1. Vectors
Chapter 4

2. A quantity with only magnitude
Scalar
A quantity with only magnitude

3. A quantity with both magnitude and direction
Vector
A quantity with both magnitude and direction

4. Vector
Tail Head

5. The sum of two or more vectors
Resultant Vector
The sum of two or more vectors

6. Vector Addition
Two addition methods:
Graphical
Algebraic

7. Graphical Vector Addition
Use the following steps

9. (1)
Draw any one of the vectors with its tail at the starting point or origin

11. Draw the 2nd vector with its tail at the head of the first vector
(2)
Draw the 2nd vector with its tail at the head of the first vector

13. (3)
Draw the resultant vector from the starting point of the 1st vector to the head of the 2nd

15. (4)
Measure the length of the resultant to determine the magnitude of the vector

16. Measure the angle to determine the direction of the vector
(5)
Measure the angle to determine the direction of the vector

17. Drill: An insect crawls 4.0 cm east, then 3.0 cm south. Calculate:
a) distance traveled
b) displacement

18. Practice: A plane flies 5.0 km west, then 2500 m south. Calculate:
a) distance traveled
b) displacement

19. Drill: A bug crawls 3.0 cm west, then 40.0 mm south. Calculate:
a) distance traveled
b) displacement

20. Drill:
A plane flies 150 m/s east in a 25 m/s wind blowing towards south. Calculate the plane’s velocity relative to the ground.

21. Review HW
Problems on page 71

22. Adding Vectors with Opposite Signs
Vector1 + (-Vector2) = Vector1 – Vector2

23. V2 V1
V2 - V1 VR

24. Practice: A bird flies 25 m west, then 57 m east. Calculate:
a) distance traveled
b) displacement

25. Practice:
A bird flies 14 m west, then 32 m east, then 21 m west. Calculate:
a) distance traveled
b) displacement

26. A boat travels upstream at 10. 0 m/s in a river flowing at 2. 5 m/s
A boat travels upstream at 10.0 m/s in a river flowing at 2.5 m/s. Calculate the velocity of the boat.

27. Multiple vectors
When adding multiple vectors, just repeat the process of head of first to tail of second etc.

28. Algebraic R B q A

29. Practice:
A car goes 3.0 km west, then 4.0 km south, then 5.0 km north. Calculate:
a) distance traveled
b) displacement

30. Algebraic
hyp
opp q
adj

31. Solving the problem
Sin q = opp/hyp
Cos q = adj/hyp
Tan q = opp/adj

32. R2 = A2 + B2 –2ABcos q otherwise
Algebraic
R2 = A2 + B2
if right angle
R2 = A2 + B2 –2ABcos q otherwise

33. A ball rolls 45 m north, then is kicked 60. 0 m west
A ball rolls 45 m north, then is kicked 60.0 m west. Calculate the distance & displacement of the ball.

34. A ball thrown at 50. 0 m/s north from a train moving 50. 0 m/s west
A ball thrown at 50.0 m/s north from a train moving 50.0 m/s west. Calculate the velocity of the ball.

35. A boat travels at 4. 0 m/s across in a river flowing at 3. 0 m/s
A boat travels at 4.0 m/s across in a river flowing at 3.0 m/s. Calculate the velocity of the boat.

36. A plane travels at 250 m/s south in a 50
A plane travels at 250 m/s south in a 50.0 m/s wind blowing east to west. Calculate the velocity of the plane.

37. A plane travels at 25 m/s south in a 15 m/s wind blowing east to west
A plane travels at 25 m/s south in a 15 m/s wind blowing east to west. Calculate the velocity of the plane.

38. Drill: A snail travels at 9. 0 cm south then 15. 0 cm west then 6
Drill: A snail travels at 9.0 cm south then 15.0 cm west then 6.0 cm south. Calculate the displacement of the snail.

39. Check HW
Problems 11 – 14
Page 74

40. Vector Resolution
Resolving any vector into its x & y components

41. y-axis
Vector = 100 units at 37o N o E
37o
x-axis

42. Determine the x & y components
y-axis
Determine the x & y components
Hypotenuse
Opposite side
37o
Adjacent side

43. Solving the problem
Sin q = opp/hyp
Cos q = adj/hyp
Tan q = opp/adj

44. Solving the problem
sin q = opp/hyp
opp = hyp x sin q

45. Solving the problem
cos q = adj/hyp
adj = hyp x cos q

46. Determine the x & y components
y-axis
Determine the x & y components
Hypotenuse = 100 m
Opposite side
= hyp(sin q)
q = 37o
Adjacent side = hyp(cos q)

47. Trig Functions x-component = 100(cos 37o) = 100(0.80) = 80 units
y-component = 100(sin 37o)
= 100(0.60) = 60 units

48. Resolve the following vector into polar or x & y components:
150 30o N o E

49. Resolve the following vector into polar or x & y components:
250 37o E o S

50. Resolve the following vector into polar or x & y components:o

51. Vector Addition Hint:
When adding multiple vectors, just add the vector components. Then solve for the final vector.

52. 50 m at 45o E o N
2) 45 m at 53o S o W
3) 80 m at 30o W o N
4) 75 m at 37o N o E
Calculate resultant

53. Equilibrium When functions applied to any system add up to zero
Steady State
Homeostasis

54. Equilibrant
The vector, when added to a set of vectors, would bring the sum of all the vectors back to the zero point or origin.

55. An automobile is driven 250 km due west, then 150 km due south
An automobile is driven 250 km due west, then 150 km due south. Calculate the resultant vector.

56. A dog walks 4. 0 miles east, then 6. 0 miles north, then 8
A dog walks 4.0 miles east, then 6.0 miles north, then 8.0 miles west. Calculate the resultant vector.

57. Drill: A cannon fires a projectile at 37o from horizontal at 1250 m/s Calculate the x & y components.

58. Check HW:

59. A jet flies 15 km due west then 25 km at 53. 1o north of west
A jet flies 15 km due west then 25 km at 53.1o north of west. Calculate the resultant vector.

60. Calculate equilibrant
9.0 m W
2) cm S
3) mm E
4) km N
Calculate equilibrant

61. Resolve a 2. 4 kN force vector that is 30
Resolve a 2.4 kN force vector that is 30.0o from horizontal into horizontal & vertical components in N:

62. Calculate equilibrant
2.0 m at 30o
2) cm at 37o
3) mm at 53o
4) km at 127o
Calculate equilibrant

63. Calculate equilibrant
The following forces are acting on a point: 1) 5.0 N at 37o
2) 8.0 N at 53o
Calculate equilibrant

64. A boat travels at 4. 0 m/s directly across a river flowing at 3. 0 m/s
A boat travels at 4.0 m/s directly across a river flowing at 3.0 m/s. Calculate the resultant vector.

65. A boy walks 4. 0 miles east, then 6. 0 miles north, then 4
A boy walks 4.0 miles east, then 6.0 miles north, then 4.0 miles east. Calculate the resultant vector.

66. A jet flies 15 km due west then 25 km at 53o north of west
A jet flies 15 km due west then 25 km at 53o north of west. Calculate the resultant vector.

67. A jet flies 28 km due west then 21 km north
A jet flies 28 km due west then 21 km north. Calculate the resultant vector.

68. A dog walks 8. 0 m due east then 15 m at 37o north of east
A dog walks 8.0 m due east then 15 m at 37o north of east. Calculate the resultant vector.

69. A jet travels 250 miles at 37o north of west
A jet travels 250 miles at 37o north of west. Resolve the displacement into north & west components.

70. 50 m at 45o E o N
2) 45 m at 53o S o W
3) 80 m at 30o W o N
4) 75 m at 37o N o E
Calculate resultant

71. A girl walks 25 m due east then 15 m at 37o north of east, the 50
A girl walks 25 m due east then 15 m at 37o north of east, the 50.0 m due south. Calculate the resultant vector.

72. A girl walks 75 m at 37o north of east, then 75 m at 53o west of north
A girl walks 75 m at 37o north of east, then 75 m at 53o west of north. Calculate the resultant vector.

73. 50 m at 45o S o W
2) 75 m at 53o E o S
3) 80 m at 37o N o E
4) 75 m at 33o W o N
Calculate resultant

74. Drill: A dog walks:
1) 0.16 km due north
2) 90.0 m due east
3) 25,000 cm at 37o N o E
Calculate: Res. & Eq.

75. Check HW
Problems 31 & 31
Page 79

76. A zombie walks:
1) 0.30 km at 30o SoW
2) 500 m at 45o NoE
Calculate resultant:

77. Drill: A snail crawls:
1) 25 cm at 37o WoS
2) 400 mm at 30o NoE
Calculate resultant:

78. A telephone pole has a wire pulling with a 3500 N force attached at 20o from the top of the pole. Calculate the force straight down.

79. A cat walks:
1) 9.0 m due south
2) 1500 cm due east
3) 5,000 mm at 37o N o E
Calculate resultant:

80. Forces act on a point:
1) 150 N at 53o EoS
2) 250 N at 37o SoW
3) 0.50 kN at 45o WoS
Calculate resultant:

81. 1) 350 N at 53o WoS
2) 150 N at 37o NoW
3) 0.25 kN at 45o WoS
4) 250 N due E
Calculate resultant:

82. 1) 0.35 kN due west
2) 150 N due south
3) 0.50 kN at 45o EoN
4) 250 N at 37o NoE
Calculate resultant:

83. Use graph paper to solve the following:
1) 250 mm due east
3) 0.50 mm 53o EoN
Calculate resultant:

84. Drill & Collect HW: Solve the following:
1) 360 m due west
3) 0.27 km due north
Calculate resultant:

85. HW: Solve with trig:
1) 0.10 MN 37o SoW
2) 250 kN 53o EoN
3) 150,000 N East
Calculate resultant:

86. Use graph paper to solve the following:
1) 3.0 m due west
3) 15 m 53o EoN
Calculate resultant:

87. 1) 0.35 km due west
2) 250 m due south
3) 0.50 km at 45o EoN
4) 150 m at 37o NoE
Calculate resultant:

88. Define the Following:
Scalar
Vector
Magnitude
Direction

89. Define the Following:
Distance
Displacement
Speed
Velocity

90. Test Review

91. Terms to Define: Equilibrant Vector Resultant Scalar Vector
Vector Resolution

92. Metric Prefixes:
Centi Kilo
Giga Mega
Micro Milli
Nano

93. Trig Functions:
Sin q Pytha-
Cos q Theorem
Tan q
Law of Cosines

94. Add the 3 Vectors Graphically:
50.0 m west
90.0 m north
170 m east

95. Add the 2 Vectors Mathematically:
20.0 m west
oNoE

96. Resolve the Vector into x & y comp:
o SoW

97. Add the 3 Vectors using vector components:
75 37o NoW
o NoE
150 53o SoW