# Vectors and Scalars

## Scalars

A scalar is a quantity that has only magnitude, (size).

When we measure scalar quantities we only give the size or magnitude, plus the appropriate SI units. The is no need to indicate a direction.

Distance travelled is a good example. If you go on a journey from Manchester to Liverpool you see the motorway signs telling you the distance you have to travel, but it doesn’t have to tell you the direction. Just how far.

Common Scalars:

**•** Mass

**•** Time

**•** Amount of a material

**•** Current

**•** Area

**•** Density

**•** Speed

**•** Length

Deciding if a quantity is a scalar is simple. Ask yourself if direction would be meaningful or important. If the answer is no, then it is a scalar quantity.

__Volume of a liquid __

You can use a measuring cylinder and get a value, its magnitude. It would be nonsense to try to add a direction description to this.

2 ml East! It is nonsense, the description of the direction has no meaning.

We have all this information we need with the 2 ml. So it is clearly a scalar quantity.

**What about time?**

Time only goes forwards, it might go backwards in Sci-Fi movies, but in the real world that is not going to happen. So we only need the magnitude of time, we do not need to add in that it is into the future. Time is a scalar.

**What about Energy?**

It takes 4200 joules (J) of energy to heat up 1 kg of water by 1 ^{o}C. Do you need any other information about the energy to understand the amount of energy used?

Answer; No. Energy is, therefore, a scalar quantity.

## Vectors

A Vector quantity has both magnitude ** and direction**.

Unlike a scalar, a vector has to have two sets of information to fully describe it.

**Size and direction**

In these examples, if we ask the question does direction matter, the answer is yes.

Force: If you push an object with 200 newtons (N) of force it is important to say in which direction you are pushing. The effect of pushing left compared to the right have opposite effects. If we miss out the direction we would only have half the information we need to understand the situation.

**Vector arrows**

A vector can be represented by an arrow.

The length of the arrow represents the magnitude, normally this is drawn to a set scale.

The head of the arrow indicated the direction.

Common Vectors in Physics:

**•** Velocity

**•** Acceleration

**•** Force

**•** Weight (which is a type of force)

**•** Displacement

**•** Momentum

For all of these, you need to give the magnitude (size), the correct SI unit and the direction.

**Representing the Direction of a Vector**

There are three main way to achieve this:

- Diagrammatically - the example of the arrow above is a diagrammatic representation of direction.
- Descriptive - using directional words, such as up, down, left, right, East or West. We can also use the headings on a compass or degrees of arc from a reference point.
- Numerically - We can assign a given direction as positive and the opposite direction as negative. The relative direction is indicated by the + or - in front of the numbers.

## Scalars and Vectors

Scalars - Magnitude only

Vectors - Magnitude and direction

**Distance and Displacement**

If someone walks from point A to point B in the diagram above and they follow the dotted line, the distance they have travelled can be measured and given in meters. As the direction is constantly changing there is no need to state it.

Distance is, therefore, a SCALAR.

The straight line distance from A to B is known as the displacement. This is the shortest distance between A and B. In this situation the length of this line also needs the direction to be stated to give a complete answer.

Displacement is, therefore, a VECTOR.

**Weight and Mass**

These two words are often used interchangeably in everyday speech. In physics, however, they have very different meanings.

Mass is a measure of the amount of material that makes up an object, measured in Kilograms. In this situation, direction has no meaning.

Mass is, therefore, a SCALAR

Weight is a measure of the pull of gravity on a mass, it is a type of force. As a force, it is measured in newtons (N). Here direction is important.

Weight is, therefore, a VECTOR.

- What is the key difference between a scalar and vectors?
- Your answer should include: direction / magnitude

Explanation:*Vectors have a direction as well as magnitude.* - What is the difference between + 10 N and - 10 N?
- Your answer should include: opposite / direction / directions