Linear ALlgebra a modern introduction 4th edition by poole solution manual

Linear ALlgebra A Modern Introduction
4th edition by Poole Solution Manual
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Linear Algebra
A Modern Introduction
FOURTH EDITION
David Poole
Trent University

Prepared by
Roger Lipsett

Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United
States


ISBN-13: 978-128586960-5
ISBN-10: 1-28586960-5

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Contents
1 Vectors
3
1.1 The Geometry and Algebra of Vectors .............................................................................................. 3
1.2 Length and Angle: The Dot Product............................................................................................. 10
Exploration: Vectors and Geometry ...................................................................................................... 25
1.3 Lines and Planes ............................................................................................................................. 27
Exploration: The Cross Product ............................................................................................................ 41
1.4 Applications .....................................................................................................................................44
Chapter Review ....................................................................................................................................... 48
2 Systems of Linear Equations
53
2.1 Introduction to Systems of Linear Equations .............................................................................. 53
2.2 Direct Methods for Solving Linear Systems .................................................................................58
Exploration: Lies My Computer Told Me ............................................................................................. 75
Exploration: Partial Pivoting.................................................................................................................. 75
Exploration: An Introduction to the Analysis of Algorithms .............................................................. 77
2.3 Spanning Sets and Linear Independence ..................................................................................... 79
2.4 Applications .....................................................................................................................................93
2.5 Iterative Methods for Solving Linear Systems ............................................................................. 112

Chapter Review ........................................................................................................................................ 123
3 Matrices
129
3.1 Matrix Operations ........................................................................................................................... 129
3.2 Matrix Algebra ................................................................................................................................ 138
3.3 The Inverse of a Matrix .................................................................................................................. 150
3.4 The LU Factorization ...................................................................................................................... 164
3.5 Subspaces, Basis, Dimension, and Rank ...................................................................................... 176
3.6 Introduction to Linear Transformations....................................................................................... 192
3.7 Applications .................................................................................................................................... 209
Chapter Review ....................................................................................................................................... 230
4 Eigenvalues and Eigenvectors
235
4.1 Introduction to Eigenvalues and Eigenvectors ............................................................................235
4.2 Determinants .................................................................................................................................. 250
Exploration: Geometric Applications of Determinants ...................................................................... 263
4.3 Eigenvalues and Eigenvectors of n × n Matrices.......................................................................... 270
4.4 Similarity and Diagonalization ...................................................................................................... 291
4.5 Iterative Methods for Computing Eigenvalues............................................................................ 308
4.6 Applications and the Perron-Frobenius Theorem ...................................................................... 326
Chapter Review ........................................................................................................................................365
1


2

CONTENTS

5 Orthogonality
371

5.1 Orthogonality in Rn ................................................................................................................................................................................. 371
5.2 Orthogonal Complements and Orthogonal Projections ................................................................ 379
5.3 The Gram-Schmidt Process and the QR Factorization ................................................................ 388
Exploration: The Modified QR Process ................................................................................................. 398
Exploration: Approximating Eigenvalues with the QR Algorithm....................................................... 402
5.4 Orthogonal Diagonalization of Symmetric Matrices ..................................................................... 405
5.5 Applications ..................................................................................................................................... 417
Chapter Review ........................................................................................................................................ 442
6 Vector Spaces
451
6.1 Vector Spaces and Subspaces ......................................................................................................... 451
6.2 Linear Independence, Basis, and Dimension ................................................................................. 463
Exploration: Magic Squares .................................................................................................................... 477
6.3 Change of Basis ............................................................................................................................... 480
6.4 Linear Transformations .................................................................................................................. 491
6.5 The Kernel and Range of a Linear Transformation ...................................................................... 498
6.6 The Matrix of a Linear Transformation ......................................................................................... 507
Exploration: Tiles, Lattices, and the Crystallographic Restriction ...................................................... 525
6.7 Applications ..................................................................................................................................... 527
Chapter Review ........................................................................................................................................ 531
7 Distance and Approximation
537
7.1 Inner Product Spaces ...................................................................................................................... 537
Exploration: Vectors and Matrices with Complex Entries .................................................................... 546
Exploration: Geometric Inequalities and Optimization Problems........................................................ 553
7.2 Norms and Distance Functions ...................................................................................................... 556
7.3 Least Squares Approximation ........................................................................................................ 568
7.4 The Singular Value Decomposition ................................................................................................ 590
7.5 Applications ..................................................................................................................................... 614
Chapter Review ........................................................................................................................................ 625

8 Codes
633
8.1 Code Vectors .......................................................................................................................................... 633
8.2 Error-Correcting Codes ................................................................................................................... 637
8.3 Dual Codes ....................................................................................................................................... 641
8.4 Linear Codes .................................................................................................................................... 647
8.5 The Minimum Distance of a Code ................................................................................................. 650


Chapter 1

Vectors
1.1

The Geometry and Algebra of Vectors
(–2, 3}

1.

(2, 3}

3

2

1

(3, 0}
–2

–1

1

2

3

–1

(3, –2}
–2

2. Since
Σ

Σ Σ Σ Σ Σ
2
3
5
+
=
,
0
−3
−3

Σ

Σ Σ Σ Σ Σ

2
2
4
+
=
,
3
0
−3

Σ

Σ Σ Σ Σ Σ
2
−2
0
+
=
,
3
0
−3

plotting those vectors gives

1

2

3

4

–1

c

b

–2

a
–3

d
–4

–5

3

5

Σ Σ Σ Σ Σ Σ
2
3
5
+
=
,

−3
−2
−5


4

CHAPTER 1. VECTORS
3.

2z

c

b

1

–2

–1

0
0

3

0

1

2

y2

1
–1

a

x
–1

d
–2

4. Since the heads are all at (3, 2, 1), the tails are at
3
0
3
2 − 2 = 0,
1
0
1

3
3
0
2 − 2 = 0,
1

1
0

3
1
2
2 − −2 = 4 ,
1
1
0

#»
5. The four vectors AB are
3

c
2

1

d
a
1

2

3

4

–1

b
–2

In standard position, the vectors are
#»
(a) AB = [4 − 1, 2 − (− 1)] = [3, 3].
#»
(b) AB = [2 − 0, −1 − (−2)] = [2, 1]
Σ Σ
Σ
#» Σ
(c) AB = 1 − 2, 3 − 3 = − 3 , 3
2
2
2 2
#»
6 Σ6
Σ 6 3 2 Σ3 Σ
(d) AB = 1 − 1 , 1 − 1 = − 1 , 1 .
3

2

a

1

c

b
d
–1

1

2

3

3
2
1

−

−1
−1
−2

4
= 3 .
3


1.1. THE GEOMETRY AND ALGEBRA OF VECTORS

5

6. Recall the notation that [a, b] denotes a move of a units horizontally and b units vertically. Then during

the first part of the walk, the hiker walks 4 km north, so a = [0, 4]. During the second part of the
walk, the hiker walks a distance of 5 km northeast. FromΣthe components,
we get
√
√ Σ
5
2
5
2
b = [5 cos 45◦ , 5 sin 45◦ ] =
,
.
2
2
Thus the net displacement vector is
c=a+b=

Σ √
5 2

√ Σ
5 2

,4+

2
7.a + b =

Σ Σ Σ Σ Σ
Σ Σ Σ

3
2
3+2
5
+
=
=
.
0
3
0 +3
3

.

2

3

2

a +b
b

1

a

Σ Σ Σ Σ Σ
Σ Σ Σ

2
−2
4
2 − (−2)
−
=
=
.
8.b −c =
3−3
3
3
0

1

2

3

4

5

3

2

—c

b
1

b—c
1

Σ Σ Σ Σ Σ Σ
3
−2
5
9.d − c =
−
=
.
3
−2
−5

2

1

2

3

3

4

5

4

d

–1

–2

d —c

–3

—c

–4

10.a + d =
Σ Σ
6
.
−2

ΣΣ
Σ Σ
Σ
Σ
3
3

3+3
+
=
=
0
−2
0 + (−2)

–5

a
1

2

3

4

5

6

d

–1

a +d
–2

11. 2a + 3c = 2[0, 2, 0] + 3[1, −2, 1] = [2 · 0, 2 · 2, 2 · 0] + [3 · 1, 3 · (−2), 3 · 1] = [3, −2, 3].
12.
3b − 2c + d = 3[3, 2, 1] − 2[1, −2, 1] + [−1, −1, −2]
= [3 · 3, 3 · 2, 3 · 1] + [−2 · 1, −2 · (−2), −2 · 1] + [−1, −1, −2]
= [6, 9, −1].


6

CHAPTER 1. VECTORS

13.u = [cos 60◦ , sin 60◦ ] =

Σ
1
,
2

√

Σ

3
2

Σ
u+v =

Σ √
Σ

, and v = [cos 210◦ , sin 210◦ ] = − 32, − 1 2, so that

Σ
√ √
1
3 3 1
−
,
−
,
2
2
2
2

Σ
u−v =

Σ
√ √
1
3 3 1
+
,
+
.
2
2
2
2

#»
AB = b − a.
#»
#»
#»
#»
(b) Since OC = AB, we have BC = OC − b = (b − a) − b = − a.
#»
(c) AD = −2a.
#»
#»
#»
(d) CF = −2OC = −2AB = −2(b − a) = 2(a − b).
# » # » #»
(e) AC = AB + BC = (b − a) + (−a) = b − 2a.
#»
#»
#»
#»
(f ) Note that F A and OB are equal, and that DE = −AB. Then

14.(a)

#» #»
#»
#» #»
BC + DE + F A = −a − AB + OB = −a − (b − a) + b = 0.

15. 2(a − 3b) + 3(2b + a)

property e.
distributivity

=

(2a − 6b) + (6b + 3a)

property b.
associativity

=

(2a + 3a) + (−6b + 6b) = 5a.

16.
property e.
distributivity

−3(a − c) + 2(a + 2b) + 3(c − b)

=

(−3a + 3c) + (2a + 4b) + (3c − 3b)

property b.
associativity

=
(−3a + 2a) + (4b − 3b) + (3c + 3c)

= −a + b + 6c.
17.x − a = 2(x − 2a) = 2x − 4a ⇒ x − 2x = a − 4a ⇒ −x = −3a ⇒ x = 3a.
18.
x + 2a − b = 3(x + a) − 2(2a − b) = 3x + 3a − 4a + 2b
x − 3x = −a − 2a + 2b + b
⇒
−2x = −3a + 3b
⇒
3
3
x = a − b.
2
2

⇒

19. We have 2u + 3v = 2[1, −1] + 3[1, 1] = [2 · 1 + 3 · 1, 2 · (−1) + 3 · 1] = [5, 1]. Plots of all three vectors are
2

w

1

v
–1

v
1

2

3

u
–1

u
–2

v

v

4

5

6


1.1. THE GEOMETRY AND ALGEBRA OF VECTORS

7

20. We have −u − 2v = −[−2, 1] − 2[2, −2] = [−(−2) − 2 · 2, −1 − 2 · (−2)] = [−2, 3]. Plots of all three
vectors are

—u
—v
3

—u
2

w
—v
1

u
–2

–1

1

2

v
–1

–2

21. From the diagram, we see that w =
−2u + 4v.

6

—u
5

—u
w

4

v

3

v

2

v

1

v
–1

1

2

3

4

5

6

u
–1

22. From the diagram, we see that w = 2u +
3v.

9

8

u
7

6

w

5

u
4

3

v

2

u

v

1

v
–2

–1

1

2

3

4

5

6

7

23. Property (d) states that u + ( −u) = 0. The first diagram below shows u along with − u. Then, as the
diagonal of the parallelogram, the resultant vector is 0.
Property (e) states that c(u + v) = cu + cv. The second figure illustrates this.

8

CHAPTER 1. VECTORS

cv
c(u + v)
cu
v

cu

u
u

u +v

u
cv

v
—u

24. Let u = [u1, u2, . . . , un] and v = [v1, v2, . . . , vn], and let c and d be scalars in R.
Property (d):
u + (−u) = [u1, u2, . . . , un] + (−1[u1, u2, . . . , un])
= [u1, u2, . . . , un] + [−u1, −u2, . . . , −un]
= [u1 − u1, u2 − u2, . . . , un − un]
= [0, 0, . . . , 0] = 0.
Property (e):
c(u + v) = c ([u1, u2, . . . , un] + [v1, v2, . . . , vn])

= c ([u1 + v1, u2 + v2, . . . , un + vn])
= [c(u1 + v1), c(u2 + v2), . . . , c(un + vn)]
= [cu1 + cv1, cu2 + cv2, . . . , cun + cvn]
= [cu1, cu2, . . . , cun] + [cv1, cv2, . . . , cvn]
= c[u1, u2, . . . , un] + c[v1, v2, . . . , vn]
= cu + cv.
Property (f):
(c + d)u = (c + d)[u1, u2, . . . , un]
= [(c + d)u1, (c + d)u2, . . . , (c + d)un]
= [cu1 + du1, cu2 + du2, . . . , cun + dun]
= [cu1, cu2, . . . , cun] + [du1, du2, . . . , dun]
= c[u1, u2, . . . , un] + d[u1, u2, . . . , un]
= cu + du.
Property (g):
c(du) = c(d[u1, u2, . . . , un])
= c[du1, du2, . . . , dun]
= [cdu1, cdu2, . . . , cdun]
= [(cd)u1, (cd)u2, . . . , (cd)un]
= (cd)[u1, u2, . . . , un]
= (cd)u.
25.u + v = [0, 1] + [1, 1] = [1, 0].
26.u + v = [1, 1, 0] + [1, 1, 1] = [0, 0, 1].


1.1. THE GEOMETRY AND ALGEBRA OF VECTORS

9

27.u + v = [1, 0, 1, 1] + [1, 1, 1, 1] = [0, 1, 0, 0].
28.u + v = [1, 1, 0, 1, 0] + [0, 1, 1, 1, 0] = [1, 0, 1, 0, 0].

29.

30.

+
0
1
2
3

0
0
1
2
3

1
1
2
3
0

2
2
3
0
1

3
3

0
1
2

·
0
1
2
3

012
0 0
0 1
0 2
0 3

3
0
2
0
2

0
3
2
1

0
0
1

2
3
4

1
1
2
3
4
0

2
2
3
4
0
1

3
3
4
0
1
2

4
4
0
1
2

3

·
0
1
2
3
4

0123 4
0 0 0
0 1 2
0 2 4
0 3 1
0 4 3

0
3
1
4
2

+
0
1
2
3
4

0

4
3
2
1

31. 2 + 2 + 2 = 6 = 0 in Z3.
32. 2 · 2 · 2 = 3 · 2 = 0 in Z3.
33. 2(2 + 1 + 2) = 2 · 2 = 3 · 1 + 1 = 1 in Z3.
34. 3 + 1 + 2 + 3 = 4 · 2 + 1 = 1 in Z4.
35. 2 · 3 · 2 = 4 · 3 + 0 = 0 in Z4.
36. 3(3 + 3 + 2) = 4 · 6 + 0 = 0 in Z4.
37. 2 + 1 + 2 + 2 + 1 = 2 in Z3, 2 + 1 + 2 + 2 + 1 = 0 in Z4, 2 + 1 + 2 + 2 + 1 = 3 in Z5.
38. (3 + 4)(3 + 2 + 4 + 2) = 2 · 1 = 2 in Z5.
39. 8(6 + 4 + 3) = 8 · 4 = 5 in Z9.
. 10Σ
40. 2100 = 210 = (1024)10 = 110 = 1 in Z .
41. [2, 1, 2] + [2, 0, 1] = [1, 1, 0] in

11

Z3 3.

42. 2[2, 2, 1] = [2 · 2, 2 · 2, 2 · 1] = [1, 1, 2] in Z33.
43. 2([3, 1, 1, 2] + [3, 3, 2, 1]) = 2[2, 0, 3, 3] = [2 · 2, 2 · 0, 2 · 3, 2 · 3] = [0, 0, 2, 2] in Z44.
2([3, 1, 1, 2] + [3, 3, 2, 1]) = 2[1, 4, 3, 3] = [2 · 1, 2 · 4, 2 · 3, 2 · 3] = [2, 3, 1, 1] in Z45.
44. x = 2 + (−3) = 2 + 2 = 4 in Z5.
45. x = 1 + (−5) = 1 + 1 = 2 in Z6
46. x = 2−1 = 2 in Z3.
47. No solution. 2 times anything is always even, so cannot leave a remainder of 1 when divided by 4.
48. x = 2−1 = 3 in Z5.

49. x = 3−14 = 2 · 4 = 3 in Z5.
50. No solution. 3 times anything is always a multiple of 3, so it cannot leave a remainder of 4 when
divided by 6 (which is also a multiple of 3).
51. No solution. 6 times anything is always even, so it cannot leave an odd number as a remainder when
divided by 8.


10

CHAPTER 1. VECTORS

52. x = 8−19 = 7 · 9 = 8 in Z11
53. x = 2−1(2 + (−3)) = 3(2 + 2) = 2 in Z5.
54. No solution. This equation is the same as 4x = 2 − 5 = − 3 = 3 in Z6. But 4 times anything is even,
so it cannot leave a remainder of 3 when divided by 6 (which is also even).
55. Add 5 to both sides to get 6x = 6, so that x = 1 or x = 5 (since 6 · 1 = 6 and 6 · 5 = 30 = 6 in Z8).
56. (a) All values.
57.(a)

(b) All values.

(c) All values.

All a ƒ= 0 in Z5 have a solution because 5 is a prime number.

(b) a = 1 and a = 5 because they have no common factors with 6 other than 1.
(c) a and m can have no common factors other than 1; that is, the greatest common divisor, gcd, of
a and m is 1.

1.2

Length and Angle: The Dot Product
Σ

Σ Σ Σ
3
·
= (−1) · 3 + 2 · 1 = −3 + 2 = −1.
2
1
Σ Σ Σ Σ
3
4
2. Following Example 1.15, u · v =
·
= 3 · 4 + (−2) · 6 = 12 − 12 = 0.
−2
6
1. Following Example 1.15, u · v =

1
3.u · v = 2 ·
3

2
3
1

−1

= 1 · 2 + 2 · 3 + 3 · 1 = 2 + 6 + 3 = 11.

4.u · v = 3.2 · 1.5 + (−0.6) · 4.1 + (−1.4) · (−0.2) = 4.8 − 2.46 + 0.28 = 2.62.
4√
√12
−
√ = 1 ·√4 +
5.u · v = √
·
0
3
−5
0
6.u · v =

1.12
−3.25
2.07
−1.83

·

√
22 · (− 2) +

3 · 0 + 0 · (−5) = 4 − 2 = 2.

−2.29
1.72
= −1.12 · 2.29 − 3.25 · 1.72 + 2.07 · 4.33 − 1.83 · (−1.54) = 3.6265.

4.33
−1.54

7. Finding a unit vector v in the same direction as a given vector u is called normalizing the vector u.
Proceed as in Example 1.19:
√
√
"u" =
(−1)2 + 22 = 5,
so a unit vector v in the same direction as u is
v=

1

1
u=√

Σ

5

"u"

Σ
1−

Σ
=

− √1

Σ
5

2

√2
5

√

√

.

8. Proceed as in Example 1.19:
"u" =

√

32 + (−2)2 =

9+4=

13,

so a unit vector v in the direction of u is
v=

1

"u"

1
u= √

Σ

Σ
3

13 −2

Σ
=

3
√
13
2
−√
13

Σ
.


1.2. LENGTH AND ANGLE: THE DOT PRODUCT
9. Proceed as in Example 1.19:

11

√
√
"u" = 12 + 22 + 32 = 14,

so a unit vector v in the direction of u is
1
1
v=
u=√
"u"
14

√1
14

1
=

2

√2

3

√

3

14

√3
14

.

10. Proceed as in Example 1.19:
√
√
√ 2
"u" =
3.2 + (−0.6)2 + (−1.4)2 = 10.24 + 0.36 + 1.96 = 12.56 ≈ 3.544,
so a unit vector v in the direction of u is
v=

"u"

11. Proceed as in Example 1.19:

1
u=

1
3.544

.
"u" =

1.5
0.4
−2.1

≈

0.903
−0.169
−0.395

.

.√ Σ 2 .√ Σ 2
√
2 +
3 + 02 = 6,

12 +

so a unit vector v in the direction of u is
√1

√1

1

1

u=√

v=

=

√
0

"u"

6

3
0
0

6
√1
√1

√
6
√2

2

√6

6

6

0

√
6
√6
3

√1

√6

=

1
3
1

0

2

=

3
√

0

3

0
0

2

0
0

2

0
0

12. Proceed as in Example 1.19:
√
√
" u" = 1.122 + ( 3.25)2 + 2.07−2 +
− ( 1.83)2 = 1.2544 + 10.5625 + 4.2849 + 3.3489
√
= 19.4507 ≈ 4.410,
so a unit vector v in the direction of u is
Σ Σ
Σ
1
1Σ
v u=
1.12 −3.25 2.07 −1.83 ≈ 0.254 −0.737 0.469 −0.415 .
4.410
"u"
Σ Σ Σ Σ Σ Σ
−1
3

−4
13. Following Example 1.20, we compute: u − v =
−
=
, so
2
1
1
.
√
d(u, v) = "u − v" = (−4)2 + 12 = 17.
Σ Σ Σ Σ Σ Σ
3
4
−1
14. Following Example 1.20, we compute: u − v =
−
=
, so
−2
6
−8


12

.
d(u, v) = "u − v" =

(−1)2 + (−8)2 =

√

65.

1
2
−1
15. Following Example 1.20, we compute: u − v = 2 − 3 = −1 , so
3
1
2
.
√
d(u, v) = "u − v" = (−1)2 + (−1)2 + 22 = 6.

CHAPTER 1. VECTORS


1.2. LENGTH AND ANGLE: THE DOT PRODUCT

13

3.2
16. Following Example 1.20, we compute: u − v = −0.6 −
−1.4

1.5
4.1 =
−0.2

1.7
−4.7 , so
−1.2

√
√
d(u, v) = "u − v" = 1.72 + (−4.7)2 + (−1.2)2 = 26.42 ≈ 5.14.
17.(a)u

· v is a real number, so "u · v" is the norm of a number, which is not defined.

(b)u · v is a scalar, while w is a vector. Thus u v +· w adds a scalar to a vector, which is not a
defined operation.
(c)u is a vector, while v w· is a scalar. Thus u (v w)
· is the dot product of a vector and a scalar,
·
which is not defined.
(d) c · (u + v) is the dot product of a scalar and a vector, which is not defined.
18. Let θ be the angle between u and v. Then
u·v
cos θ =

=√

"u" "v"

3 · (−1) + 0 · 1
√
32 + 02

√

3

(−1)2 + 12

2

=− √ =− .
2
3 2

Thus cos θ < 0 (in fact, θ = 3π 4), so the angle between u and v is obtuse.
19. Let θ be the angle between u and v. Then
u·v
cos θ =

"u" "v"

= √

2 · 1 + (−1) · (−2) + 1 · (−1)
1
√
= .
22 + (−1)2 + 12 12 + (−2)2 + 12 2

Thus cos θ > 0 (in fact, θ = 3π ), so the angle between u and v is acute.
20. Let θ be the angle between u and v. Then

u·v
cos θ =

"u" "v"

=

4 · 1 + 3 · (−1) + (−1) · 1
√

√

42 + 32 + (−1)2

12 + (−1)2 + 12

0
=√

√ = 0.
26 3

Thus the angle between u and v is a right angle.
21. Let θ be the angle between u and v. Note that we can determine whether θ is acute, right, or obtuse
by examining the sign of u·v
, which is determined by the sign of u v. Since
·
"u""v"
u · v = 0.9 · (−4.5) + 2.1 · 2.6 + 1.2 · (−0.8) = 0.45 > 0,
we have cos θ > 0 so that θ is acute.

22. Let θ be the angle between u and v. Note that we can determine whether θ is acute, right, or obtuse
by examining the sign of u·v
, which is determined by the sign of u v. Since
·
"u""v"
u · v = 1 · (−3) + 2 · 1 + 3 · 2 + 4 · (−2) = −3,
we have cos θ < 0 so that θ is obtuse.
23. Since the components of both u and v are positive, it is clear that u v >· 0, so the angle between
them is acute since it has a positive cosine.
. √ Σ
√
24. From Exercise 18, cos θ = − 2 , so that θ = cos−1 − 2 = 3π = 135◦ .
2

2

25. From Exercise 19, cos θ = , so that θ = cos
1

2

−1 1

2
π

◦

4

◦

π

= = 60 .
3

26. From Exercise 20, cos θ = 0, so that θ = 2 = 90 is a right angle.


14

CHAPTER 1. VECTORS

27. As in Example 1.21, we begin by calculating u · v and the norms of the two vectors:
u · v = 0.9 · (−4.5) + 2.1 · 2.6 + 1.2 · (−0.8) = 0.45,
√
√
" u" = 0.92 + 2.12 + 1.22 = 6.66,
√
√
"v" = (−4.5)2 + 2.62 + (−0.8)2 = 27.65.
So if θ is the angle between u and v, then
u·v
cos θ =
so that

"u" "v"

=

√

0.45
0.45
√
≈√
,
6.66 27.65
182.817

.
0.45
θ = cos−1 √

Σ
≈ 1.5375 ≈ 88.09◦.

182.817
Note that it is important to maintain as much precision as possible until the last step, or roundoff
errors may build up.
28. As in Example 1.21, we begin by calculating u · v and the norms of the two vectors:
u · v = 1 · (−3) + 2 · 1 + 3 · 2 + 4 · (−2) = −3,
√
√
" "u = 12 + 22 + 32 + 42 = 30,
√
√
"v" = (−3)2 + 12 + 22 + (−2)2 = 18.
So if θ is the angle between u and v, then

u·v
cos θ =

3

"u" "v"

=−√

.
1

√ =− √
30 18
2 15

1

−1

− √
2 15

so that θ = cos

Σ
◦

≈ 1.7 ≈ 97.42 .

29. As in Example 1.21, we begin by calculating u · v and the norms of the two vectors:
u · v = 1 · 5 + 2 · 6 + 3 · 7 + 4 · 8 = 70,
√
√
" u" = 12 + 22 + 32 + 42 = 30,
√
√
"v" = 52 + 62 + 72 + 82 = 174.
So if θ is the angle between u and v, then
u·v
cos θ =

"u" "v"

=√

70

.

35

√
= √
30 174
3 145

35

−1

so that θ = cos

√
3 145

Σ
◦

≈ 0.2502 ≈ 14.34 .

30. To show that OAB C is right, we need only show that one pair of its sides meets at a right angle. So
# »
# »
# »
let u = AB, v = BC, and w = AC. Then we must show that one of u v,· u w ·or v w is zero in
·
order to show that one of these pairs is orthogonal. Then
# »
#»
u = AB = [1 − (− 3), 0 − 2] = [4, −2] , v = BC = [4 − 1, 6 − 0] = [3, 6] ,
# »
w = AC = [4 − (−3), 6 − 2] = [7, 4] ,
and
u · v = 4 · 3 + (−2) · 6 = 12 − 12 = 0.
# » #»
Since this dot product is zero, these two vectors are orthogonal, so that AB ⊥ BC and thus O ABC is
a right triangle. It is unnecessary to test the remaining pairs of sides.

1.2. LENGTH AND ANGLE: THE DOT PRODUCT

15

31. To show that OAB C is right, we need only show that one pair of its sides meets at a right angle. So
# »
# »
# »
let u = AB, v = BC, and w = AC. Then we must show that one of u v,· u w ·or v w is zero in
·
order to show that one of these pairs is orthogonal. Then
# »
u = AB = [−3 − 1, 2 − 1, (−2) − (−1)] = [−4, 1, −1] ,
#»
v = BC = [2 − (−3), 2 − 2, −4 − (−2)] = [5, 0, −2],
#»
w = AC = [2 − 1, 2 − 1, −4 − (−1)] = [1, 1, −3],
and
u · v = −4 · 5 + 1 · 0 − 1 · (−2) = −18
u · w = −4 · 1 + 1 · 1 − 1 · (−3) = 0.
#»
#»
Since this dot product is zero, these two vectors are orthogonal, so that AB ⊥ AC and thus O ABC is
a right triangle. It is unnecessary to test the remaining pair of sides.
32. As in Example 1.22, the dimensions of the cube do not matter, so we work with a cube with side length
1. Since the cube is symmetric, we need only consider one diagonal and adjacent edge. Orient the cube
as shown in Figure 1.34; take the diagonal to be [1, 1, 1] and the adjacent edge to be [1, 0, 0]. Then the
angle θ between these two vectors satisfies
.
Σ

1
1·1 +1·0 +1·0
1
−1
◦
cos θ =
≈ 54.74 .
√ √
√
= √ ,
so
θ = cos
3
3 1
3
Thus the diagonal and an adjacent edge meet at an angle of 54.74◦.
33. As in Example 1.22, the dimensions of the cube do not matter, so we work with a cube with side length
1. Since the cube is symmetric, we need only consider one pair of diagonals. Orient the cube as shown
in Figure 1.34; take the diagonals to be u = [1, 1, 1] and v = [1, 1, 0]− [0, 0, 1] = [1, 1,− 1]. Then the
dot product is
u · v = 1 · 1 + 1 · 1 + 1 · (−1) = 1 + 1 − 1 = 1 ƒ= 0.
Since the dot product is nonzero, the diagonals are not orthogonal.
34. To show a parallelogram is a rhombus, it suffices to show that its diagonals are perpendicular (Euclid).
But
d · d = 22 · −11 = 2 · 1 + 2 · (−1) + 0 · 3 = 0.
1

2

0

3

To determine its side length, note that since the diagonals are perpendicular, one half of each diagonal
are the legs of a right triangle whose hypotenuse is one side of the rhombus. So we can use the
Pythagorean Theorem. Since
"d1" 2 = 2 22+ 2 + 0 2= 8,

2
"d2" 2= 1 +2 (−1) + 3 = 11,

we have for the side length
.
s =
2

so that s =

√
19
≈
2

"d1"
2

Σ2

.

"d2"
+
2

Σ2
=

8
4

+

11
4

=

19
4

,

2.18.

35. Since ABCD is a rectangle, opposite sides BA and CD are parallel and congruent. So we ca n use the
# »
method of Example 1.1 in Section 1.1 to find the coordinates of vertex D: we compute BA = [1 −
# »
# »
3, 2 − 6, 3 − (−2)] = [−2, −4, 5]. If BA is then translated to CD, where C = (0, 5, −4), then

D = (0 + (−2), 5 + (−4), −4 + 5) = (−2, 1, 1).


16

CHAPTER 1. VECTORS

36. The resultant velocity of the airplane is the sum of the velocity of the airplane and the velocity of the
wind:
Σ
Σ Σ
Σ Σ
Σ
200
0
200
r=p+w=
+
=
.
0
−40
−40
37. Let the x direction be east, in the direction of the current, and the y direction be north, across the
river. The speed of the boat is 4 mph north, and the current is 3 mph east, so the velocity of the boat
is
Σ Σ Σ Σ Σ Σ
0
3
3

v=
+
=
.
4
0
4
38. Let the x direction be the direction across the river, and the y direction be downstream. Since vt = d,
use the given information to find v, then solve for t and compute
Σ Σ d. Since the speed of the boat is 20
20
km/h and the speed of the current is 5 km/h, we have v =
. The width of the river is 2 km, and
Σ 5Σ
2
the distance downstream is unknown; call it y. Then d =
. Thus
y
Σ
vt =

Σ
Σ Σ
20
2
t=
.
5
y

Thus 20t = 2 so that t = 0.1, and then y = 5 · 0.1 = 0.5. Therefore
(a) Ann lands 0.5 km, or half a kilometer, downstream;
(b) It takes Ann 0.1 hours, or six minutes, to cross the river.
Note that the river flow does not increase the time required to cross the river, since its velocity is
perpendicular to the direction of travel.
39. We want to find the angle between Bert’s resultant vector, r, and his velocity vector upstream, v. Let
the first coordinate of the vector be the direction across the river, and the second
be the direction
ΣΣ
x
upstream. Bert’s velocity vector directly across the river is unknown, say u = . His velocity vector
0
Σ Σ
0
upstream compensates for the downstream flow, so v =
. So the resultant vector is r = u + v =
Σ Σ Σ Σ Σ Σ
1
x
0
x
+
=
. Since Bert’s speed is 2 mph, we have "r" = 2. Thus
0
1
1
2

x2 + 1 = "r" = 4,

√
so that x = 3.

If θ is the angle between r and v, then
cos θ =

r·v
"r" "v"

40. We have

√
=

3

2

,

so that θ = cos

−1

.√ Σ
3

= 60 ◦.

2

Σ Σ
Σ Σ Σ Σ
proj v = u · v u = (−1) · (−2) + 1 · 4 −1 = 3 −1 = −3 .
u
1
1
3
u·u
(−1) · (−1) + 1 · 1

A graph of the situation is (with proj u v in gray, and the perpendicular from v to the projection also
drawn)


1.2. LENGTH AND ANGLE: THE DOT PRODUCT

17
4

v

3

proju(v)

2

1

u

–3

41. We have

–2

3

u·v

u·u

.
Σ
· 1 + −4 · 2

5
5

proju v =

–1

u=

3

Σ

5

5

5

Σ . Σ
· + − · −4 −
3

.

3Σ

1

5
4
5

=−

4

5

1

u = −u.

A graph of the situation is (with proj u v in gray, and the perpendicular from v to the projection also
drawn)
2

v

1

proju(v)

1

u

42. We have
1

proju v =

u ·v
u=
u ·u

1

· 2 − 4 · 2 − 2 · (−2)
. 1Σ . 1Σ . 1Σ . 1Σ
· + −

−
+ −
−
2

1

2

1

2

4

4

1

1
2

1

2

8
=

−4

− 11

3

4

3

2

−4

=

1
1

2

.

4
2

−2

2

−3
−3

43. We have
proj v =

u

u·v

1
u = 1 · 2 + (−1) · (−3) + 1 · (−1) + (−1) · (−2)

44. W
e
h

ave

−1

=
1

1
6 −1

3

2

=

− 32

=

3

u.

2
3

u
·
u

1 · 1 + (−1) · (−1) + 1 · 1 +
(−1) · (−1)


18
−1

4

1
−
1

3

−2

2

CHAPTER 1. VECTORS

Σ Σ
Σ Σ Σ
Σ
proj v = u · v u = 0.5 · 2.1 + 1.5 · 1.2 0.5 = 2.85 0.5 = 0.57 = 1.14u.
u
1.71
u·u
0.5 · 0.5 + 1.5 · 1.5 1.5
2.5 1.5


1.2. LENGTH AND ANGLE: THE DOT PRODUCT

19

45. We have
proju v =

u ·v
u·u

3.01 · 1.34 − 0.33 · 4.25 + 2.52 · (−1.66)

u=

3.01
−0.33
3.01 · 3.01 − 0.33 · (−0.33) + 2.52 · 2.52
2.52
1.5523
3.01
=−

#»
46. Let u = AB =

Σ

−0.33
2.52

15.5194

≈

−0.301
1
0.033 ≈ −
u.
10
−0.252

Σ

ΣΣ
Σ
Σ Σ
Σ
#»
1
3
2 −1
4 −1
.
=
=
and
v
=
AC
=
3
2 − (−1)
0 − (−1)
1

(a) To compute the projection, we need
Σ Σ Σ Σ
1
3
u·v =
·
= 6,
3

1
Thus

Σ Σ Σ Σ
1
1
u·u=
·
= 10.
3
3

Σ Σ
proju v = u · v u = 3 1
u·u

so that

√

"u" = 12 + 32 =

,

5

9

Σ

3

=

9
5

.

√
10,

5

=

5 3
ΣΣ

Σ Σ
v − proju v = 3 − 5
1

Then

Σ3Σ

.

"v − proju v" =

12
5
4

Σ

−5

12
5

.

Σ2

√
. Σ2
4 10
4
+
= 5 ,
5

so that finally

√
1
1 √ 4 10
10

A = "u" "v − proju v" =
= 4.
2
2
·
5
√
√
√
(b) We already know u · v = 6 and "u" = 10 from part (a). Also, "v" = 32 + 12 = 10. So
u·v
cos θ =

so that

sin θ =
Thus

#»

√

6

"u" "v"

=√

√ = ,
10 10 5

.
1 − cos2 θ =

3

1−

. Σ2
3
= 4.
5
5

1
1√ √
4
A = "u" "v" sin θ =
10 10 · = 4.
2
2
5
4−3

1

6 −4

2

47. Let u = AB = −2 − (−1) =
6 −4

−1

6 −4

#»

and v = AC =
2
2

5−3

2

2−4

−2

0 − (−1) =
2−4
2−4

1 .
−2

(a) To compute the projection, we need
u·v=

Thus

1
2
−1 · 1 = −3,
2
−2

u·u =

1
1
−1 · −1 = 6.
2
2


20 1

1

u·v

−1
proju v =

u ·u

3
u=−

6

CHAPTER 1. VECTORS

2

2
−1 =
2

2

−1

,
−1


1.2. LENGTH AND ANGLE: THE DOT PRODUCT
so that

21

2

5

− 12

2

2

v − proju v =

1 −

2

=

1

−2

1

−1

−1

Then
.
"u" = √

√

12 + (−1)2 + 22 = 6,

"v − proju v" =

so that finally

. Σ 2 . Σ2
1
5
+
+ (−1)2 =
2
2

30
2

,

√
3 5
A = "u" "v − proju v" =
=
.
2
2
2
2 √
√
(b) We already know u · v = −3 and "u" = 6 from part (a). Also, "v" = 22 + 12 + (−2)2 = 3.
So
√

u·v
−3
6
1 √
6·

1

cos θ =

so that

√

√

"u" "v"

sin θ =

= √ =− ,
6
3 6
‚
.

√

30

√

. √ Σ2
6

1 − cos2 θ = ,1 −

=

6

30
6

.

Thus

√
√
1
1 √
30
3 5
A = "u" "v" sin θ =
6 ·3 ·
=
.
2
2

6
2
48. Two vectors u and v are orthogonal if and only if their dot product u v· = 0. So we set u v =· 0 and
solve for k:
Σ Σ Σ
Σ
1
2
k+1
u·v=
·
= 0 ⇒ 2(k + 1) + 3(k − 1) = 0 ⇒ 5k − 1 = 0 ⇒ k = .
3
k −1
5
Substituting into the formula for v gives
v=
As a check, we compute

Σ

1
5
+11 − 1
5

Σ

Σ
=

6
5
4

−5

Σ
.

Σ Σ Σ 6Σ
12 12
5 = −
= 0,
u·v= 2 ·
3
− 54
5 5

and the vectors are indeed orthogonal.
49. Two vectors u and v are orthogonal if and only if their dot product u v· = 0. So we set u v =· 0 and
solve for k:
1

k2

2
u·v=

−1

2

−3
·

k
−3

= 0 ⇒ k2 − k − 6 = 0 ⇒ (k + 2)(k − 3) = 0 ⇒ k = 2, −3.

Substituting into the formula for v gives
k = 2 : v1 =

(−2)2
4
−2
= −2 ,
−3
−3

k = −3 : v2 =

32
3 =
−3

As a check, we compute
1

4

1

9

9
3
−3

.


22

u · v1 = −1 · −2 = 1 · 4 − 1 · (−2) + 2 · (−3) = 0, u · v2 = −1 · 3 = 1 · 9 − 1 · 3 + 2 · (−3) = 0
2
−3
2
−3 CHAPTER 1. VECTORS
and the vectors are indeed orthogonal.


2
3

1.2. LENGTH AND ANGLE: THE DOT PRODUCT
50. Two vectors u and v are orthogonal if and only if their dot product u v· = 0. So we set u v =· 0 and
solve for y in terms of x:
Σ Σ Σ Σ
3

x
u·v=
·
= 0 ⇒ 3x + y = 0 ⇒ y = −3x.
1
y
Substituting y = −3x back into the formula for v gives
Σ Σ
Σ x Σ
1
v=
=x
.
−3
−3x
Σ Σ
Σ Σ
3
1
Thus any vector orthogonal to
is a multiple of
. As a check,
1
−3
Σ Σ Σ Σ
3
x
= 3x − 3x = 0 for any value of x,
u·v=
·

1
−3x
so that the vectors are indeed orthogonal.

51. As noted in
Σ the
Σ remarks just prior
Σ Σ to Example 1.16, the zero vector
Σ Σ 0 is orthogonal to all vectors in
a
x
a
2
R . So if = 0, any vector will do. Now assume that ƒ= 0; that is, that either a or b is
b
y
b
nonzero. Two vectors u and v are orthogonal if and only if their dot product u · v = 0. So we set
u · v = 0 and solve for y in terms of x:
Σ Σ Σ Σ
a
x
u·v=
·
= 0 ⇒ ax + by = 0.
b
y
First assume b

0. Then y = − abx, so substituting back into the expression for v we get

Σ
Σ Σ
Σ Σ
x Σ
1
x b
v=
=x
=
.
a
a

−b x

−b

b −a

Next, if b = 0, then a ƒ= 0, so that x = −ab y, and substituting back into the expression for v gives
Σ Σ
Σ
Σ
Σ Σ
b
y
b
b
−a y
−a =−

.
v=
=y
a −a
y
1
Σ Σ
Σ Σ
a
b
So in either case, a vector orthogonal to
, if it is not the zero vector, is a multiple of
. As a
b
−a
check, note that
Σ Σ Σ Σ
a
rb
·
= rab − rab = 0 for all values of r.
b
−ra
52.(a)

The geometry of the vectors in Figure 1.26 suggests that if" u + v" = " u" + "v" , then u and v
point in the same direction. This means that the angle between them must be 0. So we first prove
Lemma 1. For all vectors u and v in R2 or R3, u· v = " u" " "v if and only if the vectors point
in the same direction.
Proof. Let θ be the angle between u and v. Then

cos θ =

u·v

,

"u" "v"
so that cos θ = 1 if and only if u · v = "u" " "v . But cos θ = 1 if and only if θ = 0, which means
that u and v point in the same direction.