11.1 - Vectors and Scalars

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NateGatsby
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193139
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11.1 - Vectors and Scalars
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2013-02-15 20:33:34
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calc1
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Vectors and Scalars
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  1. P = (0,4)
    Q = (3,-2)

    Find magnitude and direction of .
    • slope = -2
    • |||| = 
  2. P = (0,4)
    Q = (3,-2)

    Write the component form of .
    <3,-6>
  3. P = (0,4)
    R = (-3,-2)

    Write the component form of .
    <3,6>
  4. Q = (3,-2)
    R = (-3,-2)

    Write the component form of .
    <-6,0>
  5. P = (0,4)
    R = (-3,-2)

    Find magnitude and direction of
    • slope = 2
    • |||| = 
  6. R = (-3,-2)
    Q = (3,-2)

    Find magnitude and direction of 
    • |||| = 6
    • slope = 0
  7. write the points of  in component form.

    u: (1,2) (5,4)
    <4,2>
  8. Find the vectors u and v whose initial and terminal points are given.

    u: (3, 2), (5, 6)
    v: (1, 5), (3, 8)

    Are they equidistant?
    • <2,4>
    • <2,3>

    No, the slopes aren't the same.
  9. Find the vectors u and v whose initial and terminal points are given.Put the vector in linear combination.

    u: (5, 3), (6, −2)
    v: (7, 10), (9, 5)

    Are they equivalent?
    • <1,-5>
    • <2,-5>

    No, they have different slopes.
  10. Write the vector as the linear combination of the standard unit vectors i and j.

    v: (2, 0),(5, 6)
    3i+6j
  11. Write the vector as the linear combination of the standard unit vectors i and j.

    v: (0, −3),(−1, −1)
    -i+2j
  12. Write the vector as the linear combination of the standard unit vectors i and j.

    v: (3,  7/3), (1/2, 3)
    -5i/2  + 2j/3
  13. u = <−3, −8>
    v = <8, 25>

    (3/4)
    <-9/4 , -6>
  14. u = <−3,−8>
    v = <8,25>

    v-u
    <11,33>
  15. u = <−3, −8>
    v = <8, 25>

    3u + 9v
    <63, 201>
  16. u = <8, −1>
    w = <1, 3>

    v = u + w
    <9,2>
  17. u = <2, −1>
    w = <2, 2>

    v = u + 2w
    <6,3>
  18. The vector v and its initial point are given. Find the terminal point.

    v = 4i − 9j
    Initial point: (5, 4)
    (9,-5)
  19. Find the magnitude of v

    v = 5i
    5
  20. Find the magnitude of v.

    v = <4, 3>
    5
  21. Find the unit vector in the direction of v.

    v = -7.3i + 3.3j
    -7.3i/ + 3.3j/
  22. Find the vector v with the given magnitude and the same direction as u.

    ||v|| = 4  
    Direction u =  <1, 1>
     >
  23. Find the component form of v given its magnitude and the angle it makes with the positive x-axis. 

    v = 6,    θ = 0°
    <6,0>
  24. Find the component form of v given its magnitude and the angle it makes with the positive x-axis. (Round your answers to four decimal places.)

    ||v|| = 4,    θ = 3.1°
    <3.9941 , 0.2163>
  25. Find the component form of u + v given the lengths of u and v and the angles that u and v make with the positive x-axis.

    ||u|| = 5,     θu = 0°
    ||v|| = 2,     θv = 60°
    < 5+2cos(/3), 2sin(/3)>
  26. Find a and b such that v = au + bw.

    u = 1, 2
    w = 1, −1
    v = -3, 6 
    • a = 1
    • b = -4
  27. Find the  unit vectors parallel and perpendicular to the equation at the point.

    f(x) = -3x2 + 5
    (1, 2)
    • < 1/, -6/ > 
    • < 6/, 1/ >
  28. Find the component form of v given the magnitudes of u and u + v and the angles that u and u + v make with the positive x-axis. 

    u = 1, θ = 45° 
    u + v = 2, θ = 90°
    < -/2, /2 >
  29. Three forces with magnitudes of 400 newtons, 280 newtons, and 340 newtons act on an object at angles of −30°, 45°, and 135° respectively, with the positive x-axis. Find the direction and magnitude of the resultant force.
    • 38.11°
    • 386.32 N
  30. Three forces with magnitudes of 51 pounds, 98 pounds, and 127 pounds act on an object at angles of 30°, 45°, and 120° respectively, with the positive x-axis. Find the direction and magnitude of the resultant force.
    • 76.29°
    • 210.78 N
  31. Forces with magnitudes of 700 pounds and 200 pounds act on a machine part at angles 30° and −45°, respectively, with the x-axis (see figure). Find the direction and magnitude of the resultant force F.
    • 15.59°
    • 776.19 lbs
  32. A gun with a muzzle velocity of 1400 feet per second is fired at an angle of 9° above the horizontal. Find the vertical and horizontal components of the velocity.
    • horizontal = 1382.76 f/s
    • vertical = 219.01 f/s
  33. A plane is flying with a bearing of 302°. Its speed with respect to the air is 900 kilometers per hour. The wind at the plane's altitude is from the southwest at 100 kilometers per hour (see figure). What is the true direction of the plane, and what is its speed with respect to the ground?
    • speed = 882.9 km/h
    • direction = 38.34°
  34. P = (0,4)
    R = (3,-2)
    Q = (-3,-2)

    Does vector PQ = RP?
    No, they dont have the same slopes.
  35. Write vector u as a linear combination of unit vectors i and j.

    u = <3,-6>
    3i-6j
  36. Write vector u as a linear combination of unit vectors i and j.

    u = <3,6>
    3i+6j
  37. Write vector u as a linear combination of unit vectors i and j.

    u = <-6,0>
    -6i
  38. Change vector u into a unit vector in the same direction as vector u.

    u = <3, -6)
    (3/)i-(6/)j
  39. Change vector u into a unit vector in the same direction as vector u.

    u = <3, 6)
    (3/)i+(6/)j
  40. Change vector u into a unit vector in the same direction as vector u.

    u = <-6,0)
    -i
  41. 3 One force of 100 lbs acts upwards while another of 20 lbs acts downwards from and object. Find direction and magnitude.
    • magnitude = 106.9 lbs
    • direction =
  42. Define scalar.
    a real number that represents a quantity that is "scaled" to appropriate units.
  43. 6. Is it a scalar, directed line segment or neither? 
    neither.
  44.  6 hours. Is it a scalar, directed line segment or neither? 
    scalar. 
  45. 80 mph. Is it a scalar, directed line segment or neither? 
    scalar. 
  46. 80 mph due east. Is it a scalar, directed line segment or neither? 
    directed line segment.
  47. An objects drops at an acceleration of 32f/s2 Is it a scalar, directed line segment or neither? 
    directed line segment. 
  48. Define directed line segment.
    anything that has magnitude and direction.
  49. What is the difference between scalar and directed line segment?
    A directed line segment is a scalar applied in a direction.
  50. How do you find magnitude? 
    use the distance formula between the 2 points.
  51. What's the distance formula? 
  52. What does direction equal?
    slope.
  53. P = (3,4)
    Q = (6,0)
    R = (-3,2)
    S = (0,-2)

    Find the magnitude and direction of vectors PQ and RS. Are they the same?
    • ||PQ|| = 5
    • direction = -4/3

    • ||RS|| = 5
    • direction = -4/3

    they are not the same, just equivalent. 
  54. What's the difference between a scalar and a directed line segment?
    a directed line segment is a scalar applied in a direction
  55. 6. Is it a scalar, directed line segment or neither?
    neither
  56. 6 hours. Is it a scalar, directed line segment or neither?
    scalar
  57. 80 miles per hour. Is it a scalar, directed line segment or neither?
    scalar
  58. 80 miles per hour due east. Is it a scalar, directed line segment or neither?
    directed line segment.
  59. object dropping at 32 feet per second. Is it a scalar, directed line segment or neither?
    directed line segment.
  60. define scalar.
    a real number that represents a quantity that is scaled to appropriate units.
  61. define direct line segment.
    anything that has magnitude and direction.
  62. Direction equals..
    slope
  63. what is the equation for magnitude?
  64. P = (3,4)
    Q = (6,0)
    R = (-3,2)
    S = (0,-2)

    Give magnitude and slope. Are vectors PQ and RS equal?
    magnitude of PQ = 5, slope = -4/3

    magnitude of RS = 5, slope = -4/3

    they are equivalent, not the same.
  65. How do you find component form for a vector?
    terminal - initial
  66. Define unit vector.
    Vector with a magnitude of 1.
  67. P = (2,3)
    Q = (2,3)

    Find vector PQ and its magnitude.
    <0,0>, 
  68. P = (-1,2)
    Q = (2,4)

    Find vector PQ.
    <3,2>
  69. What can scalar multiplication/division change?
    Length and direction.
  70. P = <1,3>

    Find 4P, P/4, -2P.
    • <4,12>
    • <1/4, 3/4>
    • <-2,-6>
  71. How do you find a unit vector?
    Take the vector in component form and divide it by the magnitude. 
  72. The USS Calculus breaks down at sea. 2 tug boats come to push it back to port. Each boat pushes the Calculus at an inward 30 degree angle with 500 lbs of force. What is the combined forward force on the boat?
    866 lbs
  73. What is the unit combination of <4,3>?
    4i + 3j
  74. How do you turn a vector into linear combination?
    Multiple the x value by i and the y by j. Then add them.
  75. How do you find a unit vector parellel or orthogonal to a function?
    differentiate the function and evaluate at the given point. Find the magnitude of those points and multiple it by the given point. 

    If it's othoganal, flip the x and y values and invert the value of the y.

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