1 Introduction

It is useful to separate physical quantities into two types: the first are called scalars ; the second are known as vectors . A scalar is a quantity that can be described by a single number which can be positive, negative or zero. An example of a scalar quantity is the mass of an object, so we might state that ‘the mass of the stone is 3 kg’. It is important to give the units in which the quantity is measured. Obvious examples of scalars are temperature and length, but there are many other engineering applications in which scalars play an important role. For example, speed, work, voltage and energy are all scalars.

On the other hand, vectors are quantities which require the specification of a magnitude and a direction . An example of a vector quantity is the force applied to an object to make it move. When the object shown in Figure 1 is moved by applying a force to it we achieve different effects depending on the direction of the force.

Figure 1 :

{ A force is a vector quantity}

In order to specify the force completely we must state not only its magnitude (its ‘strength’) but also the direction in which the force acts. For example we might state that ‘a force of 5 newtons is applied vertically from above’. Clearly this force would have a different effect from one applied horizontally. The direction in which the force acts is crucial.

There are many engineering applications where vectors are important. Force, acceleration, velocity, electric and magnetic fields are all described by vectors. Furthermore, when computer software is written to control the position of a robot, the position is described by vectors.

Sometimes confusion can arise because words used in general conversation have specific technical meanings when used in engineering calculations. An example is the use of the words ‘speed’ and ‘velocity’. In everyday conversation these words have the same meaning and are used interchangeably. However in more precise language they are not the same. Speed is a scalar quantity described by giving a single number in appropriate units; for example ‘the speed of the car is 40 kilometres per hour’. On the other hand velocity is a vector quantity and must be specified by giving a direction as well. For example ‘the velocity of the aircraft is 20 metres per second due north’.

In engineering calculations, the words speed and velocity cannot be used interchangeably. Similar problems arise from use of the words ‘mass’ and ‘weight’. Mass is a scalar which describes the amount of substance in an object. The unit of mass is the kilogramme. Weight is a vector, the direction of which is vertically downwards because weight arises from the action of gravity. The unit of weight is the newton. Displacement and distance are related quantities which can also cause confusion. Whereas distance is a scalar, displacement is ‘directed distance’, that is, distance together with a specified direction. So, referring to Figure 2, if an object is moved from point A to point B , we can state that the distance moved is 10 metres, but the displacement is 10 metres in the direction from A to B .

Figure 2 :

{ Displacement means directed distance}

You will meet many other quantities in the course of your studies and it will be helpful to know which are vectors and which are scalars. Some common quantities and their type are listed in Table  1 . The S.I. units in which these are measured are also shown.


Table 1: Some common scalar and vector quantities
quantity type S.I. unit
distance scalar metre, m
mass scalar kilogramme, kg
temperature scalar kelvin, K
pressure scalar pascal, Pa
work scalar joule, J
energy scalar joule, J
displacement vector metre m
force vector newton, N
velocity vector metres per second, m s − 1
acceleration vector metres per second per second, m s − 2
Exercise

State which of the following are scalars and which are vectors:

  1. the volume of a petrol tank,
  2. a length measured in metres,
  3. a length measured in miles,
  4. the angular velocity of a flywheel,
  5. the relative velocity of two aircraft,
  6. the work done by a force,
  7. electrostatic potential,
  8. the momentum of an atomic particle.

(1), (2), (3) (6), (7) are scalars.

(4), (5), and (8) are vectors