1. How to Add Vectors

2. How to Add Vectors
Do you remember the basic rule for adding vectors?

3. How to Add Vectors
Tip to Tail Rule

4. How to Add Vectors Tip to Tail Rule
Arrange the tip of the first vector with the tail of the second vector, then the tip of the second vector with the tail of the third vector and so on...

5. How to Add Vectors Tip to Tail Rule
Arrange the tip of the first vector with the tail of the second vector, then the tip of the second vector with the tail of the third vector and so on...
Draw a resultant vector from start to finish

6. How to Add Vectors Tip to Tail Rule
Arrange the tip of the first vector with the tail of the second vector, then the tip of the second vector with the tail of the third vector and so on...
Draw a resultant vector from start to finish
Example: Add a + b c c a b

7. How to Add Vectors
Tip to Tail Rule a + b + c a c b

8. How to Add Vectors
Tip to Tail Rule a + b + c a c b
tail
tip

9. How to Add Vectors
Tip to Tail Rule a + b + c a c b
tail
tip

10. How to Add Vectors
Tip to Tail Rule a + b + c
a + b + c a c b
tail
tip

11. Does it matter in which order we add vectors? Does a+b+c=c+b+a or a+c+b?
tail
tip

12. Does it matter in which order we add vectors? Does a+b+c=c+b+a or a+c+b?
tail
tip

13. Does it matter in which order we add vectors? Does a+b+c=c+b+a or a+c+b?
tail
tip
c+b+a

14. Does it matter in which order we add vectors? Does a+b+c=c+b+a or a+c+b?
tail
tip
c+b+a

15. Does it matter in which order we add vectors? Does a+b+c=c+b+a or a+c+b?
tail
tip
a+c+b
c+b+a

16. Vector Addition Coda

17. Vector Addition Coda
For non-collinear vectors not along the same line, the tip-to-tail rule is the fundamental way to add vectors

18. Vector Addition Coda
For non-collinear vectors not along the same line, the tip-to-tail rule is the fundamental way to add vectors
For vectors along the same line, as with the BIG FIVE equations, vectors can be added or subtracted by representing them as integers

19. Vector Addition Coda
For non-collinear vectors not along the same line, the tip-to-tail rule is the fundamental way to add vectors
For vectors along the same line, as with the BIG FIVE equations, vectors can be added or subtracted by representing them as integers
Order doesn't matter when adding vectors even with the tip-to-tail rule

20. Vector Addition Coda
For non-collinear vectors not along the same line, the tip-to-tail rule is the fundamental way to add vectors
For vectors along the same line, as with the BIG FIVE equations, vectors can be added or subtracted by representing them as integers
Order doesn't matter when adding vectors even with the tip-to-tail rule
Two vectors are equal if their length or magnitude is the same AND their direction in space is the same

21. Vector Addition Coda
For non-collinear vectors not along the same line, the tip-to-tail rule is the fundamental way to add vectors
For vectors along the same line, as with the BIG FIVE equations, vectors can be added or subtracted by representing them as integers
Order doesn't matter when adding vectors even with the tip-to-tail rule
Two vectors are equal if their length or magnitude is the same AND their direction in space is the same
We can't add different vector quantities ... example: We can't add a velocity vector and a displacement vector. We can only add vectors if the type of quantity and the units used are the same.

22. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]

23. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Why can't we say the answer is 3+4 =7.0 m [NW] ?

24. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Why can't we say the answer is 3+4 =7.0 m [NW] ?
These quantities have magnitude and direction, and are therefore vector quantities. Adding vectors like displacement, acceleration or velocity has different rules than adding scalar quantities like mass and speed.

25. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Why can't we say the answer is 3+4 =7.0 m [NW] ?
These quantities have magnitude and direction, and are therefore vector quantities. Adding vectors like displacement, acceleration or velocity has different rules than adding scalar quantities like mass and speed.
Why can't we say = +1 or 1.0 m [N] ?

26. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Why can't we say the answer is 3+4 =7.0 m [NW] ?
These quantities have magnitude and direction, and are therefore vector quantities. Adding vectors like displacement, acceleration or velocity has different rules than adding scalar quantities like mass and speed.
Why can't we say = +1 or 1.0 m [N] ?
We can represent vectors as integers, but only if the directions of each vector are along a line. [N] and [W] are not collinear.

27. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]

28. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule

29. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
3.0 m [W]

30. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
4.0 m [N]
3.0 m [W]

31. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
4.0 m [N]
3.0 m [W]

32. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule Δd
4.0 m [N]
3.0 m [W]

33. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ? Δd
4.0 m [N]
3.0 m [W]

34. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ?
= √ ( ) Δd
4.0 m [N]
3.0 m [W]

35. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ?
= √ ( )
= 5.0 m Δd
4.0 m [N]
3.0 m [W]

36. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ?
= √ ( )
= 5.0 m Δd
4.0 m [N] Θ
3.0 m [W]

37. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ?
= √ ( )
= 5.0 m
Θ = ? Δd
4.0 m [N] Θ
3.0 m [W]

38. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ?
= √ ( )
= 5.0 m
Θ = ? Δd
4.0 m [N] Θ
So/hCa/hTo/a
3.0 m [W]

39. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ?
= √ ( )
= 5.0 m
Θ = ?
Θ = tan-1( o/a ) Δd
4.0 m [N] Θ
So/hCa/hTo/a
3.0 m [W]

40. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ?
= √ ( )
= 5.0 m
Θ = ?
Θ = tan-1( o/a )
Θ = tan-1( 4/3 ) Δd
4.0 m [N] Θ
So/hCa/hTo/a
3.0 m [W]

41. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ?
= √ ( )
= 5.0 m
Θ = ?
Θ = tan-1( o/a )
Θ = tan-1( 4/3 ) = 53° Δd
4.0 m [N] Θ
So/hCa/hTo/a
3.0 m [W]

42. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ?
= √ ( )
= 5.0 m
Θ = ?
Θ = tan-1( o/a )
Θ = tan-1( 4/3 ) = 53°
Answer: Δd
4.0 m [N] Θ
So/hCa/hTo/a
3.0 m [W]

43. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ?
= √ ( )
= 5.0 m
Θ = ?
Θ = tan-1( o/a )
Θ = tan-1( 4/3 ) = 53°
Answer: m [W53°N] or [ ? ] Δd
4.0 m [N] Θ
So/hCa/hTo/a
3.0 m [W]

44. Adding vectors 90° to each other
Example #1: 3.0 m [W] m [N]
Apply the tip-to-tail rule
│Δd│ = ?
= √ ( )
= 5.0 m
Θ = ?
Θ = tan-1( o/a )
Θ = tan-1( 4/3 ) = 53°
Answer: m [W53°N] or [N37°W] Δd
4.0 m [N] Θ
So/hCa/hTo/a
3.0 m [W]

45. Adding vectors 90° to each other
Example #2: m/s [S] m/s [E]
Try this, yourself !

46. Adding vectors 90° to each other
Example #2: m/s [S] m/s [E]
Try this, yourself !
v = √( )
= m/s
Θ = tan-1( 12 / 20.8 )
= 30.0°
Answer: m/s [S30.0°E]
Or [E60.0°S] v
20.8 m/s [S] Θ
12.0 m/s [E]

47. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]

48. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
2.00 N [E30.0°N]
30.0°

49. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
8.00 N [N53.1°W]
53.1°
2.00 N [E30.0°N]
30.0°

50. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
8.00 N [N53.1°W]
53.1°
2.00 N [E30.0°N]
30.0°

51. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
8.00 N [N53.1°W] F
53.1°
2.00 N [E30.0°N]
30.0°

52. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ?
8.00 N [N53.1°W] F
53.1°
2.00 N [E30.0°N]
30.0°

53. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
8.00 N [N53.1°W] F
53.1°
2.00 N [E30.0°N]
30.0°

54. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
8.00 N [N53.1°W] F
53.1° β
2.00 N [E30.0°N]
30.0°

55. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
8.00 N [N53.1°W] F
53.1°
=36.9°
2.00 N [E30.0°N]
30.0°

56. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
8.00 N [N53.1°W] F
53.1°
=36.9° α
2.00 N [E30.0°N]
30.0°

57. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
8.00 N [N53.1°W] F
53.1°
=36.9°
30.0°
2.00 N [E30.0°N]
30.0°

58. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
│F │2 = ?
8.00 N [N53.1°W] F
53.1°
=36.9°
30.0°
2.00 N [E30.0°N]
30.0°

59. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
│F │2 = – (2)(8)(2)cos(66.9°)
8.00 N [N53.1°W] F
53.1°
=36.9°
30.0°
2.00 N [E30.0°N]
30.0°

60. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
│F │2 = – (2)(8)(2)cos(66.9°)
│F │= ?
8.00 N [N53.1°W] F
53.1°
=36.9°
30.0°
2.00 N [E30.0°N]
30.0°

61. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
│F │2 = – (2)(8)(2)cos(66.9°)
│F │= 7.45 N
8.00 N [N53.1°W] F
53.1°
=36.9°
30.0°
2.00 N [E30.0°N]
30.0°

62. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
│F │2 = – (2)(8)(2)cos(66.9°)
│F │= 7.45 N
What else is needed to specify
the vector ?
8.00 N [N53.1°W] F
53.1°
=36.9°
30.0°
2.00 N [E30.0°N]
30.0°

63. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
│F │2 = – (2)(8)(2)cos(66.9°)
│F │= 7.45 N
What else is needed to specify
the vector ?
direction specified by an
angle at the start of the resultant γ
8.00 N [N53.1°W] F
53.1°
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

64. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
│F │= ? SAS use cos law
│F │2 = – (2)(8)(2)cos(66.9°)
│F │= 7.45 N
What else is needed to specify
the vector ?
direction specified by an
angle at the start of the resultant γ
How can we find γ ?
8.00 N [N53.1°W] F
53.1°
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

65. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the Sine Law:
sin γ / 8 = ?
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

66. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the Sine Law:
sin γ / 8 = sin 66.9/ 7.45
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

67. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the Sine Law:
sin γ / 8 = sin 66.9/ 7.45
sin γ = ?
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

68. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the Sine Law:
sin γ / 8 = sin 66.9/ 7.45
sin γ = 8 sin 66.9/ 7.45
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

69. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the Sine Law:
sin γ / 8 = sin 66.9/ 7.45
sin γ = 8 sin 66.9/ 7.45
γ = sin-1 ( 8 sin 66.9/ 7.45 )
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

70. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the Sine Law:
sin γ / 8 = sin 66.9/ 7.45
sin γ = 8 sin 66.9/ 7.45
γ = sin-1 ( 8 sin 66.9/ 7.45 )
γ = 81.0° or ???
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

71. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the Sine Law:
sin γ / 8 = sin 66.9/ 7.45
sin γ = 8 sin 66.9/ 7.45
γ = sin-1 ( 8 sin 66.9/ 7.45 )
γ = 81.0° or 180 – 81 = 99.0°
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

72. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the Sine Law:
sin γ / 8 = sin 66.9/ 7.45
sin γ = 8 sin 66.9/ 7.45
γ = sin-1 ( 8 sin 66.9/ 7.45 )
γ = 81.0° or 180 – 81 = 99.0°
Which value of γ is correct?
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

73. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the Sine Law:
sin γ / 8 = sin 66.9/ 7.45
sin γ = 8 sin 66.9/ 7.45
γ = sin-1 ( 8 sin 66.9/ 7.45 )
γ = 81.0° or 180 – 81 = 99.0°
Which value of γ is correct?
We don't know.
Another method must be used here!!
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

74. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the Sine Law:
sin γ / 8 = sin 66.9/ 7.45
sin γ = 8 sin 66.9/ 7.45
γ = sin-1 ( 8 sin 66.9/ 7.45 )
γ = 81.0° or 180 – 81 = 99.0°
Note well: If the angle is opposite the
longest side, the angle could be acute
or obtuse. In this case, use the cos law.
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

75. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the Sine Law:
sin γ / 8 = sin 66.9/ 7.45
sin γ = 8 sin 66.9/ 7.45
γ = sin-1 ( 8 sin 66.9/ 7.45 )
γ = 81.0° or 180 – 81 = 99.0°
Note well: If the angle is opposite the
longest side, the angle could be acute
or obtuse. In this case, use the cos law.
Also note: you could solve for δ using sin law, then get γ δ
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

76. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ: δ
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

77. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ:
Focus on side opposite γ δ
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

78. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ:
Focus on side opposite γ
82 = ?? δ
8.00 N [N53.1°W] F
53.1°
7.45 N
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

79. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ:
Focus on side opposite γ
82 = – 2(2)(7.45)cosγ δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

80. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ:
Focus on side opposite γ
82 = – 2(2)(7.45)cosγ
64= cosγ δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

81. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ:
Focus on side opposite γ
82 = – 2(2)(7.45)cosγ
64= cosγ
+29.8cosγ= δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

82. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ:
Focus on side opposite γ
82 = – 2(2)(7.45)cosγ
64= cosγ
+29.8cosγ= – 64
29.8cosγ= -4.5 δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

83. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ:
Focus on side opposite γ
82 = – 2(2)(7.45)cosγ
64= cosγ
+29.8cosγ= – 64
29.8cosγ= -4.5
γ = ??? δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

84. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ:
Focus on side opposite γ
82 = – 2(2)(7.45)cosγ
64= cosγ
+29.8cosγ= – 64
29.8cosγ= -4.5
γ = cos-1(-4.5/29.8) δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
2.00 N [E30.0°N] γ
30.0°

85. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ:
Focus on side opposite γ
82 = – 2(2)(7.45)cosγ
64= cosγ
+29.8cosγ= – 64
29.8cosγ= -4.5
γ = cos-1(-4.5/29.8) = 98.7° δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
98.7°
2.00 N [E30.0°N]
30.0°

86. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ:
Focus on side opposite γ
82 = – 2(2)(7.45)cosγ
64= cosγ
+29.8cosγ= – 64
29.8cosγ= -4.5
γ = cos-1(-4.5/29.8) = 98.7°
Why can't 98.7° be used to specify the direction of F ? δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
98.7°
2.00 N [E30.0°N]
30.0°

87. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
Let's try the cos Law to find γ:
Focus on side opposite γ
82 = – 2(2)(7.45)cosγ
64= cosγ
+29.8cosγ= – 64
29.8cosγ= -4.5
γ = cos-1(-4.5/29.8) = 98.7°
Note 98.7°cannot specify the direction
of F b/c it is not with respect to any standard direction N,E,S,W δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
98.7°
2.00 N [E30.0°N]
30.0°

88. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
How can we write the final
answer? δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
98.7°
2.00 N [E30.0°N]
30.0°

89. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
How can we write the final
answer? δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
98.7°
2.00 N [E30.0°N] Θ
30.0°

90. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
How can we write the final
answer?
Θ = ??? δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
98.7°
2.00 N [E30.0°N] Θ
30.0°

91. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
How can we write the final
answer?
Θ = 180 – 30.0 – 98.7 δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
98.7°
2.00 N [E30.0°N] Θ
30.0°

92. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
How can we write the final
answer?
Θ = 180 – 30.0 – 98.7
= 51.3° δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
98.7°
2.00 N [E30.0°N] Θ
30.0°

93. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
How can we write the final
answer?
Θ = 180 – 30.0 – 98.7
= 51.3°
How do we write the final answer?
F = ???? δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
98.7°
2.00 N [E30.0°N] Θ
30.0°

94. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
How can we write the final
answer?
Θ = 180 – 30.0 – 98.7
= 51.3°
How do we write the final answer?
F = N [W51.3°N] or [ ??? ] δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
98.7°
2.00 N [E30.0°N] Θ
30.0°

95. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #1: N [E30.0°N] N [N53.1°W]
How can we write the final
answer?
Θ = 180 – 30.0 – 98.7
= 51.3°
How do we write the final answer?
F = N [W51.3°N] or [ ??? ]
F = N [W51.3°N]
or [N38.7W] δ
8.00 N [N53.1°W]
7.45 N
53.1° F
=36.9°
30.0°
98.7°
2.00 N [E30.0°N] Θ
30.0°

96. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #2: m/s [N30.0°E] m/s [E53.1°S]
Try this yourself !!

97. Adding Two Oblique Vectors: The Cos/sin Law Tip-to-Tail Method
Example #2: m/s [N30.0°E] m/s [E53.1°S]
|V|2 = (15)(55)cos(66.9)
|V| = 51.0 m/s
Resultant is opposite longest
side, so Ψ could be obtuse
or acute Use cos law
552 = (15)(51)cosΨ
2(15)(51)cosΨ = Ψ = 97.5°
Θ = 180°-(30.0°+Ψ )= 52.5°
Answer: m/s [S52.5°E] or [E37.5]
15.0 m/s
53.1°
30°
30° Ψ
55.0 m/s
36.9° Θ V