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1 – Introduction to Vectors

2 – 1D Horizontal Motion

3 – Motion due to Gravity

4 – Continuous Change

5 – Forces and Systems

6 – Introduction to Energy and Momentum

7 – Impacts and Collisions

8 – Uniform Circular Motion

9 – Difference Equations

10 – Introduction to Graph Theory

11 – Path Optimisation

12 – MST Optimisation

13 – Project Optimisation

Mathematical Modelling Project

Revision Material

HL Sample Papers

HL Exam Questions by Topic

OL Sample Papers

OL Exam Questions by Topic

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 a 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 \(\mbox{m}\)). Some examples of scalars are shown in the table below.

Quantity Abbreviation Units Abbreviation

Length

\(l\)

metre

\(\mbox{m}\)

Area

\(A\)

metre squared

\(\mbox{m}^2\)

Time

\(t\)

second

\(\mbox{s}\)

Mass

\(m\)

kilogram

\(\mbox{kg}\)

\[\,\]Figure 1.1.1

Key Point

Scalars have a magnitude only.

In much of Applied Maths, 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

\(\mbox{m}\)

Velocity

\(v\)

metre per second

\(\mbox{m/s}\)

Acceleration

\(a\)

metre per second squared

\(\mbox{m/s}^2\)

Force

\(F\)

Newton

\(\mbox{N}\)

\[\,\]Figure 1.1.2

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 case, how hard we kick) and a direction (in this case, the direction we kick the ball).

\[\,\]Figure 1.1.3

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

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Tip: If you ever wish to pause an animation, simply hover your mouse over it (non-touchscreens) or tap it (touchscreens).

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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 \(220{,}000\) metres.

That’s quite a large 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 (\(\mbox{km}\)) = \(1{,}000\) metres (\(\mbox{m}\)).

This then allows us to say more neatly that this distance between Dublin and Cork is approximately \(220\mbox{ km}\).

220,000 m
\[\,\]Figure 1.1.4
0.0022 m

On the smaller scale, the thickness of a \(2\) euro coin is approximately \(0.0022\) metres.

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

\[\,\]Figure 1.1.5

Below is a table of prefixes you will likely come across in this subject.

Prefix Symbol Factor

kilo

\(\mbox{k}\)

\(1{,}000\)

centi

\(\mbox{c}\)

\(0.01\)

milli

\(\mbox{m}\)

\(0.001\)

\[\,\]Figure 1.1.6

These are included in the Formulae and Tables booklet on page \(45\).

Guided Example 1.1.1

The length of an official marathon is \(42.195\mbox{ km}\).

(a) How many metres long is a marathon?

(b) How many millimetres long is a marathon?

(a) As there are \(1{,}000\) metres in a kilometre, there are \(42.195 \times 1{,}000 = 42{,}195\) metres in a marathon.

(b) As there are \(1{,}000\) millimetres in a metre, there are \(42{,}195 \times 1{,}000 = 42{,}195{,}000\) millimetres in a marathon.

For certain quantities, rather than using prefixes, we instead use completely new words.

For example, we say that are \(60\) seconds in a minute and, likewise, that there are \(60\) minutes in an hour.

Such quantities are converted in much the same way. In fact, we may sometimes need to convert a mix of several quantities!

Guided Example 1.1.2

The speed limit in the vicinity of many Irish schools is \(30\) kilometres per hour.

Using a calculator, write this speed limit in metres per second.

There are \(1{,}000\) metres in a kilometre and there are \(60\times 60=3{,}600\) seconds in an hour.

Therefore, we can rewrite this speed limit as follows:

\begin{align}30\mbox{ km/h}&= \frac{30 \mbox{ kilometres}}{1\mbox{ hour}}\\&=\frac{30\times 1{,}000\mbox{ metres}}{1\times 3{,}600\mbox { seconds}}\\&=\frac{30{,}000}{3{,}600}\frac{\mbox{metres}}{\mbox{second}}\\&=8\frac{1}{3}\mbox{ m/s}\end{align}

where the final answer was found using a calculator.

Prefixes can be applied not only to metres but to any unit in the metric system. These units, known as 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, the length of a marathon can also be stated as \(26\) miles \(385\) yards.

Likewise, you may know what your height is in SI units (metres and centimetres), you may instead know it in imperial units (feet and inches), or both!

1 metre and 48 centimetres(metric)4 feet and 10 inches(imperial)
\[\,\]Figure 1.1.7
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