1.4 Scalars & Vectors Scalar Quantity

Only has a magnitude Has no direction E.g. mass, temperature, time, length, speed, energy

Vector Quantity

Has magnitude and direction E.g. displacement, force, velocity, acceleration, momentum

Vectors On diagram, a vector can be represented by an arrow,

The length of the arrow, drawn to scale, represents its magnitude The direction of the arrow is the direction of the vector quantity

Resultant Vector When two or more vectors of the same type are acting on a body, a single

resultant vector will be experienced. Adding the vectors gives the resultant vector. When adding vectors, their directions have to be considered.

Methods of Adding Vectors 1. Graphical Method (drawn to scale)

o Parallelogram o Head-to-tail

2. Calculation (on a sketched vector diagram) o Pythagoras theorem o Trigonometric formula

3. Component method o Resolve the vectors into two perpendicular components and adding the sum

of the two perpendicular vectors to find the resultant

Head-to-tail Method Applicable to any number of vectors Order of the vectors joined in the head-to-tail method is immaterial (will not

affect the resultant) When the diagram formed is a closed polygon, the resultant is zero

Resolving Vector Two vectors can add up to form a single vector Conversely, a single vector can be considered as being made up of two component

vectors Hence, a single vector can be resolved into two perpendicular components

o E.g. If a vector A is resolved into its components A1 and A2, what is the relation between A and its components?

o E.g. If a vector A is above the horizontal, what is its horizontal component Ax and vertical component Ay in terms of A and ?

o E.g. The inclined plane

An object of mass m is placed on a slope inclined at to the horizontal, the component of its weight is

Along the slope mg sin Perpendicular to the slope mg cos

Steps of Finding Resultant Using the Component Method Resolve each of the vectors into their

components Add up all the components in the same direction Find the resultant of these two sum components

using Pythagoras theorem.

E.g. Vector A has a magnitude of 3.76 and is at an angle of 34.5 above the positive x-direction. Vector B has a magnitude of 4.53 and is at an angle of 34.1 above the negative x-direction. What is the resultant of the two vectors?

; C = A + B Cx = Ax + Bx Cy = Ay + By

Vector x-component y-component A Ax = A cos

Ax = 3.76 cos 34.5 Ax = 3.10

Ay = A sin Ay = 3.76 sin 34.5 Ay = 2.13

B Bx = B cos Bx = 4.53 cos 34.1 Bx = -3.75

By = B sin By = 4.53 sin 34.1 By = 2.54

C Cx = 3.10 + (-3.75) Cx = -0.65

Cy = 2.13 + 2.45 Cy = 4.67

C = [(Cx)2 + (Cy)2] C = [(0.65)2 + (4.67)2] C = 4.72

tan = 0.65 = 7.92 tan = 4.67 = 90 7.92 = 82.1 The resultant vector has a magnitude of 4.72 and 82.1 above the negative x-direction.

Vector Subtraction To find the difference of two vectors, say, a and b, is denoted by a b. It is equivalent to vector addition, with the direction of the second vector reversed. Thus, it can also be written as a b = a + (-b)) The vectors have the same magnitude but opposite in direction. Use the head-to-head method for diagram