1. 1

2. Chapter Objectives Parallelogram Law Cartesian vector form
Dot product and an angle between two vectors 2

3. Chapter Outline Scalars and Vectors Vector Operations
Vector Addition of Forces
Addition of a System of Coplanar Forces
Cartesian Vectors
Addition and Subtraction of Cartesian Vectors
Position Vectors
Force Vector Directed along a Line
Dot Product 3

4. 2.1 Scalars and Vectors Scalar
– A quantity characterized by a positive or negative number
– Indicated by letters in italic such as A (italics)
e.g. 4

5. 2.1 Scalars and Vectors Vector
– A quantity that has magnitude and direction
e.g.
– Arrow above a letter
– A magnitude is represented by
– In this course, we sometimes use a bold font, A, for a vector and the magnitude is represented by A 5

6. 2.2 Vector Operations
Multiplication and Division of a Vector by a Scalar
- Product of vector A and scalar a
- Magnitude =
- Law of multiplication applies e.g. A/a = ( 1/a ) A, a≠0 6

7. 2.2 Vector Operations Vector Addition
- Two vectors addition, A and B, gives a vector R, can be done by parallelogram law
- R can be obtained by triangle
- R = A + B = B + A
- For collinear vectors, i.e. A and B are on the same straight line. 7

8. 2.2 Vector Operations Vector Subtraction
- Special case of addition, similar to vector addition
e.g. R’ = A – B = A + ( - B ) 8

9. 2.3 Vector Addition of Forces
Resultant Force
Parallelogram law
Resultant,
FR = ( F1 + F2 ) 9

10. 2.3 Vector Addition of Forces
Analysis Procedure
Parallelogram Law
Draw a parallelogram, parallel to the two forces
The resultant force is a diagonal of the parallelogram 10

11. 2.3 Vector Addition of Forces
law of cosines
law of sines 11

12. Example 2.1
The screw eye is subjected to two forces, F1 and F2. Determine the magnitude and direction of the resultant force. 12

13. Solution
Parallelogram Law
Unknown: magnitude of FR and angle θ 13

14. Solution 14

15. Solution Trigonometry Direction Φ of FR measured from the horizontal15

16. 2.4 Addition of a System of Coplanar Forces
Scalar Notation
Use scalar values and use axis x and y, consider positive and negative values accordingly 16

17. 2.4 Addition of a System of Coplanar Forces
Cartesian Vector Notation
Caretesian vectors i and j for x and y, respectively
i and j is a unit vector
Examples 17

18. 2.4 Addition of a System of Coplanar Forces
Coplanar Force Resultants
Find components in x and y for all forces
Adding all the forces for each direction
The resultant force found by parallelogram or pythagorus
Cartesian vector notation: 18

19. 2.4 Addition of a System of Coplanar Forces
Coplanar Force Resultants 19

20. 2.4 Addition of a System of Coplanar Forces
Coplanar Force Resultant
Find FR by pythagorus 20

21. Example 2.5
Determine x and y components of F1 and F2 acting on the boom. Express each force as a Cartesian vector. 21

22. Solution
Scalar Notation
Cartesian Vector Notation 22

23. Solution By similar triangles we have Scalar Notation:
Cartesian Vector Notation: 23

24. Example 2.6
The link is subjected to two forces F1 and F2. Determine the magnitude and orientation of the resultant force. 24

25. Solution I
Scalar Notation: 25

26. Solution I Resultant Force From vector addition, direction angle θ is26

27. Solution II
Cartesian Vector Notation
Thus, 27

28. Right-Handed Coordinate System Right thumb z
2.5 Cartesian Vectors (3D)
Right-Handed Coordinate System
Right thumb z
Other fingers sweeping from x to y
z axis in 2D is pointing outwards from the paper 28

29. 2.5 Cartesian Vectors Rectangular Components of a Vector
Vector A can be found in x, y and z
Use parallelogram twice
A = A’ + Az
A’ = Ax + Ay
A = Ax + Ay + Az 29

30. 2.5 Cartesian Vectors Unit Vector Direction A find by unit vector
Magnitude 1
uA = A / A ดังนี้ A = A uA 30

31. 2.5 Cartesian Vectors Cartesian Vector Representations
3 componenets A in i, j and k directions 31

32. 2.5 Cartesian Vectors
Magnitude of a Cartesian Vector 32

33. 2.5 Cartesian Vectors Direction of a Cartesian Vector
Direction of A defined by α, β and γ
0° ≤ α, β and γ ≤ 180 °
The direction cosines of A is 33

34. 2.5 Cartesian Vectors Direction of a Cartesian Vector
A = Axi + Ayj + AZk
uA = A /A = (Ax/A)i + (Ay/A)j + (AZ/A)k 34

35. 2.5 Cartesian Vectors Direction of a Cartesian Vector
uA = cosαi + cosβj + cosγk
= Acosαi + Acosβj + Acosγk
= Axi + Ayj + AZk 35

36. 2.6 Addition and Subtraction of Cartesian Vectors
Concurrent Force Systems
Resultant forces
FR = ∑F = ∑Fxi + ∑Fyj + ∑Fzk 36

37. Example 2.8
Express the force F as Cartesian vector. 37

38. Solution 38

39. Solution 39

40. 2.7 Position Vectors x,y,z Coordinates
Right-handed coordinate system
All other points are relative to O 40

41. 2.7 Position Vectors Position Vector
Position vector r used to indicate a location from a reference position
E.g. r = xi + yj + zk 41

42. 2.7 Position Vectors Position Vector (between 2 points)
Vector addition rA + r = rB
Solving
r = rB – rA = (xB – xA)i + (yB – yA)j + (zB –zA)k or r = (xB – xA)i + (yB – yA)j + (zB –zA)k 42

43. 2.7 Position Vectors
Direction and magnitude of a cable can be found by position vectors A and B
Position vector r
Angles α, β and γ
Unit vector, u = r/r 43

44. Example 2.12
An elastic rubber band is attached to points A and B. Determine its length and its direction measured from A towards B.
A (1, 0, -3) m
B (-2, 2, 3) m 44

45. Solution Position vector Magnitude = length of the rubber band
Unit vector in the director of r 45

46. Solution 46

47. 2.8 Force Vector Directed along a Line
For 3D, direction of F
F = F u = F (r/r)
F with unit (N) r with unit (m)
(r/r) 47

48. 2.8 Force Vector Directed along a Line
F is force along the cable
- need x, y, z
- find position vector r along a cable
Find unit vector u = r/r
F = Fu 48

49. Example 2.13
The man pulls on the cord with a force of 350N. Represent this force acting on the support A, as a Cartesian vector and determine its direction. 49

50. Solution
End points of the cord are A (0m, 0m, 7.5m) and B (3m, -2m, 1.5m)
Magnitude = length of cord AB
Unit vector, 50

51. Solution
Force F has a magnitude of 350N, direction specified by u. 51

52. 2.9 Dot Product The result is scalar
Dot product of A and B written as A·B (A dot B)
A·B = AB cosθ where 0°≤ θ ≤180°
The result is scalar 52

53. 2.9 Dot Product Laws of Operation 1. Commutative law A·B = B·A
2. Multiplication by a scalar
a(A·B) = (aA)·B = A·(aB) = (A·B)a
3. Distribution law
A·(B + D) = (A·B) + (A·D) 53

54. 2.9 Dot Product Cartesian Vector Formulation
- Dot product of Cartesian unit vectors
i·i = (1)(1)cos0° = 1
i·j = (1)(1)cos90° = 0
- Similarly
i·i = 1 j·j = 1 k·k = 1
i·j = 0 i·k = 0 j·k = 0 54

55. 2.9 Dot Product Cartesian Vector Formulation Application
Dot product of 2 vectors A and B
A·B = AxBx + AyBy + AzBz
Application
Find angle between two vectors
θ = cos-1 [(A·B)/(AB)] 0°≤ θ ≤180°
Find vector parallel and perpendicular components
Aa = A cos θ = A·u 55

56. Example 2.17
The frame is subjected to a horizontal force F = {300j} N. Determine the components of this force parallel and perpendicular to the member AB.
A (0, 0, 0)
B (2, 6, 3) 56

57. Solution
Since
Thus 57

58. Solution
Since result is a positive scalar, FAB has the same sense of direction as uB. Express in Cartesian form
Perpendicular component 58

59. Solution
Magnitude can be determined from F┴ or from Pythagorean Theorem, or 59