − ( a1b1 + a2b2 + a3b3 ) c1, c2 , c3
=
= (a ⋅ c) b − (a ⋅ b ) c
Difficulty: Moderate
64. In many farmers' barns and in many mechanics' shops, you can find a stout steel pipe
about 2 to 3 feet in length with one end pinched so that the pinched end fits snugly over a
wrench handle. What might explain the common presense of this object in these settings?
Explain your answer in terms of the vector concepts you have learned.
Ans: Answers may vary.
The purpose of the pipe is to lengthen the wrench handle. For a given force vector,
the torque exerted by the wrench varies directly with the length of the wrench
handle.
Difficulty: Easy
Page 477
65. Find parametric equations of the line through ( –2, –7, 6 ) parallel to 3, 6, 7 .
Ans: Answers may vary.
x + 2 = 3t , y + 7 = 6t , z – 6 = 7t
Difficulty: Easy
66. Find parametric equations of the line through ( –1, –7, –9 ) and (1, –9, 6 ) .
Ans: Answers may vary.
x + 1 = 2t , y + 7 = –2t , z + 9 = 15t
Difficulty: Easy
67. Find parametric equations of the line through ( 8, 6,9 ) and perpendicular to both
6, 0, –1 and –1, –3, 7 .
Ans: Answers may vary.
x – 8 = –3t , y – 6 = –41t , z – 9 = –18t
Difficulty: Easy
68. Find symmetric equations of the line through ( –5, –9, –4 ) and parallel to –1,9, 6 .
x +1 y – 9 z – 6
=
=
–5
–9
–4
x+5 y +9 z +4
B)
=
=
–1
9
6
– x + 5 9 y + 9 6z + 4
C)
=
=
–1
9
6
x–5 y–9 z –4
D)
=
=
–1
9
6
Ans: B Difficulty: Easy
A)
69. Find symmetric equations of the line through ( 7,3, –1) and ( 4, –8, –7 ) .
x + 3 y + 11 z + 6
=
=
7
3
–1
x + 7 y + 3 z –1
=
=
B)
11
–5
–8
x – 7 y – 3 z +1
C)
=
=
–3
–11
–6
x + 7 y + 3 z –1
D)
=
=
–3
–11
–6
Ans: C Difficulty: Easy
A)
Page 478
70. Find symmetric equations of the line through ( –1, 2,3) and normal to the plane
–4 x – 8 y + 7 z = –1 .
x –1 y + 2 z + 3
A)
=
=
–4
–8
7
x +4 y +8 z – 7
B)
=
=
–1
2
3
x +1 y – 2 z – 3
C)
=
=
–4
–8
7
–4 x – 1 –8 y + 2 7 z + 3
D)
=
=
–4
–8
7
Ans: C Difficulty: Easy
71. Find the angle between the lines. Round to the nearest hundredth of a radian.
 x = –2 – 8t
 x=9–s


 y = 2+t
 y = 2 – 2s
 z = 7 – 3t
 z = 6 – 2s


A)
B)
C)
D)
Ans:
2.66
1.09
2.05
0.44
B Difficulty: Moderate
72. Determine if the lines are parallel, skew or intersect.
 x = 6 + 2t
 x =8– s


 y = –4 – 6t
 y = –10 + 3s
 z = –4 + 6t
 z = 2 – 3s


A)
B)
C)
Ans:
parallel
skew
intersect
A Difficulty: Moderate
73. Determine if the lines are parallel, skew or intersect.
 x = 8 – 5t
 x = 5 – 4s


 y = 7 + 4t
 y = 15 + 6 s
 z = –3 – 5t
 z = 18 + 8s


A)
B)
C)
Ans:
parallel
skew
intersect
C Difficulty: Moderate
Page 479
74. Determine if the lines are parallel, skew or intersect.
 x = –3 + 6t
 x = –4 + 4 s


 y = 2 + 6t
 y = 2 + 4s
 z = –9 – 5t
 z = 4 – 7s


A)
B)
C)
Ans:
parallel
skew
intersect
B Difficulty: Moderate
75. Find an equation of the given plane.
The plane containing the point ( 3,5, 4 ) with normal vector 9, –9, –4
A)
B)
C)
D)
Ans:
3x + 5y + 4z = –34
9x – 9y – 4z = 0
9x – 9y – 4z = –34
9x – 9y – 4z = 12
C Difficulty: Easy
76. Find an equation of the given plane.
The plane containing the points ( –6, –4, –4 ) , ( 4,9, –2 ) and
A)
B)
C)
D)
Ans:
( 7,5,3)
4x + 9y – 2z = –52
–3x + 4y – 5z = 22
–3x + 4y – 5z = 0
73x – 44y – 79z = 54
D Difficulty: Moderate
77. Find an equation of the given plane.
The plane containing the point ( –9, –8,1) and perpendicular to the planes –9x –
5y – 8z = 5 and 2x – 3y – z = –6
A) –19x – 25y + 37z = –16
B) –19x – 25y + 37z = 408
C) –9x – 5y – 8z = 113
D) –19x – 25y + 37z = 244
Ans: B Difficulty: Moderate
Page 480
78. Sketch the given plane.
3x – y + 2z = 6
z
y
x
A)
z
3
y
–6
2
x
B)
z
–2
y
6
–3
x
C)
Page 481
z
–2
y
–6
–3
x
D)
z
y
–1
3
2
x
Ans: A
Difficulty: Easy
Page 482
79. Sketch the given plane.
y + 2z = 6
z
y
x
A)
z
2
y
1
6
x
B)
z
3
–1
y
6
x
C)
Page 483
z
1
y
–1
2
x
D)
z
y-z plane
1
–1
y
2
x
Ans: B
Difficulty: Easy
80. Find the intersection of the planes.
9x – 4y + 6z = –74 and 6x + 4y + 3z = –28
A) x = –2 – 12t, y = 2 – 63t, z = –8 – 12t
B) x = + 12t, y = + 63t, z = + 12t
C) x = –2 – 36t, y = 2 + 9t, z = –8 + 60t
D) The planes are parallel.
Ans: C Difficulty: Moderate
Page 484
81. Find the distance between the given objects.
The point ( 8, –2, 2 ) and the plane –6x – 8y + 5z = 5
27
5 5
27
B)
125
C) 0
9
D)
2 2
Ans: A Difficulty: Moderate
A)
82. Find the distance between the given objects.
The planes –x – 8y + z = –45 and –x – 8y + z = –41
A) 4
4
B)
66
C) 0
45
D)
66
Ans: B Difficulty: Moderate
83. Find an equation of the plane containing the lines.
 x = –8 – t
 x = –18 – 6 s


and
y = 4 + t
 y = –8 – 5s
 z = 1 + 2t
 z = 3 + 3s


A)
B)
C)
D)
Ans:
13x – 9y + 11z = –129
13x – 15y + 11z = –153
–x + y + 2z = 14
The lines are skew.
A Difficulty: Moderate
84. Determine whether the given lines are the same.
x = –8 – 8t, y = 5 + 4t, z = –4 + t and x = –8 + 4t, y = 5 + 9t, z = –4 – 7t
A) The lines are the same.
B) The lines are not the same.
Ans: B Difficulty: Easy
Page 485
85. Determine whether the given lines are the same.
x = –5 – 4t, y = 8 – 3t, z = –8 – 4t and x = –13 – 8t, y = 2 – 6t, z = –16 – 8t
A) The lines are the same.
B) The lines are not the same.
Ans: A Difficulty: Easy
86. Determine whether the given planes are the same.
–7(x – 3) – 2(y + 4) + (z + 2) = 0
–7(x – 6) – 2(y – 5) + (z – 50) = 0
A) The planes are the same.
B) The planes are not the same.
Ans: A Difficulty: Easy
87. Determine whether the given planes are the same.
–5(x + 1) + 7(y – 1) + 4(z – 5) = 0
–5(x – 2) + 7(y – 1) + 4(z + 0) = 0
A) The planes are the same.
B) The planes are not the same.
Ans: B Difficulty: Easy
88. Suppose two airplanes fly paths described by the parametric equations
 x = –6 – 3t
 x = 4 + 2s


P1 :  y = 6 – 4t
P2 :  y = 16 + s ,
 z = 2 – 4t
 z = 16 + 3s


where s and t represent time. Determine whether the flight paths intersect and, if so,
whether the planes collide.
A) The paths do not intersect.
B) The paths intersect, but the planes do not collide.
C) The paths intersect and the planes collide.
Ans: C Difficulty: Moderate
89. Which one of the following statements is always true?
A) Two planes always intersect in a line.
d1 − d 2
can be used to calculate the distance between any
B) The equation d =
a 2 + b2 + c 2
two planes.
C) A plane cannot contain two skew lines.
D) The equation a1 ( x − x1 ) + a2 ( x − y1 ) + a3 ( x − z1 ) = 0 describes the plane
containing the point ( x1, y1, z1 ) and containing the vector a1, a2 , a3 .
Ans: C
Difficulty: Easy
Page 486
90. Match the equation to its graph. [The figures below are not necessarily drawn to scale.]
z = 3x 2 + 4 y 2 − 2
A)
B)
C)
D)
Page 487
Ans: D
Difficulty: Easy
Page 488
91. Match the equation to its graph. [The figures below are not necessarily drawn to scale.]
4 x 2 + 9 y 2 + 36 z 2 = 36
A)
B)
C)
D)
Page 489
Ans: A
Difficulty: Easy
Page 490
92. Match the equation to its graph. [The figures below are not necessarily drawn to scale.]
z = y 2 − 2 x2
A)
B)
C)
D)
Page 491
Ans: B
Difficulty: Easy
Page 492
93. Match the equation to its graph. [The figures below are not necessarily drawn to scale.]
4 x2 + y 2 − z 2 = 0
A)
B)
C)
D)
Page 493
Ans: C
Difficulty: Easy
Page 494
94. Match the equation to its graph. [The figures below are not necessarily drawn to scale.]
− x2 + 4 y 2 + 2 z 2 = 4
A)
B)
C)
D)
Page 495
Ans: B
Difficulty: Easy
Page 496
95. Match the equation to its graph. [The figures below are not necessarily drawn to scale.]
−2 x 2 + y 2 − 3 z 2 = 6
A)
B)
C)
D)
Page 497
Ans: A
Difficulty: Easy
Page 498
96. Match the equation to its graph. [The figures below are not necessarily drawn to scale.]
z = cos y
A)
B)
C)
D)
Page 499
Ans: B
Difficulty: Easy
Page 500
97. Match the equation to its graph. [The figures below are not necessarily drawn to scale.]
x+z =3
A)
B)
C)
D)
Page 501
Ans: A
Difficulty: Easy
98. Identify the surface.
3 x 2 + 7 y 2 + 2 z 2 = 10
A) elliptic paraboloid
B) cone
C) ellipsoid
D) hyperbolic paraboloid
Ans: C Difficulty: Easy
99. Identify the surface.
3 x 2 + 9 y 2 − z = –7
A) elliptic paraboloid
B) cone
C) hyperboloid of one sheet
D) hyperbolic paraboloid
Ans: A Difficulty: Easy
100. Identify the surface.
10 x 2 + 2 y 2 − 9 z 2 = 0
A) elliptic paraboloid
B) cone
C) hyperboloid of one sheet
D) hyperbolic paraboloid
Ans: B Difficulty: Easy
101. Identify the surface.
7 x2 + 3 y 2 = 1 + 8z 2
A) hyperboloid of two sheets
B) cone
C) hyperboloid of one sheet
D) hyperbolic paraboloid
Ans: C Difficulty: Easy
Page 502
102. Identify the surface.
5x2 = 1 + 6 y 2 + 2 z 2
A) hyperboloid of two sheets
B) cone
C) hyperboloid of one sheet
D) hyperbolic paraboloid
Ans: A Difficulty: Easy
103. A certain lunar crater is shaped approximately as a paraboloid (see figure). The cross
sections of the crater parallel to the lunar surface are circular, with a maximum radius of
300 meters. The depth of the crater is 36 meters. Find an equation for the surface of the
crater. [Write your equation such that the bottom of the crater is located at (x, y, z) = (0, 0,
0). Assume the z-axis is perpendicular to the lunar surface.]
A)
x2
y2
2
+
− z = 1, x 2 + y 2 ≤ ( 300 )
2500 2500
B)
x2
y2
2
+
= z , x 2 + y 2 ≤ ( 300 )
2500 2500
C)
x2
y2
2
+
= z 2 , x 2 + y 2 ≤ ( 300 )
2500 2500
x2
y2
z
2
+
= , x 2 + y 2 ≤ ( 300 )
300 300 36
Ans: B Difficulty: Moderate
D)
Page 503
104. If x = a sin φ cos θ , y = b sin φ sin θ , and z = sin φ for 0 ≤ θ ≤ 2π and 0 ≤ φ ≤ π , show
x2 y 2
that the point (x, y, z) lies on the upper half of the cone z 2 = 2 + 2 . Show all your
a
b
work.
Ans:
x2 y 2
z2 = 2 + 2
a
b
2
2
a cos θ sin φ )
b sin θ sin φ )
(
(
+
( sin φ ) =
2
2
2
?
a
b
?
sin 2 φ = cos 2 θ sin 2 φ + sin 2 θ sin 2 φ
?
(
sin 2 φ = sin 2 φ cos 2 θ + sin 2 θ
)
sin 2 φ = sin 2 φ
Since z = sin φ is positive for 0 ≤ φ ≤ π , the point lies on the upper half of the
cone.
Difficulty: Easy
Page 504