1

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

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

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

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

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

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

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

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

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

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

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

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

Solution 14

Solution Trigonometry Direction Φ of FR measured from the horizontal 15

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

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

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

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

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

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

Solution Scalar Notation Cartesian Vector Notation 22

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

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

Solution I Scalar Notation: 25

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

Solution II Cartesian Vector Notation Thus, 27

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

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

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

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

2.5 Cartesian Vectors Magnitude of a Cartesian Vector 32

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

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

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

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

Example 2.8 Express the force F as Cartesian vector. 37

Solution 38

Solution 39

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

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

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

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

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

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

Solution 46

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

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

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

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

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

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

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

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

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

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

Solution Since Thus 57

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

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