Study Guide
Chapter 1 Vectors
-
University:
Eastern Kentucky University -
Course:
MAT 239 | Linear Algebra and Matrices Academic year:
2021
-
Views:
117
Pages:
41
Author:
Zoie Hudson
Chapter 1
Vectors
1.1
The Geometry and Algebra of Vectors
H-2, 3L
1.
H2, 3L
3
2
1
H3, 0L
-2
1
-1
2
3
-1
H3, -2L
-2
2. Since
3
5
2
+
=
,
−3
0
−3
2
2
4
+
=
,
−3
3
0
2
−2
0
+
=
,
−3
3
0
plotting those vectors gives
1
2
3
4
-1
c
b
-2
a
-3
d
-4
-5
3
5
2
3
5
+
=
,
−3
−2
−5 4
CHAPTER 1. VECTORS
3.
z
2
c
b
1
-2
-1
0
0
3
0
1
2
2y
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]
# »
(c) AB = 21 − 2, 3 − 23 = − 32 , 23
# »
(d) AB = 61 − 13 , 21 − 13 = − 61 , 16 .
3
2
a
1
c
b
d
-1
1
2
3
3
4
−1
2 − −1 = 3 .
1
−2
3 5
1.1. THE GEOMETRY AND ALGEBRA OF VECTORS
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. From the components, we get
" √
√ #
5 2 5 2
,
b = [5 cos 45◦ , 5 sin 45◦ ] =
.
2
2
Thus the net displacement vector is
c=a+b=
7. a + b =
3
2
3+2
5
+
=
=
.
0
3
0+3
3
" √
√ #
5 2
5 2
.
, 4+
2
2
3
2
a+b
b
1
a
1
2
−2
2 − (−2)
4
8. b−c =
−
=
=
.
3
3
3−3
0
2
3
4
5
3
2
b
-c
1
b-c
1
−2
5
3
−
=
.
9. d − c =
−2
3
−5
2
1
2
3
3
4
4
5
d
-1
-2
d-c
-3
-c
-4
-5
3
3
3+3
10. a + d =
+
=
=
0
−2
0
+
(−2)
6
.
−2
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◦ ] =
h
1,
2
√
3
2
u+v =
i
"
h
, and v = [cos 210◦ , sin 210◦ ] = −
#
√ √
1
1
3
3
,
−
−
,
2
2
2
2
u−v =
"
√
3
,
2
i
− 12 , so that
#
√ √
1
3 3 1
.
+
+
,
2
2
2
2
# »
14. (a) AB = b − a.
# » # »
# » # »
(b) Since OC = AB, we have BC = OC − b = (b − a) − b = −a.
# »
(c) AD = −2a.
# »
# »
# »
(d) CF = −2OC = −2 AB = −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
# » # » # »
# » # »
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)
=
property b.
associativity
=
(−3a + 3c) + (2a + 4b) + (3c − 3b)
(−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
v
1
-1
2
3
4
v
u
-1
v
u
-2
5
6 7
1.1. THE GEOMETRY AND ALGEBRA OF VECTORS
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
6
5
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
cHu + vL
cu
u
v
cu
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 ]
= [ u 1 − u 1 , u2 − u 2 , . . . , un − u n ]
= [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]. 9
1.1. THE GEOMETRY AND ALGEBRA OF VECTORS
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
4
+
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
0 0
1 0
2 0
3 0
1
0
1
2
3
2
0
2
0
2
3
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
0 0
1 0
2 0
3 0
4 0
1
0
1
2
3
4
2
0
2
4
1
3
3
0
3
1
4
2
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 .
10
= (1024)10 = 110 = 1 in Z11 .
40. 2100 = 210
41. [2, 1, 2] + [2, 0, 1] = [1, 1, 0] in Z33.
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 = 3−1 4 = 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 = 8−1 9 = 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.
(b)
All values.
(c)
All values.
57. (a) All a 6= 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
−1
3
·
= (−1) · 3 + 2 · 1 = −3 + 2 = −1.
2
1
4
3
·
= 3 · 4 + (−2) · 6 = 12 − 12 = 0.
2. Following Example 1.15, u · v =
−2
6
1
2
3. u · v = 2 · 3 = 1 · 2 + 2 · 3 + 3 · 1 = 2 + 6 + 3 = 11.
3
1
1. Following Example 1.15, u · v =
4. u · v = 3.2 · 1.5 + (−0.6) · 4.1 + (−1.4) · (−0.2) = 4.8 − 2.46 + 0.28 = 2.62.
4
√1
√
2 − 2
√
√
√
√
5. u · v =
3 · 0 = 1 · 4 + 2 · (− 2) + 3 · 0 + 0 · (−5) = 4 − 2 = 2.
0
−5
1.12
−2.29
−3.25 1.72
6. u · v =
2.07 · 4.33 = −1.12 · 2.29 − 3.25 · 1.72 + 2.07 · 4.33 − 1.83 · (−1.54) = 3.6265.
−1.83
−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:
p
√
kuk = (−1)2 + 22 = 5,
so a unit vector v in the same direction as u is
" # "
#
− √15
1 −1
1
u=√
=
.
2
kuk
√
5
2
5
v=
8. Proceed as in Example 1.19:
kuk =
p
32 + (−2)2 =
√
9+4=
√
13,
so a unit vector v in the direction of u is
v=
#
" # "
√3
1
1
3
13
.
u=√
=
2
kuk
− √13
13 −2 11
1.2. LENGTH AND ANGLE: THE DOT PRODUCT
9. Proceed as in Example 1.19:
kuk =
so a unit vector v in the direction of u is
p
12 + 22 + 32 =
√
14,
√1
1
14
1
1
v=
u= √
2 = √2 .
kuk
14 314
3
√
14
10. Proceed as in Example 1.19:
p
√
√
kuk = 3.22 + (−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=
11. Proceed as in Example 1.19:
1. 5
0.903
1
1
0.4 ≈ −0.169 .
u=
3.544
kuk
−2.1
−0.395
kuk =
r
12 +
so a unit vector v in the direction of u is
√ 2 √ 2
√
2 +
3 + 02 = 6,
1 1 √
6
√
√
1
6
√
√62 16 √3
√ √ 3
1 2
1
6 3 √
u= √
v=
√
= √ = = 2
kuk
6
3 √36 √12 2
0
0
0
0
12. Proceed as in Example 1.19:
p
√
kuk = 1.122 + (−3.25)2 + 2.072 + (−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
1
1
u=
1.12 −3.25 2.07
v
kuk
4.410
−1.83 ≈ 0.254
−0.737
−1
3
−4
−
=
, so
2
1
1
q
√
d(u, v) = ku − vk = (−4)2 + 12 = 17.
13. Following Example 1.20, we compute: u − v =
4
−1
3
−
=
, so
−2
6
−8
q
√
d(u, v) = ku − vk = (−1)2 + (−8)2 = 65.
14. Following Example 1.20, we compute: u − v =
1
2
−1
15. Following Example 1.20, we compute: u − v = 2 − 3 = −1, so
3
1
2
q
√
2
2
d(u, v) = ku − vk = (−1) + (−1) + 22 = 6.
0.469
−0.415 . 12
CHAPTER 1. VECTORS
3. 2
1. 5
1.7
16. Following Example 1.20, we compute: u − v = −0.6 − 4.1 = −4.7 , so
−1.4
−0.2
−1.2
d(u, v) = ku − vk =
p
1.72 + (−4.7)2 + (−1.2)2 =
√
26.42 ≈ 5.14.
17. (a) u · v is a real number, so ku · vk 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
cos θ =
Thus cos θ < 0 (in fact, θ =
√
u·v
3 · (−1) + 0 · 1
3
2
.
= √
p
=− √ =−
2
2
2
2
2
kuk kvk
3 + 0 (−1) + 1
3 2
3π
),
4
so the angle between u and v is obtuse.
19. Let θ be the angle between u and v. Then
cos θ =
1
u·v
2 · 1 + (−1) · (−2) + 1 · (−1)
p
= .
= p 2
2
kuk kvk
2 + (−1)2 + 12 12 + (−2)2 + 12
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
cos θ =
4 · 1 + 3 · (−1) + (−1) · 1
0
u·v
p
= p
= √ √ = 0.
kuk kvk
26 3
42 + 32 + (−1)2 12 + (−1)2 + 12
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
u·v
, which is determined by the sign of u · v. Since
by examining the sign of kukkvk
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
u·v
by examining the sign of kukkvk
, which is determined by the sign of u · v. Since
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.
√
√
24. From Exercise 18, cos θ = − 22 , so that θ = cos−1 − 22 = 3π
= 135◦ .
4
25. From Exercise 19, cos θ = 12 , so that θ = cos−1 21 =
26. From Exercise 20, cos θ = 0, so that θ =
π
2
◦
π
3
= 60◦ .
= 90 is a right angle. 13
1.2. LENGTH AND ANGLE: THE DOT PRODUCT
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,
p
√
kuk = 0.92 + 2.12 + 1.22 = 6.66,
p
√
kvk = (−4.5)2 + 2.62 + (−0.8)2 = 27.65.
So if θ is the angle between u and v, then
cos θ =
0.45
u·v
0.45
√
≈√
= √
,
kuk kvk
6.66 27.65
182.817
so that
θ = cos−1
√
0.45
182.817
≈ 1.5375 ≈ 88.09◦ .
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,
p
√
kuk = 12 + 22 + 32 + 42 = 30,
p
√
kvk = (−3)2 + 12 + 22 + (−2)2 = 18.
So if θ is the angle between u and v, then
cos θ =
1
u·v
3
= −√ √ = − √
kuk kvk
30 18
2 15
1
≈ 1.7 ≈ 97.42◦ .
so that θ = cos−1 − √
2 15
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,
p
√
kuk = 12 + 22 + 32 + 42 = 30,
p
√
kvk = 52 + 62 + 72 + 82 = 174.
So if θ is the angle between u and v, then
cos θ =
35
u·v
70
= √
= √ √
kuk kvk
30 174
3 145
so that θ = cos−1
35
√
3 145
≈ 0.2502 ≈ 14.34◦ .
30. To show that 4ABC 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 4ABC is
a right triangle. It is unnecessary to test the remaining pairs of sides. 14
CHAPTER 1. VECTORS
31. To show that 4ABC 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 4ABC 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
1
1·1+1·0+1·0
1
√ √
=√ ,
cos θ =
≈ 54.74◦ .
so
θ = cos−1 √
3 1
3
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 6= 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
2
1
d1 · d2 = 2 · −1 = 2 · 1 + 2 · (−1) + 0 · 3 = 0.
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
2
2
kd1 k = 22 + 22 + 02 = 8,
kd2 k = 12 + (−1)2 + 32 = 11,
we have for the side length
s2 =
so that s =
√
19
2
kd 1 k
2
2
+
kd 2 k
2
2
=
8 11
19
+
=
,
4
4
4
≈ 2.18.
35. Since ABCD is a rectangle, opposite sides BA and CD are parallel and congruent. So we can
# »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). 15
1.2. LENGTH AND ANGLE: THE DOT PRODUCT
36. The resultant velocity of the airplane is the sum of the velocity of the airplane and the velocity of the
wind:
0
200
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
3
3
0
+
=
.
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
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
5
2
. Thus
the distance downstream is unknown; call it y. Then d =
y
vt =
2
20
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
x
. His velocity vector
upstream. Bert’s velocity vector directly across the river is unknown, say u =
0
0
. So the resultant vector is r = u + v =
upstream compensates for the downstream flow, so v =
1
0
x
x
+
=
. Since Bert’s speed is 2 mph, we have krk = 2. Thus
0
1
1
x2 + 1 = krk2 = 4,
so that x =
√
3.
If θ is the angle between r and v, then
cos θ =
40. We have
proju v =
√
r·v
3
=
,
krk kvk
2
so that θ = cos−1
√ !
3
= 60◦ .
2
u·v
(−1) · (−2) + 1 · 4 −1
−1
−3
=3
=
.
u=
1
1
3
(−1) · (−1) + 1 · 1
u·u
A graph of the situation is (with proju v in gray, and the perpendicular from v to the projection also
drawn) 16
CHAPTER 1. VECTORS
4
v
3
proju HvL
2
1
u
-3
-2
41. We have
proju v =
u·v
u=
u·u
3
5
-1
3
· 1 + − 54 · 2
5
· 35 + − 54 · − 45
"
3
5
− 45
#
1
= − u = −u.
1
A graph of the situation is (with proju v in gray, and the perpendicular from v to the projection also
drawn)
2
v
1
proju HvL
1
u
42. We have
u·v
proju v =
u=
u·u
43. We have
1
2
1
2
1
1
4
· 2 − 14 · 2 − 12 · (−2)
2 8 2 3
−41 = − 14 = − 23 .
3
· 21 + − 14 −41 + − 21 − 12
−21
− 12
− 43
3
1
1
2
3
1 · 2 + (−1) · (−3) + 1 · (−1) + (−1) · (−2) −1 6 −1
3
u·v
−
u=
proju v =
= = 23 = u.
u·u
1 · 1 + (−1) · (−1) + 1 · 1 + (−1) · (−1) 1 4 1 2 2
−23
−1
−1
44. We have
proju v =
2.85 0.5
0.5 · 2.1 + 1.5 · 1.2 0.5
0.57
u·v
=
=
= 1.14u.
u=
1
5
1.5
1
71
.
.
u·u
0.5 · 0.5 + 1.5 · 1.5
2.5 17
1.2. LENGTH AND ANGLE: THE DOT PRODUCT
45. We have
proju v =
3.01
u·v
3.01 · 1.34 − 0.33 · 4.25 + 2.52 · (−1.66)
−0.33
u=
3.01 · 3.01 − 0.33 · (−0.33) + 2.52 · 2.52
u·u
2.52
3.01
−0.301
1
1.5523
−0.33 ≈ 0.033 ≈ − u.
=−
15.5194
10
2.52
−0.252
# »
46. Let u = AB =
# »
1
3
2−1
4−1
=
and v = AC =
=
.
2 − (−1)
3
0 − (−1)
1
(a) To compute the projection, we need
3
1
·
= 6,
u·v =
3
1
Thus
proju v =
u·u=
" # " #
3
3 1
u·v
u=
= 59 ,
u·u
5 3
5
so that
v − proju v =
" # " # " #
3
12
3
− 59 = 54 .
−
1
5
5
Then
kuk =
so that finally
p
12
+
32
=
√
10,
1
1
·
= 10.
3
3
kv − proju vk =
s
12
5
2
+
√
2
4 10
4
=
,
5
5
√
1√
1
4 10
kuk kv − proju vk =
= 4.
10 ·
2
5
2
√
√
√
(b) We already know u · v = 6 and kuk = 10 from part (a). Also, kvk = 32 + 12 = 10. So
A=
cos θ =
6
3
u·v
= √ √ = ,
5
kuk kvk
10 10
so that
sin θ =
Thus
p
1 − cos2 θ =
s
1−
2
4
3
= .
5
5
1√ √
4
1
10 10 · = 4.
A = kuk kvk sin θ =
2
5
2
1
2
4−3
5−3
# »
# »
47. Let u = AB = −2 − (−1) = −1 and v = AC = 0 − (−1) = 1.
6−4
2
2−4
−2
(a) To compute the projection, we need
1
2
u · v = −1 · 1 = −3,
2
−2
Thus
1
1
u · u = −1 · −1 = 6.
2
2
1
−
1
u·v
3 21
proju v =
u = − −1 = 2 ,
6
u·u
2
−1 18
CHAPTER 1. VECTORS
so that
1 5
−
2
12 21
v − proju v = 1 − 2 = 2
−2
−1
−1
Then
kuk =
p
12
+
(−1)2
+
22
=
√
6,
kv − proju vk =
s
√
2 2
30
5
1
+ (−1)2 =
,
+
2
2
2
so that finally
√
√
1
1√
30 3 5
kuk kv − proju vk =
.
6·
=
2
2
2
2
p
√
(b) We already know u · v = −3 and kuk = 6 from part (a). Also, kvk = 22 + 12 + (−2)2 = 3.
So
√
−3
u·v
6
= √ =− ,
cos θ =
6
kuk kvk
3 6
so that
v
u
√ !2 √
u
p
30
6
=
sin θ = 1 − cos2 θ = t1 −
.
6
6
A=
Thus
√
√
1√
1
30 3 5
kuk kvk sin θ =
.
6·3·
=
2
2
2
6
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 :
1
k+1
2
·
= 0 ⇒ 2(k + 1) + 3(k − 1) = 0 ⇒ 5k − 1 = 0 ⇒ k = .
u·v =
3
k−1
5
A=
Substituting into the formula for v gives
v=
As a check, we compute
u·v =
and the vectors are indeed orthogonal.
"
1
5
1
5
# " #
6
+1
5
.
=
− 45
−1
" # " #
6
12 12
2
5
−
= 0,
·
=
5
5
3
− 45
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 :
2
1
k
u · v = −1 · k = 0 ⇒ k 2 − k − 6 = 0 ⇒ (k + 2)(k − 3) = 0 ⇒ k = 2, −3.
2
−3
Substituting into the formula for v gives
4
(−2)2
k = 2 : v1 = −2 = −2 ,
−3
−3
9
32
k = −3 : v2 = 3 = 3 .
−3
−3
As a check, we compute
1
4
1
9
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
and the vectors are indeed orthogonal. 19
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:
x
3
·
= 0 ⇒ 3x + y = 0 ⇒ y = −3x.
u·v =
1
y
Substituting y = −3x back into the formula for v gives
1
x
=x
.
v=
−3x
−3
Thus any vector orthogonal to
1
3
is a multiple of
. As a check,
1
−3
u·v =
x
3
·
= 3x − 3x = 0 for any value of x,
1
−3x
so that the vectors are indeed orthogonal.
51. As noted in
remarks just prior
to Example 1.16, the zero vector
0 is orthogonal to all vectors in
the
a
x
a
= 0, any vector
will do. Now assume that
6= 0; that is, that either a or b is
R2 . So if
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:
x
a
·
= 0 ⇒ ax + by = 0.
u·v =
b
y
First assume b 6= 0. Then y = − ab x, so substituting back into the expression for v we get
#
"
#
"
" #
1
x
x
b
=
x
=
v=
.
b −a
− ab x
− ab
Next, if b = 0, then a 6= 0, so that x = − ab y, and substituting back into the expression for v gives
" #
#
" #
"
−ab
b
y
− ba y
=y
=−
.
v=
a −a
y
1
So in either case, a vector orthogonal to
check, note that
b
a
, if it is not the zero vector, is a multiple of
. As a
b
−a
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 ku + vk = kuk + kvk, 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 = kuk kvk 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
,
kuk kvk
so that cos θ = 1 if and only if u · v = kuk kvk. But cos θ = 1 if and only if θ = 0, which means
that u and v point in the same direction. 20
CHAPTER 1. VECTORS
We can now show
Theorem 2. For all vectors u and v in R2 or R3 , ku + vk = kuk + kvk if and only if u and v
point in the same direction.
Proof. First assume that u and v point in the same direction. Then u · v = kuk kvk, and thus
2
ku + vk =u · u + 2u · v + v · v
2
= kuk + 2u · v + kvk
By Example 1.9
2
2
= kuk + 2 kuk kvk + kvk
2
Since w · w = kwk for any vector w
2
By the lemma
2
= (kuk + kvk) .
Since ku + vk and kuk+kvk are both nonnegative, taking square roots gives ku + vk = kuk+kvk.
For the other direction, if ku + vk = kuk + kvk, then their squares are equal, so that
2
2
2
(kuk + kvk) = kuk + 2 kuk kvk + kvk and
2
ku + vk = u · u + 2u · v + v · v
2
are equal. But kuk = u · u and similarly for v, so that canceling those terms gives 2u · v =
2 kuk kvk and thus u · v = kuk kvk. Using the lemma again shows that u and v point in the same
direction.
(b) The geometry of the vectors in Figure 1.26 suggests that if ku + vk = kuk − kvk, then u and v
point in opposite directions. In addition, since ku + vk ≥ 0, we must also have kuk ≥ kvk. If
they point in opposite directions, the angle between them must be π . This entire proof is exactly
analogous to the proof in part (a). We first prove
Lemma 3. For all vectors u and v in R2 or R3 , u · v = − kuk kvk if and only if the vectors point
in opposite directions.
Proof. Let θ be the angle between u and v. Then
cos θ =
u·v
,
kuk kvk
so that cos θ = −1 if and only if u · v = − kuk kvk. But cos θ = −1 if and only if θ = π, which
means that u and v point in opposite directions.
We can now show
Theorem 4. For all vectors u and v in R2 or R3 , ku + vk = kuk − kvk if and only if u and v
point in opposite directions and kuk ≥ kvk.
Proof. First assume that u and v point in opposite directions and kuk ≥ kvk. Then u · v =
− kuk kvk, and thus
2
ku + vk =u · u + 2u · v + v · v
2
= kuk + 2u · v + kvk
2
By Example 1.9
2
= kuk − 2 kuk kvk + kvk
2
2
Since w · w = kwk for any vector w
By the lemma
2
= (kuk − kvk) .
Now, since kuk ≥ kvk by assumption, we see that both ku + vk and kuk − kvk are nonnegative,
so that taking square roots gives ku + vk = kuk − kvk. For the other direction, if ku + vk = 21
1.2. LENGTH AND ANGLE: THE DOT PRODUCT
kuk − kvk, then first of all, since the left-hand side is nonnegative, the right-hand side must be
as well, so that kuk ≥ kvk. Next, we can square both sides of the equality, so that
2
2
2
(kuk − kvk) = kuk − 2 kuk kvk + kvk and
2
ku + vk = u · u + 2u · v + v · v
2
are equal. But kuk = u · u and similarly for v, so that canceling those terms gives 2u · v =
−2 kuk kvk and thus u · v = − kuk kvk. Using the lemma again shows that u and v point in
opposite directions.
53. Prove Theorem 1.2(b) by applying the definition of the dot product:
u · v = u1 (v1 + w1 ) + u2 (v2 + w2 ) + · · · + un (vn + wn )
= u1 v 1 + u1 w 1 + u2 v 2 + u2 w 2 + · · · + u n v n + un w n
= (u1 v1 + u2 v2 + · · · + un vn ) + (u1 w1 + u2 w2 + · · · + un wn )
= u · v + u · w.
54. Prove the three parts of Theorem 1.2(d) by applying the definition of the dot product and various
properties of real numbers:
Part 1: For any vector u, we must show u · u ≥ 0. But
u · u = u1 u1 + u2 u2 + · · · + un un = u12 + u22 + · · · + un2 .
Since for any real number x we know that x2 ≥ 0, it follows that this sum is also nonnegative, so
that u · u ≥ 0.
Part 2: We must show that if u = 0 then u · u = 0. But u = 0 means that ui = 0 for all i, so that
u · u = 0 · 0 + 0 · 0 + · · · + 0 · 0 = 0.
Part 3: We must show that if u · u = 0, then u = 0. From part 1, we know that
u · u = u21 + u22 + · · · + un2,
ui2
and that
≥ 0 for all i. So if the dot product is to be zero, each u2i must be zero, which means
that ui = 0 for all i and thus u = 0.
55. We must show d(u, v) = ku − vk = kv − uk = d(v, u). By definition, d(u, v) = ku − vk. Then by
Theorem 1.3(b) with c = −1, we have k−wk = kwk for any vector w; applying this to the vector u − v
gives
ku − vk = k−(u − v)k = kv − uk ,
which is by definition equal to d(v, u).
56. We must show that for any vectors u, v and w that d(u, w) ≤ d(u, v) + d(v, w). This is equivalent to
showing that ku − wk ≤ ku − vk + kv − wk. Now substitute u − v for x and v − w for y in Theorem
1.5, giving
ku − wk = k(u − v) + (v − w)k ≤ ku − vk + kv − wk .
57. We must show that d(u, v) = ku − vk = 0 if and only if u = v. This follows immediately from
Theorem 1.3(a), kwk = 0 if and only if w = 0, upon setting w = u − v.
58. Apply the definitions:
u · cv = [u1 , u2 , . . . , un ] · [cv1 , cv2 , . . . , cvn ]
= u1 cv1 + u2 cv2 + · · · + un cvn
= cu1 v1 + cu2 v2 + · · · + cun vn
= c(u1 v1 + u2 v2 + · · · + un vn )
= c(u · v). 22
CHAPTER 1. VECTORS
59. We want to show that ku − vk ≥ kuk − kvk. This is equivalent to showing that kuk ≤ ku − vk + kvk.
This follows immediately upon setting x = u − v and y = v in Theorem 1.5.
60. If u · v = u · w, it does not follow that v = w. For example, since 0 · v = 0 for every vector v in Rn ,
the zero vector is orthogonal to every vector v. So if u = 0 in the above equality, we know nothing
about v and w. (as an example, 0 · [1, 2] = 0 · [−17, 12]). Note, however, that u · v = u · w implies
that u · v − u · w = u(v − w) = 0, so that u is orthogonal to v − w.
2
2
61. We must show that (u + v)(u − v) = kuk − kvk for all vectors in Rn . Recall that for any w in Rn
2
that w · w = kwk , and also that u · v = v · u. Then
2
2
2
2
(u + v)(u − v) = u · u + v · u − u · v − v · v = kuk + u · v − u · v − kvk = kuk − kvk .
62. (a) Let u, v ∈ Rn . Then
2
2
ku + vk + ku − vk = (u + v) · (u + v) + (u − v) · (u − v)
= (u · u + v · v + 2u · v) + (u · u + v · v − 2u · v)
2
2
2
2
= kuk + kvk + 2u · v + kuk + kvk − 2u · v
2
2
= 2 kuk + 2 kvk .
(b) Part (a) tells us that the sum of the
squares of the lengths of the diagonals
of a parallelogram is equal to the sum
of the squares of the lengths of its four
sides.
u+v
u-v
u
v
63. Let u, v ∈ Rn . Then
1
1
1
2
2
ku + vk − ku − vk = [(u + v) · (u + v) − ((u − v) · (u − v)]
4
4
4
1
= [(u · u + v · v + 2u · v) − (u · u + v · v − 2u · v)]
4
i
1 h
2
2
2
2
kuk − kuk + kvk − kvk + 4u · v
=
4
= u · v.
64. (a) Let u, v ∈ Rn . Then using the previous exercise,
2
2
2
2
ku + vk = ku − vk ⇔ ku + vk = ku − vk
⇔ ku + vk − ku − vk = 0
1
1
⇔ ku + vk2 − ku − vk2 = 0
4
4
⇔u·v =0
⇔ u and v are orthogonal. 23
1.2. LENGTH AND ANGLE: THE DOT PRODUCT
(b) Part (a) tells us that a parallelogram is
a rectangle if and only if the lengths of
its diagonals are equal.
u+v
u-v
u
v
2
2
2
2
65. (a) By Exercise 55, (u+v)·(u− v) = kuk −kvk . Thus (u+v)·(u−v) = 0 if and only if kuk = kvk .
It follows immediately that u + v and u − v are orthogonal if and only if kuk = kvk.
(b) Part (a) tells us that the diagonals of a
parallelogram are perpendicular if and
u+v
only if the lengths of its sides are equal,
i.e., if and only if it is a rhombus.
u-v
u
v
2
2
2
2
66. From Example 1.9 and the fact that w · w = kwk , we have ku + vk = kuk + 2u · v + kvk . Taking
q
2
2
the square root of both sides yields ku + vk = kuk + 2u · v + kvk . Now substitute in the given
√
values kuk = 2, kvk = 3, and u · v = 1, giving
r
√ 2 √
√
ku + vk = 22 + 2 · 1 +
3 = 4 + 2 + 3 = 9 = 3.
67. From Theorem 1.4 (the Cauchy-Schwarcz inequality), we have |u · v| ≤ kuk kvk. If kuk = 1 and
kvk = 2, then |u · v| ≤ 2, so we cannot have u · v = 3.
68. (a) If u is orthogonal to both v and w, then u · v = u · w = 0. Then
u · (v + w) = u · v + u · w = 0 + 0 = 0,
so that u is orthogonal to v + w.
(b) If u is orthogonal to both v and w, then u · v = u · w = 0. Then
u · (sv + tw) = u · (sv) + u · (tw) = s(u · v) + t(u · w) = s · 0 + t · 0 = 0,
so that u is orthogonal to sv + tw.
69. We have
u · v
u · (v − proju v) = u · v −
u
u·u
u·v
=u·v−u·
u
·u
u · u
v
(u · u)
=u·v−
u·u
=u·v−u·v
= 0. 24
CHAPTER 1. VECTORS
u·v
70. (a) proju (proju v) = proju u·u
u =
(b) Using part (a),
u·v
u·u
u·v
proju u = u·u
u = proju v.
u · v
u · v u · v
u =
u−
proju u
proju (v − proju v) = proju v −
u·u
u·u
u·u
u · v
u·v
u−
u = 0.
=
u·u
u·u
(c) From the diagram, we see that
proju vku, so that proju (proju v) =
proju v. Also, (v − proju v) ⊥ u, so
that proju (v − proju v) = 0.
u
proju HvL
v
v - proju HvL
-proju HvL
71. (a) We have
(u21 + u22 )(v12 + v22 ) − (u1 v1 + u2 v2 )2 = u12v 21 + u12v22 + u22 v12 + u22 v22 − u21 v12 − 2u1 v1 u2 v2 − u22 v 22
= u12v 22 + u22v12 − 2u1 u2 v1 v2
= (u1 v2 − u2 v1 )2 .
But the final expression is nonnegative since it is a square. Thus the original expression is as well,
showing that (u12 + u22)(v 21 + v22) − (u1 v1 + u2 v2 )2 ≥ 0.
(b) We have
(u12 + u22 + u32)(v 21 + v22 + v 23 ) − (u1 v1 + u2 v2 + u3 v3 )2
= u21 v12 + u21 v22 + u21 v32 + u22 v 21 + u22v 22 + u22v32 + u23 v12 + u23 v22 + u32v 23
− u12v 21 − 2u1 v1 u2 v2 − u22v 22 − 2u1 v1 u3 v3 − u32v 23 − 2u2 v2 u3 v3
= u21 v22 + u21 v32 + u22 v12 + u22 v 23 + u32v 21 + u32v22
− 2u1 u2 v1 v2 − 2u1 v1 u3 v3 − 2u2 v2 u3 v3
= (u1 v2 − u2 v1 )2 + (u1 v3 − u3 v1 )2 + (u3 v2 − u2 v3 )2 .
But the final expression is nonnegative since it is the sum of three squares. Thus the original
expression is as well, showing that (u12 + u22 + u32)(v12 + v 22 + v32) − (u1 v1 + u2 v2 + u3 v3 )2 ≥ 0.
72. (a) Since proju v =
u·v
u,
u·u
we have
u·v
u·v
u· v−
u
u·u
u·u
u·v
u·v
=
u·u
u·v−
u·u
u·u
u·v
=
(u · v − u · v)
u·u
= 0,
proju v · (v − proju v) = 25
Exploration: Vectors and Geometry
so that proju v is orthogonal to v − proju v. Since their vector sum is v, those three vectors form
a right triangle with hypotenuse v, so by Pythagoras’ Theorem,
2
2
2
2
kproju vk ≤ kproju vk + kv − proju vk = kvk .
(b)
Since norms are always nonnegative, taking square roots gives kproju vk ≤ kvk.
u·v
u ≤ kvk
kproju vk ≤ kvk ⇐⇒
u·u !
u·v
⇐⇒
u ≤ kvk
2
kuk
u · v
⇐⇒
kuk ≤ kvk
kuk2
|u · v|
≤ kv k
kuk
⇐⇒ |u · v| ≤ kuk kvk ,
⇐⇒
which is the Cauchy-Schwarcz inequality.
73. Suppose proju v = cu. From the figure, we see that cos θ =
two expressions are equal, i.e.,
ckuk
.
kvk
But also cos θ =
u·v .
kukkvk
Thus these
c kuk
u·v
u·v
u·v
u·v
.
=
⇒ c kuk =
⇒c=
=
kvk
kuk kvk
kuk
kuk kuk u · u
74. The basis for induction is the cases n = 1 and n = 2. The n = 1 case is the assertion that kv1 k ≤ kv2 k,
which is obviously true. The n = 2 case is the Triangle Inequality, which is also true.
Now assume the statement holds for n = k ≥ 2; that is, for any v1 , v2 , . . . , vk ,
kv1 + v2 + · · · + vk k ≤ kv1 k + kv2 k + · · · + kvk k .
Let v1 , v2 , . . . , vk , vk+1 be any vectors. Then
kv1 + v2 + · · · + vk + vk+1 k = kv1 + v2 + · · · + (vk + vk+1 )k
≤ kv1 k + kv2 k + · · · + kvk + vk+1 k
using the inductive hypothesis. But then using the Triangle Inequality (or the case n = 2 in this
theorem), kvk + vk+1 k ≤ kvk k + kvk+1 k. Substituting into the above gives
kv1 + v2 + · · · + vk + vk+1 k ≤ kv1 k + kv2 k + · · · + kvk + vk+1 k
≤ kv1 k + kv2 k + · · · + kvk k + kvk+1 k ,
which is what we were trying to prove.
Exploration: Vectors and Geometry
# »
# »
# »
1. As in Example 1.25, let p = OP . Then p − a = AP = 13 AB = 13 (b − a), so that p = a + 31(b − a) =
# »
# »
1
1
(2a + b). More generally, if P is the point n of the way from A to B along AB, then p − a = AP =
3 # »
1
1
1
1
n AB = n (b − a), so that p = a + n (b − a) = n ((n − 1)a + b).
# »
2. Use the notation that the vector OX is written x. Then from exercise 1, we have p = 21 (a + c) and
1
q = 2 (b + c), so that
1
# »
1
1
1# »
P Q = q − p = (b + c) − (a + c) = (b − a) = AB.
2
2
2
2 26
CHAPTER 1. VECTORS
# »
# »
# » # »
# »
3. Draw AC. Then from exercise 2, we have P Q = 12 AB = SR. Also draw BD. Again from exercise 2,
# » 1# »
# »
we have P S = 2BD = QR. Thus opposite sides of the quadrilateral P QRS are equal. They are also
parallel: indeed, 4BP Q and 4BAC are similar, since they share an angle and BP : BA = BQ : BC .
Thus ∠BP Q = ∠BAC; since these angles are equal, P QkAC. Similarly, SRkAC so that P QkSR. In
a like manner, we see that P SkRQ. Thus P QRS is a parallelogram.
4. Following the hint, we find m, the point that is two-thirds of the distance from A to P . From exercise
1, we have
1
1
1
1
1
p = (b + c), so that m = (2p + a) =
2 · (b + c) + a = (a + b + c).
2
3
2
3
3
Next we find m0 , the point that is two-thirds of the distance from B to Q. Again from exercise 1, we
have
1
1
1
1
1
2 · (a + c) + b = (a + b + c).
q = (a + c), so that m0 = (2q + b) =
2
3
2
3
3
Finally we find m00 , the point that is two-thirds of the distance from C to R. Again from exercise 1,
we have
1
1
1
1
1
r = (a + b), so that m00 = (2r + c) =
2 · (a + b) + c = (a + b + c).
2
3
3
2
3
Since m = m0 = m00 , all three medians intersect at the centroid, G.
# »
# »
# » # »
5. With notation as in the figure, we know that AH is orthogonal to BC; that is, AH · BC = 0. Also
# »
# »
# » # »
# » # »
BH is orthogonal to AC; that is, BH · AC = 0. We must show that CH · AB = 0. But
# » # »
AH · BC = 0 ⇒ (h − a) · (b − c) = 0 ⇒ h · b − h · c − a · b + a · c = 0
# » # »
BH · AC = 0 ⇒ (h − b) · (c − a) = 0 ⇒ h · c − h · a − b · c + b · a = 0.
Adding these two equations together and canceling like terms gives
# » # »
0 = h · b − h · a − c · b + a · c = (h − c) · (b − a) = CH · AB,
so that these two are orthogonal. Thus all the altitudes intersect at the orthocenter H .
# »
# »
# »
# »
6. We
to CB, and must show that
# » are given that QK
# »is orthogonal to AC and that 1P K is orthogonal
1
RK is orthogonal to AB. By exercise 1, we have q = 2 (a + c), p = 2 (b + c), and r = 12 (a + b). Thus
# » # »
1
QK · AC = 0 ⇒ (k − q) · (c − a) = 0 ⇒ k − (a + c) · (c − a) = 0
2
# » # »
1
P K · CB = 0 ⇒ (k − p) · (b − c) = 0 ⇒ k − (b + c) · (b − c) = 0.
2
Expanding the two dot products gives
1
1
1
1
k·c−k·a− a·c+ a·a− c·c+ a·c=0
2
2
2
2
1
1
1
1
k · b − k · c − b · b + b · c − c · b + c · c = 0.
2
2
2
2
Add these two together and cancel like terms to get
1
1
# » # »
1
0 = k · b − k · a − b · b + a · a = k − (b + a) · (b − a) = (k − r) · (b − a) = RK · AB.
2
2
2
# »
# »
Thus RK and AB are indeed orthogonal, so all the perpendicular bisectors intersect at the circumcenter. 27
1.3. LINES AND PLANES
2
2
7. Let O, the center of the circle, be the origin. Then b = −a and kak = kck = r2 where r is the radius
# »
# »
of the circle. We want to show that AC is orthogonal to BC. But
# » # »
AC · BC = (c − a) · (c − b)
= (c − a) · (c + a)
2
2
= kck + c · a − kak − a · c
= (a · c − c · a) + (r2 − r2 ) = 0.
Thus the two are orthogonal, so that ∠ACB is a right angle.
8. As in exercise 5, we first find m, the point that is halfway from P to R. We have p = 21(a + b) and
r = 21 (c + d), so that
1
1 1
1
1
(a + b) + (c + d) = (a + b + c + d).
m = (p + r) =
2
2 2
2
4
Similarly, we find m0 , the point that is halfway from Q to S. We have q = 21 (b + c) and s = 12 (a + d),
so that
1
1
1 1
1
m0 = (q + s) =
(b + c) + (a + d) = (a + b + c + d).
2
2 2
2
4
# »
# »
Thus m = m0 , so that P R and QS intersect at their mutual midpoints; thus, they bisect each other.
1.3
Lines and Planes
0
3
· x−
= 0.
1. (a) The normal form is n · (x − p) = 0, or
2
0
x
, we get
(b) Letting x =
y
3
x
0
3
x
·
−
=
·
= 3x + 2y = 0.
2
y
0
2
y
The general form is 3x + 2y = 0.
1
3
· x−
= 0.
2. (a) The normal form is n · (x − p) = 0, or
−4
2
x
, we get
(b) Letting x =
y
x
1
3
x−1
3
·
−
=
·
= 3(x − 1) − 4(y − 2) = 0.
−4
y
2
−4
y−2
3. (a)
(b)
4. (a)
(b)
Expanding and simplifying gives the general form 3x − 4y = −5.
−1
1
+t
.
In vector form, the equation of the line is x = p + td, or x =
0
3
x
1−t
x
Letting x =
and expanding the vector form from part (a) gives
=
, which yields
y
y
3t
the parametric form x = 1 − t, y = 3t.
−4
1
In vector form, the equation of the line is x = p + td, or x =
+t
.
4
1
x
and expanding the vector form from part (a) gives the parametric form x = −4+t,
Letting x =
y
y = 4 + t. 28
CHAPTER 1. VECTORS
5. (a)
(b)
6. (a)
(b)
7. (a)
(b)
0
1
In vector form, the equation of the line is x = p + td, or x = 0 + t −1.
0
4
x
Letting x = y and expanding the vector form from part (a) gives the parametric form x = t,
z
y = −t, z = 4t.
3
2
In vector form, the equation of the line is x = p + td, or x = 0 + t 5.
−2
0
x
x
3 + 2t
Letting x = y and expanding the vector form from part (a) gives y = 5t , which
z
z
−2
yields the parametric form x = 3 + 2t, y = 5t, z = −2.
3
0
2
· x − 1 = 0.
The normal form is n · (x − p) = 0, or
1
0
x
Letting x = y, we get
z
3
x
0
3
x
2 · y − 1 = 2 · y − 1 = 3x + 2(y − 1) + z = 0.
1
z
0
1
z
Expanding and simplifying gives the general form 3x + 2y + z = 2.
2
3
8. (a) The normal form is n · (x − p) = 0, or 5 · x − 0 = 0.
0
−2
x
(b) Letting x = y, we get
z
2
x
3
2
x−3
5 · y − 0 = 5 · y = 2(x − 3) + 5y = 0.
0
z
−2
0
z+2
Expanding and simplifying gives the general form 2x + 5y = 6.
9. (a) In vector form, the equation of the line is x = p + su + tv, or
0
2
−3
x = 0 + s 1 + t 2 .
0
2
1
x
(b) Letting x = y and expanding the vector form from part (a) gives
z
x
2s − 3t
y = s + 2t
z
2s + t
which yields the parametric form the parametric form x = 2s − 3t, y = s + 2t, z = 2s + t. 1.3. LINES AND PLANES
29
10. (a) In vector form, the equation of the line is x = p + su + tv, or
6
0
−1
x = −4 + s 1 + t 1 .
−3
1
1
x
(b) Letting x = y and expanding the vector form from part (a) gives the parametric form x = 6 −t,
z
y = −4 + s + t, z = −3 + s + t.
11. Any pair of points on ` determine a direction vector, so we use P and Q. We choose P to represent
# »
the point on the line. Then a direction vector forthe line is
d
= P Q = (3, 0) − (1, −2) = (2, 2). The
2
1
+t
.
vector equation for the line is x = p + td, or x =
−2
2
12. Any pair of points on ` determine a direction vector, so we use# P»and Q. We choose P to represent the
point on the line. Then a direction vector for the line
is d
= P Q= (−2,
1, 3) − (0, 1, −1) = (−2, 0, 4).
0
−2
The vector equation for the line is x = p + td, or x = 1 + t 0.
−1
4
13. We must find two direction vectors, u and v. Since P , Q, and R lie in a plane, we compute We get
two direction vectors
# »
u = P Q = q − p = (4, 0, 2) − (1, 1, 1) = (3, −1, 1)
# »
v = P R = r − p = (0, 1, −1) − (1, 1, 1) = (−1, 0, −2).
Since u and v are not scalar multiples of each other, they will serve as direction vectors (if they were
parallel to each other, wewould
have
but a line). So the vector equation for the plane is
not aplane
1
3
−1
x = p + su + tv, or x = 1 + s −1 + t 0.
1
1
−2
14. We must find two direction vectors, u and v. Since P , Q, and R lie in a plane, we compute We get
two direction vectors
# »
u = P Q = q − p = (1, 0, 1) − (1, 1, 0) = (0, −1, 1)
# »
v = P R = r − p = (0, 1, 1) − (1, 1, 0) = (−1, 0, 1).
Since u and v are not scalar multiples of each other, they will serve as direction vectors (if they were
parallel to each other, wewould
have
but a line). So the vector equation for the plane is
not aplane
1
0
−1
x = p + su + tv, or x = 1 + s −1 + t 0.
0
1
1
15. The parametric and associated vector forms x = p + td found below are not unique.
(a) As in the remarks prior to Example 1.20, we start by letting x = t. Substituting x = t into
y = 3x − 1 gives
3t − 1. So
we get parametric equations x = t, y = 3t − 1, and corresponding
y=
0
1
x
=
+t
.
vector form
y
−1
3
(b) In this case since the coefficient of y is 2, we start by letting x = 2t. Substituting x = 2t into
3x + 2y = 5 gives 3 · 2t + 2y = 5, which gives y = −3t + 25 . So we get parametric equations x = 2t,
y = 52 − 3t, with corresponding vector equation
" # " #
" #
0
x
2
= 5 +t
.
y
−3
2 30
CHAPTER 1. VECTORS
Note that theequation
was of the form ax + by = c with a = 3, b = 2, and that a direction vector
b
. This is true in general.
was given by
−a
16. Note that x = p + t(q − p) is the line that passes through p (when t = 0) and q (when t = 1). We
write d = q − p; this is a direction vector for the line through p and q.
(a) As noted above, the line p + td passes through P at t = 0 and through Q at t = 1. So as t varies
from 0 to 1, the line describes the line segment P Q.
(b) As shown in Exploration: Vectors and Geometry, to find the midpoint of P Q, we start at P
# »
and travel half the length of P Q in the direction of the vector P Q = q − p. That is, the midpoint
1
of P Q is the head of the vector p + 2 (q − p). Since x = p + t(q − p), we see that this line passes
through the midpoint at t = 21 , and that the midpoint is in fact p + 12 (q − p) = 12 (p + q).
(c) From part (b), the midpoint is
(d) From part (b), the midpoint is
1
2
1
2
([2, −3] + [0, 1]) = 12 [2, −2] = [1, −1].
([1, 0, 1] + [4, 1, −2]) = 12 [5, 1, −1] = 52 , 21 , − 21 .
(e) Again from Exploration: Vectors and Geometry, the vector whose head is 13 of the way from
P to Q along P Q is x1 = 31 (2p + q). Similarly, the vector whose head is 32 of the way from P to
Q along P Q is also the vector one third of the way from Q to P along QP ; applying the same
formula gives for this point x2 = 31 (2q + p). When p = [2, −3] and q = [0, 1], we get
1
(2[2, −3] + [0, 1]) =
3
1
x2 = (2[0, 1] + [2, −3]) =
3
x1 =
4
1
5
[4, −5] = , −
3
3
3
1
1
2
[2, −1] = , − .
3
3
3
(f) Using the formulas from part (e) with p = [1, 0, −1] and q = [4, 1, −2] gives
1
(2[1, 0, −1] + [4, 1, −2]) =
3
1
x2 = (2[4, 1, −2] + [1, 0, −1]) =
3
x1 =
1 4
1
[6, 1, −4] = 2, , −
3
3
3
2 5
1
[9, 2, −5] = 3, , −
.
3
3
3
17. A line `1 with slope m1 has equation y = m1 x + b1 , or −m1 x + y = b1 . Similarly, a line `2 with slope
−m1
m2 has equation y = m2 x + b2 , or −m2 x + y = b2 . Thus the normal vector for `1 is n1 =
, and
1
−m2
the normal vector for `2 is n2 =
. Now, `1 and `2 are perpendicular if and only if their normal
1
vectors are perpendicular, i.e., if and only if n1 · n2 = 0. But
−m1
−m2
n1 · n2 =
·
= m1 m2 + 1,
1
1
so that the normal vectors are perpendicular if and only if m1 m2 +1 = 0, i.e., if and only if m1 m2 = −1.
18. Suppose the line ` has direction vector d, and the plane P has normal vector n. Then if d · n = 0 (d
and n are orthogonal), then the line ` is parallel to the plane P. If on the other hand d and n are
parallel, so that d = n, then ` is perpendicular to P .
2
(a) Since the general form of P is 2x + 3y − z = 1, its normal vector is n = 3. Since d = 1n, we
−1
see that ` is perpendicular to P . 31
1.3. LINES AND PLANES
4
(b) Since the general form of P is 4x − y + 5z = 0, its normal vector is n = −1 . Since
5
2
4
d · n = 3 · −1 = 2 · 4 + 3 · (−1) − 1 · 5 = 0,
−1
5
` is parallel to P .
1
(c) Since the general form of P is x − y − z = 3, its normal vector is n = −1. Since
−1
2
1
d · n = 3 · −1 = 2 · 1 + 3 · (−1) − 1 · (−1) = 0,
−1
−1
` is parallel to P .
4
(d) Since the general form of P is 4x + 6y − 2z = 0, its normal vector is n = 6. Since
−2
2
4
1
1
d = 3 = 6 = n,
2
2
−1
−2
` is perpendicular to P .
19. Suppose the plane P1 has normal vector n1 , and the plane P has normal vector n. Then if n1 · n = 0
(n1 and n are orthogonal), then P1 is perpendicular to P. If on the other hand n1 and n are
parallel, so that n1 = cn, thenP1is parallel to P. Note that in this exercise, P1 has the equation
4
4x − y + 5z = 2, so that n1 = −1.
5
2
(a) Since the general form of P is 2x + 3y − z = 1, its normal vector is n = 3. Since
−1
4
2
n1 · n = −1 · 3 = 4 · 2 − 1 · 3 + 5 · (−1) = 0,
5
−1
the normal vectors are perpendicular, and thus P1 is perpendicular to P .
4
(b) Since the general form of P is 4x − y + 5z = 0, its normal vector is n = −1. Since n1 = n,
5
P1 is parallel to P.
1
(c) Since the general form of P is x − y − z = 3, its normal vector is n = −1. Since
−1
4
1
n1 · n = −1 · −1 = 4 · 1 − 1 · (−1) + 5 · (−1) = 0,
5
−1
the normal vectors are perpendicular, and thus P1 is perpendicular to P . 32
CHAPTER 1. VECTORS
4
(d) Since the general form of P is 4x + 6y − 2z = 0, its normal vector is n = 6. Since
−2
4
4
n1 · n = −1 · 6 = 4 · 4 − 1 · 6 + 5 · (−2) = 0,
5
−2
the normal vectors are perpendicular, and thus P1 is perpendicular to P .
20. Since the vector form is x = p + td, we use the given information to determine
p and d. The general
2
. Our line is perpendicular
equation of the given line is 2x − 3y = 1, so its normal vector is n =
−3
2
. Furthermore, since our line passes through the
to that line, so it has direction vector d = n =
−3
2
. Thus the vector form of the line perpendicular to 2x − 3y = 1
point P = (2, −1), we have p =
−1
through the point P = (2, −1) is
2
2
x
=
+t
.
y
−1
−3
21. Since the vector form is x = p + td, we use the given information todetermine
p and d. The general
2
equation of the given line is 2x − 3y = 1, so its normal vector is n =
. Our line is parallel to that
−3
3
(note that d · n = 0). Since our line passes through the point
line, so it has direction vector d =
2
2
, so that the vector equation of the line parallel to 2x − 3y = 1 through
P = (2, −1), we have p =
−1
the point P = (2, −1) is
2
3
x
=
+t
.
y
−1
2
22. Since the vector form is x = p + td, we use the given information to determine p and d. A line is
perpendicular to a plane if its direction vector d is the normal vector n of
the
plane. The general
1
equation of the given plane is x − 3y + 2z = 5, so its normal vector is n = −3. Thus the direction
2
1
vector of our line is d = −3. Furthermore, since our line passes through the point P = (−1, 0, 3), we
2
−1
have p = 0. So the vector form of the line perpendicular to x − 3y + 2z = 5 through P = (−1, 0, 3)
3
is
x
1
−1
y = 0 + t −3 .
z
3
2
23. Since the vector form is x = p + td, we use the given information to determine p and d. Since the
given line has parametric equations
x
1
−1
x = 1 − t, y = 2 + 3t, z = −2 − t, it has vector form y = 2 + t 3 .
z
−2
−1 1.3. LINES AND PLANES
33
−1
So its direction vector is 3, and this must be the direction vector d of the line we want, which is
−1
−1
parallel to the given line. Since our line passes through the point P = (−1, 0, 3), we have p = 0.
3
So the vector form of the line parallel to the given line through P = (−1, 0, 3) is
x
−1
−1
y = 0 + t 3 .
z
3
−1
24. Since the normal form is n · x = n · p, we use the given information to determine n and p. Note that
a plane is parallel to a given plane if their normal vectors
equal. Since the general form of the
are
6
given plane is 6x − y + 2z = 3, its normal vector is n = −1, so this must be a normal vector of the
2
desired
as well. Furthermore, since our plane passes through the point P = (0, −2, 5), we have
plane
0
p = −2. So the normal form of the plane parallel to 6x − y + 2z = 3 through (0, −2, 5) is
5
6
x
6
0
6
x
−1 · y = −1 · −2 or −1 · y = 12.
2
z
2
5
2
z
25. Using Figure 1.34 in Section 1.2 for reference, we will find a normal vector n and a point vector p for
each of the sides, then substitute into n · x = n · p to get an equation for each plane.
(a) Start with
P1 determined by the face of the cube in the xy-plane. Clearly a normal vector for
1
P1 is n = 0, or any vector parallel to the x-axis. Also, the plane passes through P = (0, 0, 0),
0
0
so we set p = 0. Then substituting gives
0
1
x
1
0
0 · y = 0 · 0 or x = 0.
0
z
0
0
So the general equation for P1 is x = 0. Applying the same argument above to the plane P2
determined by the face in the xz-plane gives a general equation of y = 0, and similarly the plane
P3 determined by the face in the xy-plane gives a general equation of z = 0.
Now consider P4 , the plane containing the face parallel to the face
in
the yz-plane but passing
1
through (1, 1, 1). Since P4 is parallel to P1 , its normal vector is also 0; since it passes through
0
1
(1, 1, 1), we set p = 1 . Then substituting gives
1
1
x
1
1
0 · y = 0 · 1 or x = 1.
0
z
0
1
So the general equation for P4 is x = 1. Similarly, the general equations for P5 and P6 are
y = 1 and z = 1. 34
CHAPTER 1. VECTORS
x
(b) Let n = y be a normal vector for the desired plane P. Since P is perpendicular to the
z
xy-plane, their normal vectors must be orthogonal. Thus
x
0
y · 0 = x · 0 + y · 0 + z · 1 = z = 0.
z
1
x
Thus z = 0, so the normal vector is of the form n = y. But the normal vector is also
0
perpendicular to the plane in question, by definition. Since that plane contains both the origin
and (1, 1, 1), the normal vector is orthogonal to (1, 1, 1) − (0, 0, 0):
x
1
y · 1 = x · 1 + y · 1 + z · 0 = x + y = 0.
0
1
x
Thus x + y = 0, so that y = −x. So finally, a normal vector to P is given by n = −x for
0
1
any nonzero x. We may as well choose x = 1, giving n = −1. Since the plane passes through
0
(0, 0, 0), we let p = 0. Then substituting in n · x = n · p gives
1
x
1
0
−1 · y = −1 · 0 , or x − y = 0.
0
z
0
0
Thus the general equation for the plane perpendicular to the xy-plane and containing the diagonal
from the origin to (1, 1, 1) is x − y = 0.
(c) As in Example 1.22 (Figure 1.34)
in Section 1.2, use u = [0, 1, 1] and v = [1, 0, 1] as two vectors
x
in the required plane. If n = y is a normal vector to the plane, then n · u = 0 = n · v:
z
x
0
x
1
n · u = y · 1 = y + z = 0 ⇒ y = −z,
n · v = y · 0 = x + z = 0 ⇒ x = −z.
z
1
z
1
1
−z
Thus the normal vector is of the form n = −z for any z. Taking z = −1 gives n = 1.
z
−1
Now, the side diagonals pass through (0, 0, 0), so set p = 0. Then n · x = n · p yields
1
x
1
0
1 · y = 1 · 0 , or x + y − z = 0.
−1
z
−1
0
The general equation for the plane containing the side diagonals is x + y − z = 0.
26. Finding the distance between points A and B is equivalent to finding d(a, b), where a is the vector
from the origin to A, and similarly for b. Given x = [x, y, z], p = [1, 0, −2], and q = [5, 2, 4], we want
to solve d(x, p) = d(x, q); that is,
p
p
d(x, p) = (x − 1)2 + (y − 0)2 + (z + 2)2 = (x − 5)2 + (y − 2)2 + (z − 4)2 = d(x, q). 35
1.3. LINES AND PLANES
Squaring both sides gives
(x − 1)2 + (y − 0)2 + (z + 2)2 = (x − 5)2 + (y − 2)2 + (z − 4)2
⇒
x2 − 2x + 1 + y 2 + z 2 + 4z + 4 = x2 − 10x + 25 + y 2 − 4y + 4 + z 2 − 8z + 16
8x + 4y + 12z = 40 ⇒
2x + y + 3z = 10.
⇒
Thus all such points (x, y, z) lie on the plane 2x + y + 3z = 10.
27. To calculate d(Q, `) =
|ax0 +by0 −c|
√ 2
,
a +b2
we first put ` into general form. With d =
since then n · d = 0. Then we have
n·x=n·p ⇒
1
1
, we get n =
−1
1
1
x
1 −1
·
=
= 1.
1
y
1
2
Thus x + y = 1 and thus a = b = c = 1. Since Q = (2, 2) = (x0 , y0 ), we have
d(Q, `) =
√
3 2
3
|1 · 2 + 1 · 2 − 1|
√
= √ =
.
2
2
12 + 12
−2
28. Comparing the given equation to x = p + td, we get P = (1, 1, 1) and d = 0. As suggested by
3
# »
Figure 1.63, we need to calculate #the
» length of RQ, where R is the point on the line at the foot of the
perpendicular from Q. So if v = P Q, then
# »
# »
P R = projd v,
RQ = v − projd v.
0
1
−1
# »
Now, v = P Q = q − p = 1 − 1 = 0, so that
0
1
−1
projd v =
Thus
d·v
d·d
d=
−2 · (−1) + 3 · (−1)
−2 · (−2) + 3 · 3
2
−2
13
0 = 0 .
3
− 13
3
2
−1
− 15
13 13
v − projd v = 0 − 0 = 0 .
3
10
− 13
−13
−1
Then the distance d(Q, `) from Q to ` is
kv − projd vk =
|ax +by +cz −d|
5
13
√
3
p
0 = 5 32 + 22 = 5 13 .
13
13
2
0
0
0
√
29. To calculate d(Q, P) =
, we first note that the plane has equation x + y − z = 0, so
a 2 +b 2 +c 2
that a = b = 1, c = −1, and d = 0. Also, Q = (2, 2, 2), so that x0 = y0 = z0 = 2. Hence
d(Q, P) =
√
2 3
2
|1 · 2 + 1 · 2 − 1 · 2 − 0|
.
p
=√ =
2
2
2
3
3
1 + 1 + (−1) 36
CHAPTER 1. VECTORS
|ax +by +cz −d|
0
0
0
√
30. To calculate d(Q, P) =
, we first note that the plane has equation x − 2y + 2z = 1, so
a 2 +b 2 +c 2
that a = 1, b = −2, c = 2, and d = 1. Also, Q = (0, 0, 0), so that x0 = y0 = z0 = 0. Hence
1
|1 · 0 − 2 · 0 + 2 · 0 − 1|
p
= .
3
12 + (−2)2 + 22
d(Q, P) =
# »
# »
31. Figure 1.66 suggests that we let v = P Q;
w = P R = projd v. Comparing
given
line ` to
then
the
# »
−1
3
1
2
. Then v = P Q = q − p =
−
=
. Next,
x = p + td, we get P = (−1, 2) and d =
−1
2
2
0
w = projd v =
3
1 · 3 + (−1) · 0
1
1
d·v
=
.
d=
d·d
1 · 1 + (−1) · (−1) −1
2 −1
So
" # " # " #
1
3
−1
# »
2 = 2 .
r = p + P R = p + projd v = p + w =
+
1
− 32
2
2
So the point R on ` that is closest to Q is 21 , 21 .
# »
# »
32. Figure 1.66 suggests that we let
= P Q; then P R = projd v. Comparing
givenline ` to x = p+td,
v
the
0
1
−2
−1
# »
we get P = (1, 1, 1) and d = 0. Then v = P Q = q − p = 1 − 1 = 0. Next,
3
0
1
−1
projd v =
d·v
d·d
d=
−2 · (−1) + 3 · (−1)
(−2)2 + 32
So
2
−2
13
0 = 0 .
3
− 13
3
2 15
1
# »
13 13
r = p + P R = p + projd v = 1 + 0 = 1 .
3
10
− 13
1
13
15
10
So the point R on ` that is closest to Q is 13 , 1, 13 .
# »
# »
33. Figure 1.67 suggests we let v = P Q, where Pis some
point on the plane; then QR = projn v. The
1
equation of the plane is x + y − z = 0, so n = 1. Setting y = 0 shows that P = (1, 0, 1) is a point
−1
on the plane. Then
2
1
1
# »
v = P Q = q − p = 2 − 0 = 2 ,
2
1
1
so that
projn v =
Finally,
n · v
n·n
n=
1·1+1·1−1·1
12 + 12 + (−1)2
2
1
3
= 2
1 3 .
−1
−32
2
4
1
1
3
3
# » # »
+ − 2 = 4
r = p + P Q + P R = p + v − projn v = 0 2 3 3 .
8
1
1
− 23
3
4 4 8
Therefore, the point R in P that is closest to Q is 3 , 3 , 3 . 37
1.3. LINES AND PLANES
# »
# »
34. Figure 1.67 suggests we let v = P Q, where P is some
point on the plane; then QR = projn v. The
1
equation of the plane is x − 2y + 2z = 1, so n = −2 . Setting y = z = 0 shows that P = (1, 0, 0) is a
2
point on the plane. Then
0
1
−1
# »
v = P Q = q − p = 0 − 0 = 0 ,
0
0
0
so that
1
−1
n · v
1 · (−1)
92
n=
projn v =
−2 = 9 .
2
2
2
1 + (−2) + 2
n·n
− 29
2
Finally,
− 19
− 19
1
−1
# » # »
+ − 2 = 2
r = p + P Q + P R = p + v − projn v = 0 0 9 9 .
0
0
− 29
− 29
Therefore, the point R in P that is closest to Q is − 19 , 92 , − 29 .
35. Since the given lines `1 and `2 are parallel, choose arbitrary points Q on `1 and P on `2 , say Q = (1, 1)
and P = (5, 4). The direction vector of `2 is d = [−2, 3]. Then
# »
5
−4
1
v = PQ = q − p =
−
=
,
1
4
−3
so that
1 −2
−2 · (−4) + 3 · (−3) −2
d·v
=−
.
d=
2
2
3
3
d·d
(−2) + 3
13
Then the distance between the lines is given by
√
−4
1 −2
= 1 −54 = 18 3 = 18 13.
kv − projd vk =
−3 + 13
3
13 2 13
13 −36
projd v =
36. Since the given lines `1 and `2 are parallel, choose arbitrary points Q on `1 and P on `2 , say Q =
(1, 0, −1) and P = (0, 1, 1). The direction vector of `2 is d = [1, 1, 1]. Then
1
0
1
# »
v = P Q = q − p = 0 − 1 = −1 ,
−1
1
−2
so that
projd v =
d·v
d·d
d=
1 · 1 + 1 · (−1) + 1 · (−2)
12 + 12 + 12
1
1
1 = − 2 1 .
3
1
1
Then the distance between the lines is given by
5
1
√
1
3
2
42
1p 2
.
5 + (−1)2 + (−4)2 =
kv − projd vk = −1 + 1 = − 13 =
3
3
3
4
−2
1 −3
37. Since P1 and P2 are parallel, we choose an arbitrary point on P1 , say Q = (0, 0, 0), and compute
d(Q, P2 ). Since the equation of P2 is 2x + y − 2z = 5, we have a = 2, b = 1, c = −2, and d = 5; since
Q = (0, 0, 0), we have x0 = y0 = z0 = 0. Thus the distance is
d(P1 , P2 ) = d(Q, P2 ) =
5
|ax0 + by0 + cz0 − d|
|2 · 0 + 1 · 0 − 2 · 0 − 5|
√
√
=
= .
3
22 + 12 + 22
a 2 + b2 + c 2 38
CHAPTER 1. VECTORS
38. Since P1 and P2 are parallel, we choose an arbitrary point on P1 , say Q = (1, 0, 0), and compute
d(Q, P2 ). Since the equation of P2 is x + y + z = 3, we have a = b = c = 1 and d = 3; since
Q = (1, 0, 0), we have x0 = 1, y0 = 0, and z0 = 0. Thus the distance is
√
2 3
2
|ax0 + by0 + cz0 − d|
|1 · 1 + 1 · 0 + 1 · 0 − 3|
.
√
√
=
= √ =
d(P1 , P2 ) = d(Q, P2 ) =
2
2
2
2
2
2
3
3
1
+
1
+
1
a +b +c
# »
a
|ax +by −c|
, n · a = c, and B = (x0 , y0 ). If v = AB =
39. We wish to show that d(B, `) = √0 2 0 2 , where n =
a +b
b
b − a, then
x
a
· 0 − c = ax0 + by0 − c.
n · v = n · (b − a) = n · b − n · a =
b
y0
Then from Figure 1.65, we see that
|n · v|
|ax0 + by0 − c|
n·v
√
d(B, `) = kprojn vk =
.
=
n =
knk
n·n
a 2 + b2
a
|ax0 +by0 +cz0 −d|
40. We wish to show that d(B, `) = √
, where n = b , n · a = d, and B = (x0 , y0 , z0 ). If
a 2 +b 2 +c 2
c
# »
v = AB = b − a, then
a
x0
n · v = n · (b − a) = n · b − n · a = b · y0 − d = ax0 + by0 + cz0 − d.
c
z0
Then from Figure 1.65, we see that
|ax0 + by0 + cz0 − d|
n · v |n · v|
√
d(B, `) = kprojn vk =
.
=
n =
knk
n·n
a 2 + b2 + c 2
41. Choose B = (x0 , y0 ) on `1 ; since `1 and
`2 are parallel, the distance between them is d(B, `2 ). Then
x
a
· 0 = ax0 + by0 = c1 . Choose A on `2 , so that n · a = c2 . Set
since B lies on `1 , we have n · b =
b
y0
v = b − a. Then using the formula in Exercise 39, the distance is
d(`1 , `2 ) = d(B, `2 ) =
|n · v|
|n · (b − a)| |n · b − n · a|
|c1 − c2 |
.
=
=
=
knk
knk
knk
knk
42. Choose B = (x0 , y0 , z0 ) on P1 ; since P1 and
are
parallel, the distance between them is d(B, P2 ).
2
P
a
x0
Then since B lies on P1 , we have n · b = b · y0 = ax0 + by0 + cz0 = d1 . Choose A on P2 , so
c
z0
that n · a = d2 . Set v = b − a. Then using the formula in Exercise 40, the distance is
|n · v|
|n · (b − a)| |n · b − n · a|
|d1 − d2 |
.
=
=
=
knk
knk
knk
knk
1
2
43. Since P1 has normal vector n1 = 1 and P2 has normal vector n2 = 1, the angle θ between
1
−2
the normal vectors satisfies
d(P1 , P2 ) = d(B, P2 ) =
cos θ =
Thus
n1 · n2
1
1 · 2 + 1 · 1 + 1 · (−2)
p
= √
= √ .
kn1 k kn2 k
3 3
12 + 12 + 12 22 + 12 + (−2)2
θ = cos−1
1
√
3 3
≈ 78.9◦ . 39
1.3. LINES AND PLANES
3
1
44. Since P1 has normal vector n1 = −1 and P2 has normal vector n2 = 4 , the angle θ between
2
−1
the normal vectors satisfies
cos θ =
3
1
3 · 1 − 1 · 4 + 2 · (−1)
n1 · n2
p
= −√ √ = − √ .
= p 2
kn 1 k kn 2 k
14 18
3 + (−1)2 + 22 12 + 42 + (−1)2
28
This is an obtuse angle, so the acute angle is
1
≈ 79.1◦ .
π − θ = π − cos−1 − √
28
45. First, to see that P and ` intersect, substitute the parametric equations for ` into the equation for P ,
giving
x + y + 2z = (2 + t) + (1 − 2t) + 2(3 + t) = 9 + t = 0,
so that t = −9 represents the point of intersection,
which is thus (2 + (−9), 1 − 2(−
9), 3+ (−9)) =
1
1
(−7, 19, −6). Now, the normal to P is n = 1, and a direction vector for ` is d = −2. So if θ is
2
1
the angle between n and d, then θ satisfies
cos θ =
so that
1
1 · 1 + 1 · (−2) + 2 · 1
n·d
p
= √
= ,
6
knk kdk
12 + 12 + 12 12 + (−2)2 + 12
θ = cos−1
1
≈ 80.4◦ .
6
Thus the angle between the line and the plane is 90◦ − 80.4◦ ≈ 9.6◦ .
46. First, to see that P and ` intersect, substitute the parametric equations for ` into the equation for P ,
giving
4x − y − z = 4 · t − (1 + 2t) − (2 + 3t) = −t − 3 = 6,
so that t = −9 represents the point of intersection,
· (−9)) =
which is thus (−9, 1 + 2 · (−9) , 2 +3
4
1
(−9, −17, −25). Now, the normal to P is n = −1, and a direction vector for ` is d = 2. So if θ
−1
3
is the angle between n and d, then θ satisfies
cos θ =
1
n·d
4·1−1·2−1·3
√
= −√ √ .
= √ 2
knk kdk
4 + 12 + 12 12 + 22 + 32
18 14
This corresponds to an obtuse angle, so the acute angle between the two is
1
≈ 86.4◦ .
θ = π − cos−1 −√ √
18 14
Thus the angle between the line and the plane is 90◦ − 86.4◦ ≈ 3.6◦ .
47. We have p = v − c n, so that c n = v − p. Take the dot product of both sides with n, giving
(c n) · n = (v − p) · n
c(n · n) = v · n − p · n
⇒
⇒
c(n · n) = v · n (since p and n are orthogonal)
n·v
c=
.
n·n
⇒ 40
CHAPTER 1. VECTORS
Note that another interpretation of the figure is that c n = projn v =
c = n·v
.
n·n
Now substitute this value of c into the original equation, giving
n · v
p = v − cn = v −
n.
n·n
n·v
n·n
n, which also implies that
1
48. (a) A normal vector to the plane x + y + z = 0 is n = 1. Then
1
so that c = −31. Then
1
1
n · v = 1 · 0 = 1 · 1 + 1 · 0 + 1 · (−2) = −1
1
−2
1
1
n · n = 1 · 1 = 1 · 1 + 1 · 1 + 1 · 1 = 3,
1
1
4
1
1
3
1 1
p=v−
n = 0 + 1 = 3 .
n·n
3
5
−3
1
−2
n · v
3
(b) A normal vector to the plane 3x − y + z = 0 is n = −1. Then
1
so that c =
1
.
11
3
1
n · v = −1 · 0 = 3 · 1 − 1 · 0 + 1 · (−2) = 1
1
−2
3
3
n · n = −1 · −1 = 3 · 3 − 1 · (−1) + 1 · 1 = 11,
1
1
Then
p=v−
8
1
3
1 11
1
n = 0 −
−1 = 11
.
n·n
11
23
1
−2
− 11
n · v
1
(c) A normal vector to the plane x − 2z = 0 is n = 0. Then
−2
1
1
n · v = 0 · 0 = 1 · 1 + 0 · 0 − 2 · (−2) = 5
−2
−2
1
1
n · n = 0 · 0 = 1 · 1 + 0 · 0 − 2 · (−2) = 5,
−2
−2 41
Exploration: The Cross Product
so that c = 1. Then
1
1
p=v−
n = 0 − 0 = 0.
n·n
−2
−2
n · v
Note that the projection is 0 because the vector is normal to the plane, so its projection onto the
plane is a single point.
2
(d) A normal vector to the plane 2x − 3y + z = 0 is n = −3. Then
1
2
1
n · v = −3 · 0 = 2 · 1 − 3 · 0 + 1 · (−2) = 0
1
−2
2
2
n · n = −3 · −3 = 2 · 2 − 3 · (−3) + 1 · 1 = 14,
1
1
1
so that c = 0. Thus p = v = 0. Note that the projection is the vector itself because the
−2
vector is parallel to the plane, so it is orthogonal to the normal vector.
Exploration: The Cross Product
3
u2 v 3 − u3 v 2
1 · 3 − 0 · (−1)
1. (a) u × v = u3 v1 − u1 v3 = 1 · 3 − 0 · 2 = 3.
u1 v 2 − u2 v 1
0 · (−1) − 1 · 3
−3
−1 · 1 − 2 · 1
−3
u2 v 3 − u3 v 2
(b) u × v = u3 v1 − u1 v3 = 2 · 0 − 3 · 1 = −3.
u1 v 2 − u2 v 1
3 · 1 − (−1) · 0
3
u2 v 3 − u3 v 2
2 · (−6) − 3 · (−4)
(c) u × v = u3 v1 − u1 v3 = 3 · 2 − (−1) · (−6) = 0.
u1 v 2 − u2 v 1
−1 · (−4) − 2 · 2
1
u2 v 3 − u3 v 2
1·3−1·2
(d) u × v = u3 v1 − u1 v3 = 1 · 1 − 1 · 3 = −2 .
u1 v 2 − u2 v 1
1·2−1·1
1
2. We have
1
0
0
0·0−0·1
e1 × e2 = 0 × 1 = 0 · 0 − 1 · 0 = 0 = e3
0
0
1·1−0·0
1
0
0
1
1·1−0·0
e2 × e3 = 1 × 0 = 0 · 0 − 0 · 1 = 0 = e1
0
1
0·0−1·0
0
0
1
0
0·0−1·0
e3 × e1 = 0 × 0 = 1 · 1 − 0 · 0 = 1 = e2 .
1
0
0·0−0·1
0 42
CHAPTER 1. VECTORS
3. Two vectors are orthogonal if their dot product equals zero. But
u1
u 2 v 3 − u3 v 2
(u × v) · u = u3 v1 − u1 v3 · u2
u 1 v 2 − u2 v 1
u3
= (u2 v3 − u3 v2 )u1 + (u3 v1 − u1 v3 )u2 + (u1 v2 − u2 v1 )u3
= (u2 v3 u1 − u1 v3 u2 ) + (u3 v1 u2 − u2 v1 u3 ) + (u1 v2 u3 − u3 v2 u1 ) = 0
v1
u 2 v 3 − u3 v 2
(u × v) · v = u3 v1 − u1 v3 · v2
u 1 v 2 − u2 v 1
v3
= (u2 v3 − u3 v2 )v1 + (u3 v1 − u1 v3 )v2 + (u1 v2 − u2 v1 )v3
= (u2 v3 v1 − u2 v1 v3 ) + (u3 v1 v2 − u3 v2 v1 ) + (u1 v2 v3 − u1 v3 v2 ) = 0.
4. (a) By Exercise 1, a vector normal to the plane is
0
3
3
1 · 2 − 1 · (−1)
n = u × v = 1 × −1 = 1 · 3 − 0 · 2 = 3 .
1
2
0 · (−1) − 1 · 3
−3
So the normal form for the equation of this plane is n · x = n · p, or
3
x
3
1
3 · y = 3 · 0 = 9.
−3
z
−3
−2
This simplifies to 3x + 3y − 3z = 9, or x + y − z = 3.
2
1
# »
# »
(b) Two vectors in the plane are u = P Q = 1 and v = P R = 3. So by Exercise 1, a vector
1
−2
normal to the plane is
2
1
−5
1 · (−2) − 1 · 3
3 = 1 · 1 − 2 · (−2) = 5 .
n=u×v = 1 ×
1
−2
2·3−1·1
5
So the normal form for the equation of this plane is n · x = n · p, or
x
0
−5
−5
5 · y = 5 · −1 = 0.
5
z
5
1
5. (a)
(b)
(c)
(d)
This simplifies to −5x + 5y + 5z = 0, or x − y − z = 0.
v 2 u3 − v 3 u2
u2 v 3 − u3 v 2
v × u = v3 u1 − u3 v1 = − u3 v1 − u1 v3 = −(u × v).
v 1 u2 − v 2 u1
u1 v 2 − u2 v 1
u1
0
0
u2 · 0 − u3 · 0
u × 0 = u2 × 0 = u3 · 0 − u1 · 0 = 0 = 0.
u3
0
u1 · 0 − u2 · 0
0
0
u2 u 3 − u 3 u2
u × u = u3 u1 − u1 u3 = 0 = 0.
u1 u 2 − u 2 u1
0
u2 kv3 − u3 kv2
u2 v 3 − u3 v 2
u × kv = u3 kv1 − u1 kv3 = k u3 v1 − u1 v3 = k(u × v).
u1 kv2 − u2 kv1
u1 v 2 − u2 v 1 43
Exploration: The Cross Product
(e) u × ku = k(u × u) = k(0) = 0 by parts (d) and (c).
(f) Compute the cross-product:
u2 (v3 + w3 ) − u3 (v2 + w2 )
u × (v + w) = u3 (v1 + w1 ) − u1 (v3 + w3 )
u1 (v2 + w2 ) − u2 (v1 + w1 )
(u2 v3 − u3 v2 ) + (u2 w3 − u3 w2 )
= (u3 v1 − u1 v3 ) + (u3 w1 − u1 w3 )
(u1 v2 − u2 v1 ) + (u1 w2 − u2 w1 )
u2 v 3 − u3 v 2
u2 w 3 − u 3 w 2
= u3 v1 − u1 v3 + u3 w1 − u1 w3
u1 v 2 − u2 v 1
u1 w 2 − u 2 w 1
= u × v + u × w.
6. In each case, simply compute:
(a)
u1
v2 w 3 − v3 w2
u · (v × w) = u2 · v3 w1 − v1 w3
u3
v1 w 2 − v2 w1
= u1 v 2 w 3 − u1 v 3 w 2 + u 2 v 3 w 1 − u2 v 1 w 3 + u 3 v 1 w 2 − u3 v 2 w 1
= (u2 v3 − u3 v2 )w1 + (u3 v1 − u1 v3 )w2 + (u1 v2 − u2 v1 )w3
= (u × v) · w.
(b)
u1
v2 w3 − v3 w2
u × (v × w) = u2 × v3 w1 − v1 w3
u3
v1 w2 − v2 w1
u2 (v1 w2 − v2 w1 ) − u3 (v3 w1 − v1 w3 )
= u3 (v2 w3 − v3 w2 ) − u1 (v1 w2 − v2 w1 )
u1 (v3 w1 − v1 w3 ) − u2 (v2 w3 − v3 w2 )
(u1 w1 + u2 w2 + u3 w3 )v1 − (u1 v1 + u2 v2 + u3 v3 )w1
= (u1 w1 + u2 w2 + u3 w3 )v2 − (u1 v1 + u2 v2 + u3 v3 )w2
(u1 w1 + u2 w2 + u3 w3 )v3 − (u1 v1 + u2 v2 + u3 v3 )w3
v1
w1
= (u1 w1 + u2 w2 + u3 w3 ) v2 − (u1 v1 + u2 v2 + u3 v3 ) w2
v3
w3
= (u · w)v − (u · v)w.
(c)
u2 v3 − u3 v2 2
ku × vk = u3 v1 − u1 v3
u1 v 2 − u 2 v 1
2
= (u2 v3 − u3 v2 )2 + (u3 v1 − u1 v3 )2 + (u1 v2 − u2 v1 )2
= (u12 + u22 + u32)2 (v 21 + v22 + v32 )2 − (u1 v1 + u2 v2 + u3 v3 )2
2
2
= kuk kvk − (u · v)2 .
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