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Vectors and the geometry of space




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Partes: 1, 2

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    C H A P T E R 1 1
    Vectors and the Geometry of Space
    Section 11.1

    Section 11.2

    Section 11.3

    Section 11.4

    Section 11.5

    Section 11.6

    Section 11.7
    Vectors in the Plane……………………………………………………………………..2

    Space Coordinates and Vectors in Space ……………………………………..13

    The Dot Product of Two Vectors…………………………………………………22

    The Cross Product of Two Vectors in Space ………………………………..30

    Lines and Planes in Space …………………………………………………………..37

    Surfaces in Space……………………………………………………………………….50

    Cylindrical and Spherical Coordinates …………………………………………57
    Review Exercises …………………………………………………………………………………………….68

    Problem Solving ……………………………………………………………………………………………..76

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    C H A P T E R 1 1
    Vectors and the Geometry of Space

    Section 11.1 Vectors in the Plane
    1. (a) v
    (b)
    5 1, 4 2
    4, 2
    7. u
    v
    6 0, 2 3
    9 3, 5 10
    6, 5
    6, 5
    5
    4
    u
    v
    (4, 2)
    8. u
    11 4 , 4 1
    15, 3
    2
    1
    v
    x
    v
    u
    25 0, 10 13
    v
    15, 3
    1
    2
    3
    4
    5
    9. (b) v
    5 2, 5 0
    3, 5
    2. (a) v
    (b)
    3 3, 2 4
    y
    0, 6
    (c) v
    (a), (d)
    3i 5j
    y
    -3 -2 -1
    -1
    1
    2
    3
    x
    5
    4
    (3, 5)
    (5, 5)
    -2
    -3
    v
    3
    2
    v
    -4
    -5
    1
    (2, 0)
    x
    -6
    (0, – 6)
    -1
    -1
    1
    2
    3
    4
    5
    3. (a) v
    4 2, 3 3
    6, 0
    10. (b) v
    3 4, 6 6
    1, 12
    (b)
    4
    (c) v
    (a), (d)
    i 12 j
    y
    (-1, 12)
    -8
    (-6, 0)
    -6
    -4
    v
    -2
    2
    x
    v
    8
    6
    (3, 6)
    -2
    4
    2
    -4
    -8 – 6 – 4 -2
    2
    6 8 10
    x
    – 4
    4. (a) v
    1 2, 3 1
    3, 2
    – 6
    (4, -6)
    (b)
    11. (b) v
    (c) v
    6 8, 1 3
    2i 4 j
    2, 4
    (- 3, 2)
    2
    (a), (d)
    y
    6
    v
    1
    4
    2
    (8, 3)
    v
    -3
    -2
    -1
    x
    -4 -2
    2
    4
    (6, – 1)
    8
    x
    5. u
    v
    u

    6. u
    v
    u
    5 3, 6 2
    3 1, 8 4
    v

    1 4 , 8 0
    7 2, 7 1
    v
    2, 4
    2, 4

    5, 8
    5, 8
    (-2, -4)
    -6
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    6
    6
    2
    2 2
    2
    7
    2
    v
    3
    3
    5
    y
    3
    2
    3
    y
    2
    3
    2 3
    y
    Section 11.1
    Vectors in the Plane
    12. (b) v
    5 0, 1 4
    5, 3
    17. (a) 2 v
    2 3, 5
    6, 10
    (c) v 5i 3j
    (a) and (d).
    (- 5, 3)
    4
    y
    10
    8
    6
    y
    (3, 5)
    (6, 10)
    v
    2
    4
    v
    2v
    2
    x
    -6 -4
    (- 5, -1)
    -2
    -2
    2
    -2
    -2
    2
    4
    6
    8
    10
    x
    (0, – 4)
    (b) 3v
    9, 15
    13. (b) v
    6 6, 6 2
    0, 4
    (3, 5)
    (c) v
    4 j
    3
    v
    x
    (a) and (d).
    (6, 6)
    -15 -12 -9 -6 -3
    -3v -6
    -9
    (-9, – 15)
    -12
    4
    (0, 4)
    v
    -15
    2
    (6, 2)
    (c)
    7 v
    21, 35
    x
    y
    2
    4
    6
    18
    (21, 35 (
    14. (b) v
    (c) v
    3 7, 1 1
    10i
    10, 0
    15
    12
    9
    6
    (3, 5)
    (a) and (d).
    y
    3
    v
    x
    3
    -3
    -3
    3
    6
    9
    12 15 18
    2
    (-10, 0)
    v
    1
    x
    (d)
    2 v
    2, 10
    -8 -6 -4 -2
    (-3, -1)
    -2
    -3
    2 4 6 8
    (7, -1)
    4
    (3, 5)
    3
    v
    (2, 10 (
    15. (b) v
    1
    2
    3 , 3
    4
    3
    1, 5
    2
    1
    2
    3
    v
    (c) v
    i
    5
    3
    j
    -1
    -1
    1
    2
    3
    4
    5
    x
    (a) and (d)
    3
    ( 1 , 3(
    (- 1, 5 ( 2
    v

    ( 3 , 4 (
    – 2
    -1
    1
    2
    x
    16. (b) v
    (c) v
    0.84 0.12, 1.25 0.60
    0.72i 0.65 j
    0.72, 0.65
    (a) and (d).
    1.25
    1.00
    0.75
    0.50
    0.25
    (0.12, 0.60)
    (0.84, 1.25)

    (0.72, 0.65)

    v
    x
    0.25 0.50 0.75 1.00 1.25
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    2
    2
    2
    3
    3
    2
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    3
    3
    v
    1
    1
    x
    2
    y
    2
    y
    -1
    Chapter 11
    Vectors and the Geometry of Space
    18. (a) 4 v
    4 2, 3
    8, 12
    21.
    y
    (- 8, 12) 12
    10
    4v
    8
    6
    4
    u – v
    – v
    x
    (-2, 3)
    v
    x
    -8 – 6 – 4 -2

    (b) 1 v
    2

    1, 3
    4
    6
    22.
    y
    u + 2v
    y
    (-2, 3)
    3
    2
    2v
    v
    x
    u
    x
    -3 -2 -1
    – 1 v
    3
    -2
    -3
    (1, – 3(
    23. (a)
    2 u
    2
    3
    4, 9
    8 , 6
    (b) v u
    2, 5 4, 9
    2, 14
    (c) 0 v
    0, 0
    (c) 2u 5v
    2 4, 9 5 2, 5
    18, 7
    (-2, 3)
    24. (a)
    2 u
    2
    3
    3, 8
    2, 16
    2
    (b) v u
    8, 25 3, 8
    11, 33
    -3
    -2
    -1
    0v
    (c) 2u 5v
    2 3, 8 5 8, 25
    34, 109
    -1
    (d) 6u
    12, 18
    25. v
    3
    2
    2i
    j
    3i
    3 j
    3, 3
    y
    (- 2, 3)
    1
    -6
    v
    -2
    2
    6
    10
    14
    x
    2
    3
    x
    -6
    -10
    -14
    – 6v
    -1

    -2
    3
    2
    u
    u
    -18
    (12, -18)
    -3
    19.
    26. v
    2i
    j i 2 j
    3i j
    3, 1
    y
    – u
    2
    x
    1
    w
    v
    x
    1
    2
    3
    u
    20. Twice as long as given vector u.
    y

    u
    2u

    x

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    y
    4
    § 3 · § 5 ·
    ©
    2 ¹
    © 2 ¹
    v
    2
    © 2 ¹ © 2 ¹
    v
    2
    2
    2
    2
    Section 11.1
    Vectors in the Plane
    27. v
    2i
    j 2 i 2 j
    4i 3j
    4, 3
    38. v
    5, 15
    v
    25 225
    250
    5 10
    2w
    u
    v
    v
    5, 15
    5 10

    10 3 10
    ,
    10 10
    unit vector
    2
    u + 2w
    u
    4
    6
    x
    39. v
    3 5
    ,
    2 2
    -2

    28. v
    5u 3w
    5 2, 1 3 1, 2
    7, 11
    2
    ¨ ¸ ¨ ¸
    34
    2
    -4 -2
    2
    y
    4
    6
    8
    10
    x
    u
    v
    v
    § 3 · § 5 ·
    ¨ ¸, ¨ ¸
    34
    3
    34
    ,
    5
    34
    -3w
    5u
    2
    -6
    -8
    -10
    -12
    40. v
    6.2, 3.4
    3 34 5 34
    ,
    34 34
    unit vector
    29. u1 4
    u2 2
    1
    3
    u1
    u2
    Q
    3
    5
    3, 5
    v

    u
    v
    v
    6.2 3.4
    6.2, 3.4
    5 2

    50 5 2

    31 2 17 2
    ,
    50 50
    unit vector
    30. u1 5
    u2 3
    4
    9
    u1
    u2
    9
    6
    41. u
    1, 1 , v
    1, 2
    Q
    9, 6
    Terminal point
    (a)
    u
    1 1
    2
    31.
    v
    0 72
    7
    (b)
    v
    1 4
    5
    (c)
    u v
    0, 1
    32.
    v
    3
    2
    0
    3
    u v
    0 1
    1
    33.

    34.
    v

    v
    42 32

    122 5
    2
    5
    13
    (d)
    u
    u

    u
    u
    1
    1
    2
    1, 1
    35.
    v
    62 5
    2
    61
    (e)
    v
    v
    1
    5
    1, 2
    36.
    v
    10 2
    32
    109
    v
    v
    1
    37. v

    v
    3, 12

    3 12
    153
    (f )
    u v
    u v
    0, 1
    u
    v
    v
    3, 12
    153
    3
    153
    ,
    12
    153
    u v
    u v
    1
    17 4 17
    ,
    17 17
    unit vector
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    1
    x

    6 0, 1
    ©
    ¹
    Chapter 11
    Vectors and the Geometry of Space
    42. u
    0, 1 , v
    3, 3
    44. u
    2, 4 , v
    5, 5
    (a)

    (b)

    (c)
    u

    v
    u v
    0 1
    9 9
    3, 2
    1

    3 2
    (a)

    (b)

    (c)
    u

    v
    u v
    4 16
    25 25
    7, 1
    2 5

    5 2
    u v
    9 4
    13
    u v
    49 1
    5 2
    (d)
    u
    u
    0, 1
    (d)
    u
    u
    1
    2 5
    2, 4
    u
    u
    1
    u
    u
    1
    (e)
    v
    v
    1
    3 2
    3, 3
    (e)
    v
    v
    1
    5 2
    5, 5
    v
    v
    1
    v
    v
    1
    (f )
    u v
    u v
    1
    13
    3, 2
    (f )
    u v
    u v
    1
    5 2
    7, 1
    u v
    u v
    1
    u v
    u v
    1
    43. u
    1,
    1
    2
    , v
    2, 3
    45.
    u
    u
    2, 1
    5 | 2.236
    7
    y
    (a)

    (b)
    u

    v
    1
    4
    4 9
    5
    2
    13
    v
    v
    u v
    5, 4
    41 | 6.403
    7, 5
    6
    5
    4
    3
    2
    1
    v
    u
    u + v
    (c)
    u v

    u v
    7
    3,
    2

    9
    49
    4
    85
    2
    u v 74 | 8.602
    u v d u v
    74 d 5 41
    -1
    1
    2
    3
    4
    5
    6
    7
    (d)

    (e)
    u
    u

    u
    u

    v
    v

    v
    v
    1

    1
    2
    5

    1
    13
    1
    1,
    2

    2, 3
    46.
    u 3, 2
    u 13 | 3.606
    v 1, 2
    v 5 | 2.236
    u v 2, 0
    u v 2
    u+ v d u v
    u

    u+v
    -3 -2 -1
    3
    2
    1

    -1
    -2
    -3
    y
    1
    v
    2
    3
    x
    (f )
    u v
    u v
    u v
    u v
    1
    2
    85
    3,
    7
    2
    47.
    2 d 13

    u 1
    0, 3 0, 1
    u 3
    § u ·
    ¨ u ¸
    5

    0, 6
    v
    0, 6
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    7

    ©
    ¹
    ¬ ¼ ¬ ¼

    ©
    ¹
    1
    S
    ,
    ,

    ©
    ¹
    3i
    52. v
    ¬ ¼
    ¬ ¼
    5
    ,
    j
    a
    ¬ ¼
    ¬ ¼
    u
    v
    j
    ¸i
    ¨
    ©
    ¹
    ,
    j
    u
    3
    3
    v
    u
    v
    Section 11.1
    Vectors in the Plane
    48.
    u
    u
    § u ·
    ¨ u ¸
    1
    1, 1
    2

    2 2 1, 1
    58.
    u

    v
    u v
    5ªcos 0.5 º i 5ªsin 0.5 º j
    5 cos 0.5 i 5 sin 0.5 j
    5 cos 0.5 i 5 sin 0.5 j
    10 cos 0.5 i 10 cos 0.5, 0
    v
    2 2, 2 2
    59. Answers will vary. Sample answer: A scalar is a real
    49.
    u
    u
    § u ·
    ¨ u ¸
    1
    5
    1, 2

    5
    5

    2
    5
    1
    5
    2
    5

    5, 2 5
    number such as 2. A vector is represented by a directed
    line segment. A vector has both magnitude and direction.
    For example 3, 1 has direction and a magnitude
    6
    of 2.
    v

    5, 2 5
    60. See page 766:
    (ku1, ku2)
    50.
    u
    u
    1
    2 3
    3, 3
    (u1, u2)
    (u1 + v1, u2 + v2)
    u + v
    ku
    ku2
    § u ·
    ¨ u ¸
    1
    3
    3, 3
    u
    (v1, v2)
    v
    u2

    v2
    u1
    u
    (u1, u2)
    u2
    v
    1,
    3
    v1
    u1
    ku1
    51. v
    3ª cos 0q i sin 0q jº

    5ª cos 120q i sin 120q jº
    3, 0
    61. (a) Vector. The velocity has both magnitude and
    direction.
    (b) Scalar. The price is a number.
    i
    2
    5 3
    2
    5 5 3
    2 2
    62. (a) Scalar. The temperature is a number.
    (b) Vector. The weight has magnitude and direction.
    53. v
    2ª cos 150q i sin 150q jº
    3i j 3, 1
    For Exercises 63–68,
    au bw a i 2j b i j
    b i 2a b j.
    54. v
    4ª cos 3.5q i sin 3.5q jº
    63. v 2i j. So, a b 2, 2a b
    simultaneously, you have a 1, b
    1. Solving
    1.
    | 3.9925i 0.2442 j
    3.9925, 0.2442
    64. v
    3j. So, a b
    0, 2a b
    3. Solving
    simultaneously, you have a
    1, b
    1.
    55.
    cos 0q i sin 0q j i
    3 cos 45q i 3 sin 45q j
    3 2
    2
    i
    3 2
    2
    65. v 3i. So, a b 3, 2a b
    simultaneously, you have a 1, b
    0. Solving
    2.
    u v
    § 2 3 2 ·
    ¨ 2 ¸
    3 2
    2
    2 3 2 3 2
    2 2
    66. v 3i 3j. So, a b
    simultaneously, you have a
    3, 2a b
    2, b
    1.
    3. Solving
    56.
    4 cos 0q i 4 sin 0q j
    4i
    67. v
    i j. So, a b
    1, 2a b
    1. Solving
    2 cos 30q i 2 sin 30q j
    i
    3j
    simultaneously, you have a
    2 , b
    1 .
    u v
    5i
    3j
    5,
    3
    68. v
    i 7 j. So, a b
    1, 2a b
    7. Solving
    57.
    2 cos 4 i 2 sin 4 j
    simultaneously, you have a
    2, b
    3.
    cos 2 i sin 2 j
    u v
    2 cos 4 cos 2 i 2 sin 4 sin 2 j
    2 cos 4 cos 2, 2 sin 4 sin 2
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    8
    6
    y
    8
    6
    x
    6
    y
    4
    3
    y
    2
    1
    1
    y
    4
    x
    y
    3
    1
    2
    1
    y
    2
    4
    x
    1
    Chapter 11
    Vectors and the Geometry of Space
    69. f x
    x 2 , f c x
    2 x, f c 3
    10
    (a) m
    6. Let w
    1, 6 , w
    37, then r
    w
    w
    r
    1
    37
    1, 6 .
    (a)
    4
    (b)
    (b) m
    1 . Let w
    6, 1 , w
    37, then r
    w
    w
    r
    1
    37
    6, 1 .
    -2
    2
    2
    (3, 9)
    4
    6
    8
    10
    70. f x
    x 2 5, f c x
    2 x, f c 1
    2
    (a) m
    2. Let w
    1, 2 , w
    5, then r
    w
    w
    r
    1
    5
    1, 2 .
    (a)

    (1, 4)
    (b)
    (b) m
    1
    2
    . Let w
    2, 1 , w
    5, then r
    w
    w
    r
    1
    5
    2, 1 .
    -3
    -1
    2
    1

    -1
    1
    2
    3
    x
    71. f x
    x3 , f c x
    3x 2
    3 at x
    1.
    (a) m
    3. Let w
    1, 3 , w
    10, then
    w
    w
    r
    1
    10
    1, 3 .
    (a)
    (b) m
    . Let w
    3
    3, 1 , w
    10, then
    w
    w
    r
    1
    10
    3, 1 .
    (1, 1)
    (b)
    x
    1
    2
    72. f x
    x3 , f c x
    3x 2
    12 at x
    2.
    (a) m
    12. Let w
    1, 12 , w
    145, then
    w
    w
    r
    1
    145
    1, 12 .
    -6
    -4
    -2
    -4
    2
    (a)
    (b) m

    1
    12
    . Let w
    12, 1 , w
    145, then
    w
    w
    r
    1
    145
    12, 1 .
    -6
    (b)
    -10
    73. f x
    25 x 2
    f c x
    x
    25 x 2
    3
    4
    at x
    3.
    4
    3
    (a)
    (3, 4)
    (b)
    (a) m
    . Let w
    4
    4, 3 , w
    5, then
    w
    w
    r 4, 3 .
    5
    (b) m
    4
    3
    . Let w
    3, 4 , w
    5, then
    w
    w
    r
    1
    5
    3, 4 .
    -1
    1
    2
    3
    4
    5
    x
    74. f x
    tan x
    f c x
    sec2 x
    2 at x
    S
    4
    2.0
    1.5
    (a)
    (a) m
    2. Let w
    1, 2 , w
    5, then
    w
    w
    r
    1
    5
    1, 2 .
    – p
    – p
    1.0
    0.5
    p
    4
    (b)

    p
    2
    (b) m
    . Let w
    2
    2, 1 , w
    5, then
    w
    w
    r
    1
    5
    2, 1 .
    -1.0
    © 2010 Brooks/Cole, Cengage Learning

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    9
    2
    j
    y
    1
    2
    6
    x
    R
    R
    81.
    2
    2
    § 90 ·
    © 430.88 ¹
    2
    arctan « »
    D
    Section 11.1
    Vectors in the Plane
    75.
    u

    u v
    i
    2
    2 j
    2
    2
    78. (a) v
    (b) v
    (c)
    9 3, 1 4
    6i 5 j
    6, 5
    v
    u v u

    2
    2
    i
    2
    2
    j

    2
    2
    ,
    2
    2
    6
    5
    4
    (6, 5)
    3
    v
    76.
    u
    2 3i 2 j
    2
    1
    u v
    v
    3i 3 3j
    u v u
    3 2 3 i 3
    3 2 j
    (d)
    -1

    v
    3 4 5

    62 52
    61
    3 2 3, 3 3 2

    77. (a)–(c) Programs will vary.
    (d) Magnitude | 63.5
    79.
    F1
    F2
    F3
    2, TF1
    3, TF2
    2.5, TF3
    33q
    125q
    110q
    Direction | 8.26q
    T R
    F1 F2 F3 | 1.33
    TF1 F2 F3 | 132.5q
    80.
    F1
    F2
    F3
    2, TF1
    4, TF2
    3, TF3
    10q
    140q
    200q
    F1 F2 F3 | 4.09
    T R
    TF1 F2 F3 | 163.0q
    F1 F2
    500 cos 30qi 500 sin 30q j 200 cos 45q i 200 sin 45q j
    250
    3 100 2 i 250 100 2 j
    F1 F2
    250
    3 100 2
    250 100 2
    | 584.6 lb
    tan T
    250 100 2
    250 3 100 2
    ? T | 10.7q
    82. (a) 180 cos 30qi sin 30q j 275i | 430.88i 90 j
    Direction: D | arctan¨ ¸ | 0.206 | 11.8q
    Magnitude:
    430.882 902 | 440.18 newtons
    (b) M
    275 180 cos T 2
    180 sin T
    ª 180 sin T º
    ¬ 275 180 cos T ¼
    (c)
    T
    M
    D
    0q
    455
    0q
    30q
    440.2
    11.8q
    60q
    396.9
    23.1q
    90q
    328.7
    33.2q
    120q
    241.9
    40.1q
    150q
    149.3
    37.1q
    180q
    95
    0
    (d)
    500
    M
    50
    a
    0
    0
    180
    0
    0
    180
    (e) M decreases because the forces change from acting in the same direction to acting in the opposite direction as T increases
    from 0q to 180q.

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    10
    125
    125
    75 3 50 2 i 75 50 2
    2
    2
    2
    2
    R
    ¬
    ¼
    ¬
    ¼
    ¬ ¼ ¬ ¼ ¬ ¼
    2
    R
    2
    arctan¨
    T R
    q.
    © ¹
    S .
    3
    P
    1
    P2
    Chapter 11

    83. F1 F2 F3

    T R

    84. F1 F2 F3
    Vectors and the Geometry of Space

    75 cos 30qi 75 sin 30q j 100 cos 45qi 100 sin 45q j 125 cos 120qi 125 sin 120q j
    3 j
    F1 F2 F3 | 228.5 lb
    TF1 F2 F3 | 71.3q
    ª400 cos 30q i sin 30q j º ª280 cos 45q i sin 45q j º ª350 cos 135q i sin 135q j º
    ª200 3 140 2 175 2 ºi ª 200 140 2 175 2 º j
    200
    3 35 2
    200 315 2
    | 385.2483 newtons
    § 200 315 2 ·
    ¨ 200 3 35 2 ¸ | 0.6908 | 39.6q

    85. (a) The forces act along the same direction. T
    (b) The forces cancel out each other. T
    180q.
    (c) No, the magnitude of the resultant can not be greater than the sum.
    86. F1

    (a)
    20, 0 , F2
    F1 F2
    10 cos T sin T
    20 10 cos T , 10 sin T
    400 400 cos T 100 cos2 T 100 sin 2 T
    500 400 cos T
    (b)
    0
    40
    2
    0
    (c) The range is 10 d F1 F2 d 30.
    The maximum is 30, which occur at T
    The minimum is 10 at T
    (d) The minimum of the resultant is 10.
    87. 4, 1 , 6, 5 , 10, 3
    0 and T
    2S .
    8
    y
    8
    y
    8
    y
    6
    4
    (1, 2)
    (8, 4)
    6
    4
    (1, 2)
    (6, 5)
    (8, 4)
    6
    4
    (1, 2)
    (8, 4)
    2

    (- 4, -1)
    2
    4
    (3, 1)
    6
    8
    x
    2
    -4 -2
    -2
    2
    4
    (3, 1)
    6
    8
    x
    2
    -2
    -2
    (3, 1)
    4
    6
    8
    (10, 3)
    x
    10
    -4
    -4
    -4
    88.
    u
    1 u
    7 1, 5 2
    2, 1
    6, 3
    1, 2 2, 1 3, 3
    1, 2 2 2, 1 5, 4

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    11
    CB
    CA
    JJJ
    JJJ
    G
    G
    y
    A
    u
    v
    0
    u ¨
    ©
    ¹
    100.
    0 or
    § 24 ·
    T2
    § 24 ·
    © 10 ¹
    u
    v
    § 1 ·
    0.
    © 2 ¹
    3 and adding to the
    § 1 ·
    0 gives
    © 2 ¹
    5000
    0
    v
    v
    u v
    § 547.64 ·
    © 692.53 ¹
    2
    2
    Section 11.1
    Vectors in the Plane
    89. u
    v
    50°
    u cos 30qi sin 30q j
    v cos 130qi sin 130q j

    130° 30° B
    91. Horizontal component

    Vertical component
    v cos T
    1200 cos 6q | 1193.43 ft sec

    v sin T
    1200 sin 6q | 125.43 ft sec
    v
    C
    u
    30°
    x
    92. To lift the weight vertically, the sum of the vertical
    components of u and v must be 100 and the sum of the
    horizontal components must be 0.
    u cos 60qi sin 60q j
    Vertical components: u sin 30q v sin 130q
    3000
    v cos 110qi sin 110q j
    Horizontal components: u cos 30q v cos 130q
    So, u sin 60q v sin 110q
    100, or
    Solving this system, you obtain
    u | 1958.1 pounds
    v | 2638.2 pounds

    90. T1 arctan¨ ¸ | 0.8761 or 50.2q
    © 20 ¹
    arctan¨ ¸ S | 1.9656 or 112.6q
    u cos T1 i sin T1 j
    v cos T 2 i sin T 2 j
    Vertical components: u sin T1 v sin T2
    § 3 ·
    ¨ 2 ¸ v sin 110q
    And u cos 60q v cos 110q

    u ¨ ¸ v cos 110q

    Multiplying the last equation by
    first equation gives
    u sin 110q 3 cos 110q 100 ? v | 65.27 lb.

    Then, u ¨ ¸ 65.27 cos 110q
    u | 44.65 lb.
    Horizontal components: u cos T1 v cos T 2
    (a) The tension in each rope: u
    44.65 lb,
    Solving this system, you obtain
    u | 2169.4 and v | 3611.2.
    y
    65.27 lb
    (b) Vertical components: u sin 60q | 38.67 lb,
    v sin 110q | 61.33 lb
    A
    v
    C
    ?2

    ?1
    u
    B
    x
    20°

    v
    30°

    u
    100 lb
    93. u
    900 cos 148q i sin 148q j
    100 cos 45q i sin 45q j
    900 cos 148q 100 cos 45q i
    900 sin 148q 100 sin 45q j
    | 692.53i 547.64 j
    T | arctan¨ ¸ | 38.34q; 38.34q North of West
    u v |
    692.53
    547.64
    | 882.9 km h
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    12
    u v
    v
    ¸i j
    ¨
    y
    c
    u
    2
    a
    x
    u
    v
    1
    x
    2
    ¬
    ¼
    ¬
    ¼
    u
    ¸i sin¨
    v «cos¨ u
    ¸ cos¨ ¸ cos¨ ¸ j
    2 u
    ¬ ¼
    ¼
    § T
    ©
    § T
    ©
    ¸ cos¨
    ¸ cos¨
    ¸
    ¸
    tan¨ u
    ©
    ¸
    Chapter 11
    Vectors and the Geometry of Space
    94.
    u
    400i plane

    400 25 2 i 25 2 j | 364.64i 35.36 j
    50 cos 135qi sin 135q j 25 2i 25 2 j wind
    tan T
    35.36
    364.64
    ? T | 5.54q
    Direction North of East: | N 84.46q E
    Speed: | 336.35 mi h
    95. True

    96. True

    97. True
    102. Let the triangle have vertices at 0, 0 , a, 0 , and
    b, c . Let u be the vector joining 0, 0 and b, c , as
    indicated in the figure. Then v, the vector joining the
    midpoints, is
    98. False
    a

    99. False
    b
    0
    v
    § a b a · c
    © 2 2 ¹ 2
    b
    c
    i + j
    2
    2
    (b, c)
    ( a + b , 2 (
    ai bj

    100. True
    2 a
    1
    2
    bi cj
    1
    2
    u.
    (0, 0)
    v
    ( 2 , 0 (
    (a, 0)
    101.
    cos2 T sin 2 T
    sin 2 T cos2 T
    1,
    103. Let u and v be the vectors that determine the
    parallelogram, as indicated in the figure. The two
    diagonals are u v and v u. So,
    r x u v , s 4 v u . But,
    r s
    u
    x u v y v u
    y u x y v.
    So, x y
    1 and x y
    0. Solving you have
    1 .
    x
    y
    s
    u
    r
    v
    104. w
    u v v u
    u ª v cos T v i v sin T v jº v ª u cos Tu i u sin Tu jº
    v ª cos Tu cos T v i sin Tu sin T v jº
    ª § T T v · § Tu T v · § Tu T v · § Tu T v · º
    ¬ © 2 ¹ © 2 ¹ © 2 ¹ © 2 ¹ »
    tan T w
    sin¨ u

    cos¨ u
    T v · § Tu
    2 ¹ ©
    T v · § Tu
    2 ¹ ©
    T v ·
    2 ¹
    T v ·
    2 ¹
    § T T v ·
    2 ¹
    So, T w
    Tu
    T v 2 and w bisects the angle between u and v.
    105. The set is a circle of radius 5, centered at the origin.
    u
    x, y
    x 2 y 2
    5 ? x 2 y 2
    25
    © 2010 Brooks/Cole, Cengage Learning

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    13
    gt .
    y
    §
    ¸ g ¨
    ¸
    t
    2
    g 2
    a
    2
    gx 2
    2v0
    x
    2v0
    2v0
    2 g
    2 g
    v0
    2
    2v0
    2 g
    v0
    2
    2
    gx 2 gx 2 § v 2 ·
    2
    gx 2
    2v0
    gt
    2
    v2 § g · v 2
    2 g © v0
    ¹
    2 g
    2
    2v0
    2v0
    2 g
    v0
    2
    3
    z
    4
    2
    6
    z
    2
    Section 11.2
    Space Coordinates and Vectors in Space
    106. Let x
    v0t cos D and y
    v0t sin D
    1 2
    2
    x
    v0 cos D
    ? y
    v0 sin D ¨
    ©
    x · 1 § x ·
    v0 cos D ¹ 2 © v0 cos D ¹
    x tan D

    x tan D
    2v0
    2 x sec2 D

    1 tan 2 D
    (x, y)
    gx 2 gx 2 v 2
    2 2 tan 2 D x tan D 0
    gx 2 gx 2 ª § v 2 · v 4 º
    2 2 «tan 2 D 2 tan D ¨ 0 ¸ 20 2 »
    2v0 ¬ © gx ¹ g x ¼
    v0
    2 g
    2 2 ¨ tan D 0 ¸
    2v0 2v0 © gx ¹
    2
    If y d
    v0
    2 g
    2 , then D can be chosen to hit the point x, y . To hit 0, y : Let D
    90q. Then
    y
    v0t
    1 2
    2
    v0
    2 g
    2
    0 ¨ t 1¸ , and you need y d 0 .
    The set H is given by 0 d x, 0 y and y d
    v0
    2 g
    gx 2
    2
    Note: The parabola y
    gx 2
    2 is called the “parabola of safety.”
    Section 11.2 Space Coordinates and Vectors in Space
    1. A 2, 3, 4
    B 1, 2, 2
    2. A 2, 3, 1
    B 3, 1, 4
    5.
    (5, – 2, 2) – 3
    3
    4
    x
    (5, – 2, – 2)
    2
    1
    3
    2
    1

    – 2
    – 3
    z
    1 2
    3
    y
    3.
    6
    z
    6.
    z
    5
    4
    8
    (2, 1, 3)
    (-1, 2, 1)
    6
    (4, 0, 5)
    2
    1
    x
    4
    3
    2
    2 3
    4
    y
    x
    6
    – 2
    – 4
    6
    y
    – 6
    (0, 4, – 5)
    4.
    8
    7. x
    3, y
    4, z
    5: 3, 4, 5
    (3, -2, 5) 6
    8. x
    7, y
    2, z
    1:
    7, 2, 1
    x
    6
    y
    ( 3 , 4, -2(
    9. y
    0, x
    12: 12, 0, 0
    10. x
    0, y
    3, z
    2: 0, 3, 2
    © 2010 Brooks/Cole, Cengage Learning

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    14
    4.
    2
    2
    35. ¨
    ¸
    ,
    ,
    2 2
    § 3 ·
    © 2
    ¹
    2
    2
    ,
    ,
    © 2
    ¹
    2
    2
    2
    2
    2
    2
    2
    2
    2
    2
    4
    2
    2
    2
    2
    2
    25
    2
    2
    2
    2
    2
    2
    2
    2
    2
    Chapter 11
    Vectors and the Geometry of Space
    11. The z-coordinate is 0.
    30. A 3, 4, 1 , B 0, 6, 2 , C 3, 5, 6
    12. The x-coordinate is 0.
    13. The point is 6 units above the xy-plane.
    14. The point is 2 units in front of the xz-plane.
    AB
    AC
    BC
    9 4 1
    0 1 25
    9 1 16
    14
    26
    26
    15. The point is on the plane parallel to the yz -plane that
    Because AC
    BC , the triangle is isosceles.
    passes through x 3.
    16. The point is on the plane parallel to the xy-plane that
    31. A 1, 0, 2 , B 1, 5, 2 , C 3, 1, 1
    passes through z
    5 2.
    AB
    0 25 16
    41
    17. The point is to the left of the xz-plane.
    18. The point is in front of the yz-plane.
    AC

    BC
    4 1 9
    4 36 1
    14

    41
    19. The point is on or between the planes y
    y 3.
    3 and
    Because AB
    BC , the triangle is isosceles.
    20. The point is in front of the plane x
    32. A 4, 1, 1 , B 2, 0, 4 , C 3, 5, 1
    21. The point x, y, z is 3 units below the xy-plane, and
    below either quadrant I or III.
    22. The point x, y, z is 4 units above the xy-plane, and
    above either quadrant II or IV.
    AB
    AC
    BC
    Neither
    4 1 9
    1 36 0
    1 25 9
    14
    37
    35
    23. The point could be above the xy-plane and so above
    quadrants II or IV, or below the xy-plane, and so below
    quadrants I or III.
    24. The point could be above the xy-plane, and so above
    quadrants I and III, or below the xy-plane, and so below
    quadrants II or IV.
    33. The z-coordinate is changed by 5 units:
    0, 0, 9 , 2, 6, 12 , 6, 4, 3

    34. The y-coordinate is changed by 3 units:
    3, 7, 1 , 0, 9, 2 , 3, 8, 6
    25. d
    4 0 2 0 7 0
    16 4 49 69
    § 5 2 9 3 7 3 ·
    © 2 ¹
    ¨ , 3, 5¸
    26. d
    2 2
    5 3 2 2
    § 4 8 0 8 6 20 ·
    36. ¨ ¸
    6, 4, 7
    16 64 16
    96
    4 6
    37. Center: 0, 2, 5
    27. d
    6 1 2 2 2 4
    25 0 36 61
    Radius: 2
    x 0 y 2 z 5
    28. d
    4 2 2
    5 2 6 3
    38. Center: 4, 1, 1
    4 49 9 62
    29. A 0, 0, 4 , B 2, 6, 7 , C 6, 4, 8
    Radius: 5
    x 4 y 1 z 1
    AB

    AC
    22 62 32
    62 42 12
    49
    7

    196
    14
    39. Center:

    Radius:
    2, 0, 0 0, 6, 0
    2
    10
    1, 3, 0
    BC
    BC
    4 2 15
    245 49 196
    2
    245
    AB AC
    7 5
    x
    1 y 3 z 0
    10
    Right triangle

    © 2010 Brooks/Cole, Cengage Learning

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    15
    2
    2
    2
    2
    2
    2
    2
    § 2 81 ·
    2
    2
    ©
    4 ¹
    § 9 ·
    2
    ©
    2 ¹
    2
    § 9 ·
    © 2
    ¹
    2
    2
    2
    3
    3
    9
    2
    2
    2
    3
    0
    3
    2
    2
    2
    40. Center: 3, 2, 4
    Section 11.2
    Space Coordinates and Vectors in Space
    r
    3
    tangent to yz-plane
    x
    3 y 2 z 4
    2
    9
    41.
    x
    x 2 y 2 z 2 2 x 6 y 8 z 1
    2 x 1 y 6 y 9 z 8 z 16
    0
    1 1 9 16
    x
    1 y 3 z 4
    2
    25
    Center: 1, 3, 4
    Radius: 5
    42.
    x 2 y 2 z 2 9 x 2 y 10 z 19
    0
    ¨ x 9 x ¸ y 2 y 1 z 10 z 25
    2
    ¨ x ¸ y 1 z 5
    19

    109
    4
    81
    4
    1 25
    Center: ¨ , 1, 5¸
    Radius:
    109
    2
    43.
    9 x 2 9 y 2 9 z 2 6 x 18 y 1
    0
    x y z
    2 x
    2 y
    1
    9
    0
    x
    2

    2 x

    1
    9
    y
    2
    2 y 1 z
    2
    1
    1
    9
    1
    x 1 y 1
    z 0
    1
    Center: 1 , 1, 0
    Radius: 1

    44. 4 x 2 4 y 2 4 z 2 24 x 4 y 8 z 23
    x
    2
    6 x 9 y y
    1
    4
    z
    2
    2 z 1
    23
    4
    9
    1
    4
    1
    x
    3 y
    1
    2

    2
    z 1
    2
    16
    Center: 3, 1 , 1
    Radius: 4

    45. x 2 y 2 z 2 d 36
    Solid sphere of radius 6 centered at origin.

    46. x 2 y 2 z 2 ! 4
    Set of all points in space outside the ball of radius 2 centered at the origin.

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    16
    47.
    2
    2
    2
    x
    48.
    2
    2
    2
    x
    ,
    ,
    5
    4
    3
    2
    1
    z
    ,
    ,
    z
    8
    6
    4
    2
    ,
    3
    2
    v
    Chapter 11
    Vectors and the Geometry of Space

    x 2 y 2 z 2 4 x 6 y 8 z 13
    x2 4 x 4 y 2 6 y 9 z 2 8 z 16
    4 9 16 13
    2 y 3 z 4 16
    Interior of sphere of radius 4 centered at 2, 3, 4 .

    x 2 y 2 z 2 ! 4 x 6 y 8 z 13
    x2 4 x 4 y 2 6 y 9 z 2 8 z 16
    ! 13 4 9 16
    2 y 3 z 4 ! 16
    Set of all points in space outside the ball of radius 4 centered at 2, 3, 4 .
    49. (a) v
    2 4, 4 2, 3 1
    2, 2, 2
    53.
    4 3, 1 2, 6 0
    1, 1, 6
    (b) v
    2i 2 j 2k
    1, 1, 6
    1 1 36
    38
    (c)
    -2
    – 3
    < – 2, 2, 2>
    54.
    1, 1, 6
    Unit vector:
    38

    1 4, 7 5 , 3 2
    1
    38
    1 6
    38 38

    5, 12, 5
    3
    2
    1
    1
    2
    3
    4
    y
    5, 12, 5
    25 144 25
    194
    x

    50. (a) v
    4 0, 0 5, 3 1
    4, 5, 2
    Unit vector:
    5, 12, 5
    194
    5
    194
    12
    194
    5
    194
    (b) v
    (c)
    4i 5 j 2h
    55.
    5 4 , 3 3, 0 1
    1, 0, 1
    1 1
    1, 0, 1
    2
    < 4, – 5, 2 >
    Unit vector:
    1, 0, 1
    2
    1
    2
    , 0,
    1
    2
    x
    6
    4
    2
    2
    4
    6
    y
    56. 2 1, 4 2 , 2 4
    1, 6, 6
    51. (a) v
    (b) v
    (c)
    0 3, 3 3, 3 0
    3i 3k
    z
    < -3, 0, 3>
    5
    4
    -3
    3, 0, 3
    1, 6, 6 1 36 36
    1, 6, 6 1
    Unit vector: ,
    73 73

    57. (b) v 3 1 , 3 2, 4 3
    4i j k
    (c) v
    73
    6 6
    73 73

    4, 1, 1
    -2
    2
    3
    1
    1
    1
    2
    3
    4
    y
    (a), (d)
    5
    z
    (3, 3, 4)
    x
    4
    3
    (0, 0, 0) 2
    (-1, 2, 3)
    52. (a) v
    2 2, 3 3, 4 0
    0, 0, 4
    – 2
    (b) v
    4k
    (4, 1, 1) 2
    4
    2
    4
    y
    (c)
    4
    3
    2
    1
    z
    < 0, 0, 4 >
    x
    x
    3
    2
    1
    1
    2
    3
    y
    © 2010 Brooks/Cole, Cengage Learning

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    17
    z
    z
    4
    3
    z
    8
    6
    4
    2
    Q
    2
    x
    6
    y
    3
    3 2
    2
    2
    2
    5
    8
    6
    4
    2
    – 2
    1
    2
    3
    y
    x
    2
    2
    3
    2
    2
    2
    z
    2
    2
    Section 11.2
    Space Coordinates and Vectors in Space
    58. (b) v
    4 2, 3 1 , 7 2
    6, 4, 9
    62. (a) v
    2, 2, 1
    (c) v
    (a), (d)
    6i 4 j 9k

    (- 4, 3, 7)
    12
    9
    6
    (- 6, 4, 9)
    2
    1
    3
    x
    3
    3
    >- 2, 2, -1>
    y
    x
    9
    9
    y
    (b) 2 v
    4, 4, 2
    (2, -1, – 2)
    59. q1 , q2 , q3 0, 6, 2
    3, 5, 6
    Q
    3, 1, 8
    >4, – 4, 2>
    60. q1 , q2 , q3 0, 2,
    1, 4 , 3
    5
    2
    1, 2 , 1
    (c)
    1 v
    6

    1, 1,
    1
    2
    z
    61. (a) 2 v
    2, 4, 4
    z
    5
    4
    >1, -1, 1 >
    1
    3
    2
    < 2, 4, 4>
    x
    y
    4
    -2
    2
    3
    1
    1
    2
    y
    (d)
    5 v
    5, 5,
    5
    2
    x
    z
    (b) v
    1, 2, 2
    z

    3
    < 5, -5, 2 <
    2
    – 2
    – 3
    x
    6
    y
    – 3
    < – 1, -2, -2>
    2
    3
    63. z
    u v
    1, 2, 3 2, 2, 1
    1, 0, 4
    – 2
    – 3
    64. z
    u v 2w
    1, 2, 3 2, 2, 1 8, 0, 8
    7, 0, 4
    (c)
    3 v
    3 , 3, 3

    z
    65. z
    2u 4v w
    2, 4, 6 8, 8, 4 4, 0, 4
    6, 12, 6
    – 3
    -2
    -2
    – 3
    < 3 , 3, 3>
    66. z
    5u 3v 1 w
    x
    3
    2
    – 2
    – 3
    1
    y
    5, 10, 15 6, 6, 3 2, 0, 2
    3, 4, 20
    (d) 0v
    0, 0, 0
    67. 2z 3u
    2 z1 3
    2 z1 , z2 , z3 3 1, 2, 3
    4 ? z1
    7
    2
    4, 0, 4
    -3

    x
    -2
    2
    3
    1
    3
    2
    1

    -1
    -2
    -3
    – 3
    – 2
    < 0, 0, 0>
    1
    2
    3
    y
    2 z2 6 0 ? z2
    2 z3 9 4 ? z3
    z 7 , 3, 5
    3
    5
    2
    © 2010 Brooks/Cole, Cengage Learning

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    18
    0
    2
    i 3 k 2 1 i 3 k and
    3
    2
    2
    3
    4
    j 9 k 3 k .
    4
    8
    3 i 3 1 i 2 j
    3 3
    K
    K
    K
    K
    K
    KK
    2
    K
    K
    v
    2
    K
    K
    K
    K
    K
    Chapter 11
    Vectors and the Geometry of Space
    68. 2u v w 3z
    2 1, 2, 3 2, 2, 1 4, 0, 4 3 z1 , z2 , z3
    0, 0, 0
    0, 6, 9 3z1 , 3z2 , 3z3
    0, 0, 0
    0 3z1
    0 ? z1
    6 3z2
    9 3z3
    0 ? z2
    0 ? z3
    2
    3
    z
    0, 2, 3
    69. (a) and (b) are parallel because
    6, 4, 10 2 3, 2, 5 and
    2, 4 , 10 3 3, 2, 5 .

    70. (b) and (d) are parallel because
    4 j 2 j

    2 2 3 4
    71. z 3i 4 j 2k
    (a) is parallel because 6i 8j 4k 2z.
    72. z 7, 8, 3
    (b) is parallel because z z 14, 16, 6 .
    73. P 0, 2, 5 , Q 3, 4, 4 , R 2, 2, 1
    JJJK
    PQ 3, 6, 9
    JJJ
    PR 2, 4, 6
    77. A 2, 9, 1 , B 3, 11, 4 , C 0, 10, 2 , D 1, 12, 5
    JJJ
    AB 1, 2, 3
    JJJK
    CD 1, 2, 3
    JJJK
    AC 2, 1, 1
    JJJK
    BD 2, 1, 1
    JJJ JJJK JJJK JJJK
    Because AB CD and AC BD, the given points
    form the vertices of a parallelogram.
    78. A 1, 1, 3 B 9, 1, 2 , C 11, 2, 9 , D 3, 4, 4
    JJJ
    AB 8, 2, 5
    JJJK
    DC 8, 2, 5
    JJJK
    AD 2, 3, 7
    JJJ
    BC 2, 3, 7
    JJJ JJJK JJJK JJJ
    Because AB DC and AD BC , the given points
    form the vertices of a parallelogram.
    3, 6, 9 3 2, 4, 6
    JJJK JJJ
    So, PQ and PR are parallel, the points are collinear.

    74. P 4, 2, 7 , Q 2, 0, 3 , R 7, 3, 9
    JJJK
    PQ 6, 2, 4
    JJJ
    PR 3, 1, 2
    79. v
    v

    80. v
    v

    81. v
    0, 0, 0
    0

    1, 0, 3
    1 0 9 10

    3j 5k 0, 3, 5
    3, 1, 2 1 6, 2, 4
    JJJK JJJ
    So, PQ and PR are parallel. The points are collinear.

    75. P 1, 2, 4 , Q 2, 5, 0 , R 0, 1, 5
    JJJK
    PQ 1, 3, 4
    JJJ
    PR 1, 1, 1
    JJJK JJJ
    Because PQ and PR are not parallel, the points are not
    collinear.
    76. P 0, 0, 0 , Q 1, 3, 2 , R 2, 6, 4
    JJJK
    PQ 1, 3, 2
    JJJ
    PR 2, 6, 4
    v

    82. v
    v

    83. v
    v

    84. v
    0 9 25

    2i 5 j k
    4 25 1

    i 2 j 3k
    1 4 9

    4i 3j 7k
    16 9 49
    34

    2, 5, 1
    30

    1, 2, 3
    14

    4, 3, 7

    74
    JJJK JJJ
    Because PQ and PR are not parallel, the points are not
    collinear.

    © 2010 Brooks/Cole, Cengage Learning

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    19
    7
    3
    2
    7
    ,
    , ,
    ,
    ,
    ¬ ¼
    1
    z
    y
    v
    z
    8
    6
    3
    Section 11.2
    Space Coordinates and Vectors in Space
    85. v
    2, 1, 2
    91.
    cv
    c 2i 2 j k
    4c 2 4c 2 c 2
    v
    4 1 4
    3
    9c
    2
    7
    (a)
    v
    v
    1
    3
    2, 1, 2
    9c
    2

    c
    49
    r 7
    (b)
    v
    v

    1
    3
    2, 1, 2
    92.
    cu
    14c
    c i 2 j 3k
    4
    c 2 4c 2 9c 2
    4
    86. v
    6, 0, 8
    14c 2
    16
    v
    36 0 64
    10
    c
    r 8
    (a)
    v
    v
    1
    10
    6, 0, 8
    93. v
    10
    u
    u
    10
    0, 3, 3
    3 2
    (b)
    v
    v

    1
    10
    6, 0, 8
    10 0,
    1
    2
    1
    2
    0,
    10 10
    ,
    2 2
    87. v
    v
    3, 2, 5
    9 4 25
    38
    94. v
    3
    u
    u
    3
    1, 1, 1
    3
    (a)
    v
    v
    1
    38
    3, 2, 5
    3
    1
    3
    ,
    1
    3
    ,
    1
    3
    3
    3
    ,
    3
    3
    ,
    3
    3
    (b)
    v
    v

    1
    38
    3, 2, 5
    95. v
    3 u
    2 u
    3 2, 2, 1
    2 3
    3 2 2 1
    2 3 3 3
    1, 1,
    1
    2
    88. v
    v
    8, 0, 0
    8
    96. v
    7
    u
    u
    7
    4, 6, 2
    2 14
    14 21 7
    14 14 14
    (a)
    v
    v
    1
    8
    1, 0, 0
    97. v
    2ªcos r30q j sin r30q k º
    3j r k 0, 3, r1
    (b)
    v
    v
    1, 0, 0
    8
    2
    -2
    89. (a)–(d) Programs will vary.

    (e) u v
    4, 7.5, 2
    -2
    2
    -1
    1
    1

    -1
    < 0,
    3, 1>
    u v | 8.732
    x
    -2
    < 0,
    3, – 1>
    u | 5.099
    v | 9.019
    98. v
    5 cos 45qi sin 45qk
    5 2
    2
    i k or
    90. The terminal points of the vectors tu, u tv and
    5 cos 135qi sin 135qk
    5 2
    2
    i k
    su tv are collinear.
    su + tv
    5 2
    2
    (i + k)
    5 2
    2
    (- i + k)
    4
    2
    su
    u + tv
    6
    x
    6
    y
    u
    v
    tv
    99.
    v
    3, 6, 3
    2 v
    2, 4, 2
    4, 3, 0
    2, 4, 2
    2, 1, 2
    © 2010 Brooks/Cole, Cengage Learning

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    20
    2
    2
    3
    3

    3
    2
    2
    2
    4
    3
    z
    1
    v
    0
    8
    h
    8
    8
    L2 182
    8L
    T
    T
    2
    2
    2
    2
    2
    2
    2
    2
    0
    K K
    K K K
    K
    Chapter 11
    Vectors and the Geometry of Space
    100.
    v
    5, 6, 3
    108.
    r r0
    x
    1 y 1 z 1
    2
    2
    1, 2, 5
    10 ,
    2 v
    4, 2
    10 , 4, 2
    13 , 6, 3
    x 1 y 1 z 1
    This is a sphere of radius 2 and center 1, 1, 1 .
    101. (a)
    109. (a) The height of the right triangle is h
    JJJK
    The vector PQ is given by
    JJJK
    PQ 0, 18, h .
    L2 182 .
    x
    1
    u
    1
    y
    The tension vector T in each wire is
    (b) w
    au bv
    ai a b j bk
    T
    c 0, 18, h where ch
    24
    3
    8.
    a 0, a b 0, b 0
    So, a and b are both zero.
    So, T
    0, 18, h and
    (c) ai a b j bk
    a 1, a b 2, b
    w u v
    (d) ai a b j bk
    a 1, a b 2, b
    Not possible
    i 2 j k
    1

    i 2 j 3k
    3
    182 h 2
    h
    182 L2 182

    , L ! 18.
    L2 182
    Q (0, 0, h)
    102. A sphere of radius 4 centered at x1 , y1 , z1 .
    v
    x x2 y y1 , z z1
    L
    x
    x1 y y1 z z1
    2
    4
    (0, 18, 0)
    x
    x1 y y1 z z1
    2
    16
    (0, 0, 0)
    18
    P
    103. x0 is directed distance to yz-plane.
    y0 is directed distance to xz-plane.
    (b)
    L
    T
    20
    18.4
    25
    11.5
    30
    10
    35
    9.3
    40
    9.0
    45
    8.7
    50
    8.6
    z0 is directed distance to xy-plane.
    104. d
    x2
    x1 y2 y1 z2 z1
    2
    (c)
    30
    L = 18
    105.
    x
    x0 y y0 z z0
    2
    r 2
    T =8
    0
    100
    106. Two nonzero vectors u and v are parallel if u
    some scalar c.
    cv for
    0
    x 18 is a vertical asymptote and y
    horizontal asymptote.
    8 is a
    107.
    B
    (d)
    lim
    L o18
    8L
    L2 182
    f
    C
    lim
    L of
    8L
    L2 182
    lim
    L of
    8
    1 18 L
    2
    8
    A
    JJJ JJJ JJJK
    AB BC AC
    JJJ JJJ JJJ
    So, AB BC CA
    JJJK JJJ
    AC CA
    (e) From the table, T

    110. As in Exercise 109(c), x
    asymptote. So, lim T
    r0 o a
    10 implies L 30 inches.

    a will be a vertical
    f.
    © 2010 Brooks/Cole, Cengage Learning

    Monografias.com

    21
    K
    z
    69
    23
    69
    x
    C
    AB
    AC
    K
    K
    K
    F
    2
    2
    2
    2
    0
    § 2 8
    ©
    3
    16 · § 2 1 ·
    2
    2
    9 ¹
    ©
    9 ¹
    1
    9
    § 4 · § 1 ·
    2
    ©
    3 ¹
    ©
    3 ¹
    § 4 1 ·
    © 3
    3 ¹
    Section 11.2
    Space Coordinates and Vectors in Space
    111. Let D be the angle between v and the coordinate axes.
    v cos D i cos D j cos D k
    v 3 cos D 1
    JJJ
    113. AB
    JJJK
    AC
    JJJK
    AD
    0, 70, 115 , F1
    60, 0, 115 , F2
    45, 65, 115 , F3
    C1 0, 70, 115
    C2 60, 0, 115
    C3 45, 65, 115
    cos D

    v
    1
    3
    3
    3
    3
    3
    i j k
    3
    3
    1, 1, 1
    F
    So:
    F1 F2 F3 0, 0, 500
    60C2 45C3 0
    70C1 65C3 0
    115 C1 C2 C3 500
    0.6
    Solving this system yields C1
    104 , C
    2
    28 ,
    and
    0.4
    0.2
    (
    3
    3
    ,
    3
    3
    ,
    3
    3
    (
    C3
    112 . So:
    0.6
    0.4
    0.2
    0.4
    y
    F1 | 202.919 N
    F2 | 157.909 N
    F3 | 226.521N
    112.
    550
    302,500
    c 2
    c 75i 50 j 100k
    18,125c 2
    16.689655
    c | 4.085
    F | 4.085 75i 50 j 100k
    | 306i 204 j 409k
    114. Let A lie on the y-axis and the wall on the x-axis. Then A
    JJJ JJJK
    AB 8, 10, 6 , AC 10, 10, 6 .

    AB
    10 2, AC
    2 59
    JJJ
    JJJK
    420 JJJJK , F2
    650 JJJJJ
    Thus, F1
    AB
    AC
    0, 10, 0 , B
    8, 0, 6 ,
    10, 0, 6 and
    F1 F2 | 237.6, 297.0, 178.2 423.1, 423.1, 253.9 | 185.5, 720.1, 432.1
    F | 860.0 lb
    115. d AP
    2d BP
    x 2 y 1 z 1
    x 1 2
    y 2 z 2
    x 2 y 2 z 2 2 y 2 z 2
    4 x 2 y 2 z 2 2 x 4 y 5
    3x 2 3 y 2 3z 2 8 x 18 y 2 z 18
    6
    16
    9
    9
    ¨ x x
    ¸ y 6 y 9 ¨ z 3 z ¸
    44
    9

    Sphere; center: ¨ , 3, ¸, radius:
    2
    ¨ x ¸ y 3 ¨ z ¸

    2 11
    3
    2
    © 2010 Brooks/Cole, Cengage Learning

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    22
    i
    2
    2
    2
    2
    1
    6
    2
    2
    2
    2
    2
    2
    2
    2
    2
    Chapter 11
    Vectors and the Geometry of Space
    Section 11.3 The Dot Product of Two Vectors
    1. u
    3, 4 , v
    1, 5
    6. u
    i, v
    2

    (d) u ? v v 17 1, 5 17, 85
    (e) u ? 2v 2 u ? v 2 17 34

    (d) u ? v v 22 2, 3 44, 66
    (e) u ? 2v 2 u ? v 2 22 44
    (a) u ? v 3 1 4 5 17
    (b) u ? u 3 3 4 4 25
    (c) u 32 42 25

    2. u 4, 10 , v 2, 3
    (a) u ? v 4 2 10 3 22
    (b) u ? u 4 4 10 10 116
    (c) u 42 102 116
    (a) u ? v 1
    (b) u ? u 1
    (c) u 1
    (d) u ? v v i
    (e) u ? 2v 2 u ? v 2

    7. u 2i j k , v i k
    (a) u ? v 2 1 1 0 1 1
    (b) u ? u 2 2 1 1 1 1
    (c) u 22 1 12 6
    (d) u ? v v v i k
    (e) u ? 2v 2 u ? v 2
    3. u = 6, 4 , v 3, 2
    (a) u ? v 6 3 4 2 26
    (b) u ? u 6 6 4 4 52
    (c) u 62 4 52
    (d) u ? v v 26 3, 2 78, 52
    (e) u ? 2v 2 u ? v 2 26 52
    8. u 2i j 2k , v i 3j 2k
    (a) u ? v 2 1 1 3 2 2
    (b) u ? u 2 2 1 1 2 2
    (c) u 22 12 2 9
    (d) u ? v v 5 i 3j 2k
    (e) u ? 2v 2 u ? v 2 5
    5
    9

    5i 15 j 10k
    10
    4. u
    4, 8 , v
    7, 5
    9.
    u ? v
    u v
    cos T
    (a) u ? v
    (b) u ? u
    4 7 8 5
    4 4 8 8
    12
    80
    u ? v
    8 5 cos
    S
    3
    20
    (c) u
    (d) u ? v v
    (e) u ? 2v
    4 2 82
    12 7, 5
    2 u ? v
    80

    84, 60
    2 12
    24
    10.
    u ? v
    u v

    u ? v
    cos T

    40 25 cos
    5S
    6
    500 3
    5. u
    2, 3, 4 , v
    0, 6, 5
    11. u
    1, 1 , v
    2, 2
    (a) u ? v
    (b) u ? u

    (c) u
    2 0 3 6 4 5
    2 2 3 3 4 4
    2 3 4 29
    2

    29
    cos T

    T
    u ? v
    u v
    S
    2
    0
    2 8
    0
    (d)
    u ? v v
    2 0, 6, 5
    0, 12, 10
    12. u
    3, 1 , v
    2, 1
    (e) u ? 2v
    2 u ? v
    2 2
    4
    cos T
    u ? v
    u v
    5
    10 5
    1
    2
    T
    S
    4
    © 2010 Brooks/Cole, Cengage Learning

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    23
    3
    2
    ,
    6
    § 1 ·
    arccos¨ ¸ | 98.1q
    1 ,
    2
    3
    2
    T
    § S · § S ·
    ©
    6 ¹
    © 6 ¹
    § 3S · § 3S ·
    © 4 ¹
    © 4 ¹
    1
    2
    u
    v
    j
    3
    6
    ¨
    2 ©
    ¹
    ©
    ¹
    3

    T
    cos T
    S
    0
    0
    arccos¨
    ©
    ¹
    T
    cos T
    T
    arccos¨
    ©
    ¹
    5 3
    5
    AC
    CA
    BC
    CB
    3
    5
    3
    3 21
    14
    2 2
    K K
    K
    2
    K
    2 2 2 2
    K
    K K
    2
    K K
    Section 11.3
    The Dot Product of Two Vectors
    13. u
    3i j, v
    2i 4 j
    20. u
    2, 18 , v
    1
    cos T
    u ? v
    u v
    2
    10 20
    1
    5 2
    u z cv ? not parallel
    u ? v 0 ? orthogonal
    © 5 2 ¹
    21. u 4, 3 , v
    u z cv ? not parallel
    14.
    cos¨ ¸i sin¨ ¸ j

    cos¨ ¸i sin¨ ¸ j
    3
    2

    i j
    2

    i
    2
    2
    2
    u ? v

    22. u
    u
    0 ? orthogonal
    1 i 2 j , v 2i 4 j
    1 v ? parallel
    cos T
    u ? v
    u v
    3 § 2 · 1 § 2 ·
    ¨ 2 ¸ 2 ¨ 2 ¸
    2
    4
    1
    23. u j 6k , v i 2 j k
    u z cv ? not parallel
    u ? v 8 z 0 ? not orthogonal
    Neither
    ª 2
    arccos «
    ¬ 4
    1
    º
    3 »
    ¼
    105q
    24. u 2i 3j k , v
    u z cv ? not parallel
    2i j k
    15. u
    1, 1, 1 , v
    2, 1, 1
    u ? v
    0 ? orthogonal
    cos T
    u ? v
    u v
    2
    3 6
    2
    3
    25. u 2, 3, 1 , v 1, 1, 1
    u z cv ? not parallel
    u ? v 0 ? orthogonal
    T
    arccos
    2
    3
    | 61.9q
    26. u
    cos T , sin T , 1 ,
    16. u 3i 2 j k , v
    u ? v
    u v

    T =
    2
    2i 3j
    3 2 2 3 0
    u v
    v sin T , cos T , 0
    u z cv ? not parallel
    u ? v 0 ? orthogonal
    27. The vector 1, 2, 0 joining 1, 2, 0 and 0, 0, 0 is
    perpendicular to the vector 2, 1, 0 joining
    17. u
    3i 4 j, v = 2 j 3k
    2, 1, 0 and 0, 0, 0 :
    1, 2, 0 ? 2, 1, 0
    cos T
    u ? v
    u v
    8
    5 13
    8 13
    65
    The triangle has a right angle, so it is a right triangle.
    28. Consider the vector 3, 0, 0 joining 0, 0, 0 and
    § 8 13 ·
    ¨ 65 ¸ | 116.3q

    18. u 2i 3j k , v i 2 j k
    u ? v 9 9
    u v 14 6 2 21
    § 3 21 ·
    ¨ 14 ¸ | 10.9q

    19. u 4, 0 , v 1, 1
    u z cv ? not parallel
    u ? v 4 z 0 ? not orthogonal
    Neither
    3, 0, 0 , and the vector 1, 2, 3 joining 0, 0, 0 and
    1, 2, 3 : 3, 0, 0 ? 1, 2, 3 3 0
    The triangle has an obtuse angle, so it is an obtuse triangle.
    29. A 2, 0, 1 , B 0, 1, 2 , C 1 , 3 , 0
    JJJ

    Partes: 1, 2

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