### Weekly Grinds

1.1 – Scalars and Vectors

## Scalars and Vectors

In both mathematics and science during your Junior Certificate, you will have dealt with physical quantities that have a magnitude, i.e. a value, associated with them. Quantities which only have a magnitude are referred to as scalars (because they can be measured using a scale or ruler).

For example, we can say that your table has a particular length, e.g. $$1.5$$ metres. The quantity in this case is length (abbreviated as $$l$$) and the unit associated with that quantity is metres (abbreviated as m). Some examples of scalars are shown in the table below.

Quantity Abbreviation Units Abbreviation
Length
$$l$$
metre
m
Area
$$A$$
metre squared
m$$^2$$
Time
$$t$$
second
s
Mass
$$m$$
kilogram
kg

### Key Point

Scalars have a magnitude only.

In much of applied mathematics, we shall also be dealing with what are referred to as vectors. Whereas scalars have only a magnitude, vectors have both a magnitude and a direction. Some examples of vectors are shown below (don’t worry, we are going to properly define what these quantities actually are in later lessons!).

Quantity Abbreviation Units Abbreviation
Displacement
$$s$$
metre
m
Velocity
$$v$$
metres per second
m/s
Acceleration
$$a$$
metres per second squared
m/s$$^2$$
Force
$$F$$
Newton
N

### Key Point

Vectors have both a magnitude and a direction.

For example, if we kick a ball along the ground, we are applying a force to that ball in a particular direction. Force is therefore a vector rather than a scalar – it has a magnitude (in this example, how hard we kick) and a direction (in this case, the direction we kick the ball).

###### Figure 1.1.3

However, it would not make sense to say your table’s length has a direction. Length therefore only has a magnitude (value) and is therefore instead considered a scalar.

When dealing with both scalars and vectors, it is quite common to have to deal with both very large and very small numbers. On the large scale, we could consider distances across Ireland. For example, the straight line distance between Cork and Dublin is approximately $$220000$$ metres. ###### Figure 1.1.4

That’s quite a long number, and it can be quite easy to accidentally leave out a zero when writing it down.

Therefore, we instead introduce what are known as prefixes to our units. The prefix that is most useful here is kilo, i.e. a thousand of a particular quantity. Thus, we say that $$1$$ kilometre (km) = $$1000$$ metres (m) and this allows us to instead say more neatly that this distance is $$220$$ km.

On the smaller scale, the thickness of a €2 coin is approximately 0.0022 metres. ###### Figure 1.1.5

Again, we could instead use another prefix, mili (‘thousandth’), to instead say that this coin’s thickness is 2.2 mm (millimetres).

Below is a table of prefixes you will likely come across in this course, and more extreme prefixes that describe even larger and smaller values can be found in your Formulae and Tables booklet on page 47.

Prefix Symbol Factor
kilo
k
1000
centi
c
0.01
milli
m
0.001
###### Figure 1.1.6

Such prefixes can be applied not only to metres but to any unit in the metric system. These SI (système international) units have been adopted and are used in the majority of the world, especially in science and mathematics. You may however sometimes still hear the older imperial system being used in your daily life. For example, you may know what your height is in SI units (metres and centimetres) or you may instead know it in imperial units (feet and inches). 