2.1 Scalars and Vectors

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1 2.1 Scalars and Vectors Scalar A quantity characterized by a positive or negative number Indicated by letters in italic such as A e.g. Mass, volume and length

2 2.1 Scalars and Vectors Vector A quantity that has magnitude and direction e.g. Position, force and moment Represent by a letter with an arrow over it, Magnitude is designated as In this subject, vector is presented as A and its magnitude (positive quantity) as A A A

3 2.2 Vector Operations Multiplication and Division of a Vector by a Scalar - Product of vector A and scalar a = aa - Magnitude = aa - Law of multiplication applies e.g. A/a = ( 1/a ) A, a 0

4 2.2 Vector Operations Vector Addition - Addition of two vectors A and B gives a resultant vector R by the parallelogram law - Result R can be found by triangle construction - Communicative e.g. R = A + B = B + A - Special case: Vectors A and B are collinear (both have the same line of action)

5 2.2 Vector Operations Vector Subtraction - Special case of addition e.g. R = A B = A + ( - B ) - Rules of Vector Addition Applies

6 2.3 Vector Addition of orces inding a Resultant orce Parallelogram law is carried out to find the resultant force Resultant, R = ( )

7 2.3 Vector Addition of orces Procedure for Analysis Parallelogram Law Make a sketch using the parallelogram law 2 components forces add to form the resultant force Resultant force is shown by the diagonal of the parallelogram The components is shown by the sides of the parallelogram

8 2.3 Vector Addition of orces Procedure for Analysis Trigonometry Redraw half portion of the parallelogram Magnitude of the resultant force can be determined by the law of cosines Direction if the resultant force can be determined by the law of sines Magnitude of the two components can be determined by the law of sines

9 2.4 Addition of a System of Coplanar orces Scalar Notation x and y axes are designated positive and negative Components of forces expressed as algebraic scalars x x y cos and sin y

10 2.4 Addition of a System of Coplanar orces Cartesian Vector Notation Cartesian unit vectors i and j are used to designate the x and y directions Unit vectors i and j have dimensionless magnitude of unity ( = 1 ) Magnitude is always a positive quantity, represented by scalars x and y i x y j

11 2.4 Addition of a System of Coplanar orces Coplanar orce Resultants To determine resultant of several coplanar forces: Resolve force into x and y components Addition of the respective components using scalar algebra Resultant force is found using the parallelogram law Cartesian vector notation: i j x 3x 2x i 1y i 3 y 2 y j j

12 2.4 Addition of a System of Coplanar orces Coplanar orce Resultants Vector resultant is therefore R i j Rx Ry If scalar notation are used Rx Ry 1x 1y 2x 2 y 3x 3y

13 2.4 Addition of a System of Coplanar orces Coplanar orce Resultants In all cases we have Rx Ry x y * Take note of sign conventions Magnitude of R can be found by Pythagorean Theorem R 2 Rx 2 Ry and tan -1 Ry Rx

14 2.5 Cartesian Vectors Right-Handed Coordinate System A rectangular or Cartesian coordinate system is said to be right-handed provided: Thumb of right hand points in the direction of the positive z axis, when the right-hand fingers are curled about this axis and directed from the pozitive x towards the pozitive y axis. z-axis for the 2D problem would be perpendicular, directed out of the page.

15 2.5 Cartesian Vectors Rectangular Components of a Vector A vector A may have one, two or three rectangular components along the x, y and z axes, depending on orientation By two successive application of the parallelogram law A = A + A z A = A x + A y Combing the equations, A can be expressed as A = A x + A y + A z

16 2.5 Cartesian Vectors Unit Vectors Direction of A can be specified using a unit vector Unit vector has a magnitude of 1 If A is a vector having a magnitude of A 0, unit vector having the same direction as A is expressed by u A = A / A. So that A = A u A

17 2.5 Cartesian Vectors Cartesian Vector Representations Since the three components of A act in the positive i, j and k directions A = A x i + A y j + A Z k *Note the magnitude and direction of each components are separated, easing vector algebraic operations.

18 2.5 Cartesian Vectors Magnitude of a Cartesian Vector rom the blue colored triangle, rom the gray colored triangle, Combining the equations gives magnitude of A z y x A A A A 2 2 ' y x A A A 2 2 ' z A A A

19 2.5 Cartesian Vectors Direction of a Cartesian Vector Orientation of A is defined by the coordinate direction angles α, β and γ measured between the tail of A and the positive x, y and z axes 0 α, β, γ 180 The direction cosines of A is cos A x A cos A z A cos A y A

20 2.5 Cartesian Vectors Direction of a Cartesian Vector Once the direction cosines have been obtained, angles α, β and γ can be determined by the inverse cosines An easy way of obtaining these direction cosines is to form a unit vector u A in the direction of A. Given, A = A x i + A y j + A z k then, u A = A /A = (A x /A)i + (A y /A)j + (A Z /A)k where A A A A x y z

21 2.5 Cartesian Vectors Direction of a Cartesian Vector It can be noticed that the i, j,k components of u A represent the direction cosines of A. u A = cosαi + cosβj + cosγk Since the magnitude of a vector is equal to the positive square root of the sum of the squares of the magnitudes of its components, and u A has a magnitude of one, we have cos 2 cos cos 1 A as expressed in Cartesian vector form is A = Au A = Acosαi + Acosβj + Acosγk = A x i + A y j + A Z k 2 2

22 2.6 Addition and Subtraction of Cartesian Vectors Concurrent orce Systems orce resultant is the vector sum of all the forces in the system R = = x i + y j + z k

23 2.7 Position Vectors Position vector is of importance in formulating a Cartesian force vector directed between two points in space. x,y,z Coordinates Right-handed coordinate system Positive z axis points upwards, measuring the height of an object or the altitude of a point Points are measured relative to the origin, O.

24 2.7 Position Vectors Position Vector Position vector r is defined as a fixed vector which locates a point in space relative to another point. E.g. r = xi + yj + zk

25 2.7 Position Vectors Position Vector The position vector may be directed from point A to point B in space, ig. By the head-to-tail vector addition: r A + r = r B Solving for r yields, r = r B r A = (x B x A )i + (y B y A )j + (z B z A )k

26 2.8 orce Vector Directed along a Line In 3D problems, direction of is specified by two points, through which its line of action passes can be formulated as a Cartesian vector = u = (r/r) Note that has units of forces (N) unlike r, with units of length (m)

27 2.9 Dot Product Through the dot product operation, one can find the angle between two lines or the components of a force parallel and perpendicular to a line. Dot product of vectors A and B is written as A B (Read A dot B) It is defined as the product of the magnitudes of A and B and the cosine of the angle θ between their tails, A B = AB cosθ where 0 θ 180 It is often referred to as the scalar product of vectors since the result is a scalar.

28 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)

29 2.9 Dot Product Cartesian Vector ormulation - 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

30 2.9 Dot Product Cartesian Vector ormulation Dot product of two vectors A and B, A B = (A x i + A y j + A z k) (B x i + B y j + B z k) A B = A x B x + A y B y + A z B z Thus, to determine the dot product of two Cartesian vectors, multiply their corresponding x, y, z components and sum these products algebraically.

31 2.9 Dot Product Applications The angle formed between two vectors or intersecting lines. θ = cos -1 [(A B)/(AB)] 0 θ 180

32 2.9 Dot Product Applications The components of a vector parallel and perpendicular to a line. The component of vector A parallel to or collinear with the line aa is defined as A a = A cos θ. If the direction of the line is specified by u a, since u a =1, the magnitude of A a can be determined from the dot product as A a = A cos θ = A u

33 2.9 Dot Product Applications The components of a vector parallel and perpendicular to a line. The component A a represented as a vector is therefore A a = A a u a The component of A that is perpendicular to line aa can also be obtained. Since, A = A a + A, then A = A - A a By Pythagorean s theorem: A 2 2 A Aa

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