Complete Solutions Manual for
Multivariable Calculus
Part 2: Chapter 11-16
Calculus
ELEVENTH EDITION
Ron Larson
Bruce Edwards
,Contents
Chapter 11: Vectors and the Geometry of Space ....................................................................... 1090
Chapter 12: Vector-Valued Functions ....................................................................................... 1175
Chapter 13: Functions of Several Variables .............................................................................. 1250
Chapter 14: Multiple Integration................................................................................................ 1376
Chapter 15: Vector Analysis ...................................................................................................... 1470
Chapter 16: Additional Topics in Differential Equations .......................................................... 1568
, C H A P T E R 1 1
Vectors and the Geometry of Space
Section 11.1 Vectors in the Plane ..........................................................................1091
Section 11.2 Space Coordinates and Vectors in Space ........................................1101
Section 11.3 The Dot Product of Two Vectors.....................................................1110
Section 11.4 The Cross Product of Two Vectors in Space ..................................1118
Section 11.5 Lines and Planes in Space ................................................................1125
Section 11.6 Surfaces in Space ..............................................................................1137
Section 11.7 Cylindrical and Spherical Coordinates ............................................1141
Review Exercises ......................................................................................................1152
Problem Solving .......................................................................................................1159
© 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
, C H A P T E R 1 1
Vectors and the Geometry of Space
Section 11.1 Vectors in the Plane
1. Answers will vary. Sample answer: A scalar is a real 9. (b) v = 5 − 2, 5 − 0 = 3, 5
number such as 2. A vector is represented by a directed
line segment. A vector has both magnitude and direction. (c) v = 3i + 5 j
π (a), (d) y
For example 3, 1 has direction and a magnitude of
6 5
(3, 5) (5, 5)
2. 4
3
v
2. Notice that v = 6, − 7 = 2 − ( − 4), −1 − 6 = QP. 2
1
Hence, Q is the initial point and P is the terminal point. (2, 0)
x
−1 1 2 3 4 5
−1
3. (a) v = 5 − 1, 4 − 2 = 4, 2
(b) y
10. (b) v = 3 − 4, 6 − ( −6) = −1, 12
5
(c) v = −i + 12 j
4
y
3
(a), (d)
(4, 2) (− 1, 12)
2 v
8
1 v
6 (3, 6)
x 4
1 2 3 4 5 2
x
−8 −6 −4 −2 2 6 8 10
4. (a) v = −4 − 2, −3 − ( −3) = −6, 0 −4
−6 (4, − 6)
y
(b)
4 11. (b) v = 6 − 8, −1 − 3 = −2, − 4
2
(− 6, 0)
(c) v = −2i − 4 j
v
x y
−8 −6 −4 −2 (a), (d)
−2 6
−4 4 (8, 3)
2 v
x
5. u = 5 − 3, 6 − 2 = 2, 4 −4 −2 2 4
(6, − 1)
8
v = 3 − 1, 8 − 4 = 2, 4 (− 2, − 4)
−6
u = v
12. (b) v = −5 − 0, −1 − ( −4) = −5, 3
6. u = 1 − ( −4), 8 − 0 = 5, 8
(c) v = −5i + 3j
v = 7 − 2, 7 − ( −1) = 5, 8
(a) and (d). y
u = v
(−5, 3) 4
7. u = 6 − 0, − 2 − 3 = 6, − 5 v 2
v = 9 − 3, 5 − 10 = 6, − 5 x
−6 −4 −2 2
u = v (−5, −1) −2
(0, −4)
8. u = 11 − ( −4), − 4 − ( −1) = 15, − 3
v = 25 − 0, 10 − 13 = 15, − 3
u = v
© 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 1091
Multivariable Calculus
Part 2: Chapter 11-16
Calculus
ELEVENTH EDITION
Ron Larson
Bruce Edwards
,Contents
Chapter 11: Vectors and the Geometry of Space ....................................................................... 1090
Chapter 12: Vector-Valued Functions ....................................................................................... 1175
Chapter 13: Functions of Several Variables .............................................................................. 1250
Chapter 14: Multiple Integration................................................................................................ 1376
Chapter 15: Vector Analysis ...................................................................................................... 1470
Chapter 16: Additional Topics in Differential Equations .......................................................... 1568
, C H A P T E R 1 1
Vectors and the Geometry of Space
Section 11.1 Vectors in the Plane ..........................................................................1091
Section 11.2 Space Coordinates and Vectors in Space ........................................1101
Section 11.3 The Dot Product of Two Vectors.....................................................1110
Section 11.4 The Cross Product of Two Vectors in Space ..................................1118
Section 11.5 Lines and Planes in Space ................................................................1125
Section 11.6 Surfaces in Space ..............................................................................1137
Section 11.7 Cylindrical and Spherical Coordinates ............................................1141
Review Exercises ......................................................................................................1152
Problem Solving .......................................................................................................1159
© 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
, C H A P T E R 1 1
Vectors and the Geometry of Space
Section 11.1 Vectors in the Plane
1. Answers will vary. Sample answer: A scalar is a real 9. (b) v = 5 − 2, 5 − 0 = 3, 5
number such as 2. A vector is represented by a directed
line segment. A vector has both magnitude and direction. (c) v = 3i + 5 j
π (a), (d) y
For example 3, 1 has direction and a magnitude of
6 5
(3, 5) (5, 5)
2. 4
3
v
2. Notice that v = 6, − 7 = 2 − ( − 4), −1 − 6 = QP. 2
1
Hence, Q is the initial point and P is the terminal point. (2, 0)
x
−1 1 2 3 4 5
−1
3. (a) v = 5 − 1, 4 − 2 = 4, 2
(b) y
10. (b) v = 3 − 4, 6 − ( −6) = −1, 12
5
(c) v = −i + 12 j
4
y
3
(a), (d)
(4, 2) (− 1, 12)
2 v
8
1 v
6 (3, 6)
x 4
1 2 3 4 5 2
x
−8 −6 −4 −2 2 6 8 10
4. (a) v = −4 − 2, −3 − ( −3) = −6, 0 −4
−6 (4, − 6)
y
(b)
4 11. (b) v = 6 − 8, −1 − 3 = −2, − 4
2
(− 6, 0)
(c) v = −2i − 4 j
v
x y
−8 −6 −4 −2 (a), (d)
−2 6
−4 4 (8, 3)
2 v
x
5. u = 5 − 3, 6 − 2 = 2, 4 −4 −2 2 4
(6, − 1)
8
v = 3 − 1, 8 − 4 = 2, 4 (− 2, − 4)
−6
u = v
12. (b) v = −5 − 0, −1 − ( −4) = −5, 3
6. u = 1 − ( −4), 8 − 0 = 5, 8
(c) v = −5i + 3j
v = 7 − 2, 7 − ( −1) = 5, 8
(a) and (d). y
u = v
(−5, 3) 4
7. u = 6 − 0, − 2 − 3 = 6, − 5 v 2
v = 9 − 3, 5 − 10 = 6, − 5 x
−6 −4 −2 2
u = v (−5, −1) −2
(0, −4)
8. u = 11 − ( −4), − 4 − ( −1) = 15, − 3
v = 25 − 0, 10 − 13 = 15, − 3
u = v
© 2018 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 1091
