# Scalar quantities and vector quantities

Physics, as a discipline that explores nature and its fundamental principles, uses a wide range of terms and concepts to describe the world around us. Among these, two fundamental concepts are scalar quantities and vector quantities. These two categories represent not only the foundations of classical physics but also profoundly influence our understanding of complex phenomena in modern and quantum physics.

In everyday life, often unconsciously, we come across examples of both magnitudes. When we control temperature, we are considering a scalar quantity, while when we control the speed and direction of the wind, we are working with a vector quantity. These apparently simple distinctions become of fundamental importance when we delve into more advanced studies, such as fluid dynamics or the theory of relativity.

Scalar and vector quantities differ mainly in the nature of the information they convey. While scalar quantities provide an absolute value, such as mass or energy, vector quantities offer a more complete picture, giving information on both magnitude and direction, as in the case of force or velocity. This distinction becomes even more crucial when we tackle problems that require vector analysis, as in the case of rigid body mechanics or studies of electromagnetism.

## The DNA of physical data

The DNA of data: what does physical data look like inside? We have seen how making a measurement means, for a physical phenomenon, associating a number with a unit of measurement. This is not enough for us: how are these data really made?

We look inside the data as we would look inside a living being to discover the fundamental information that defines it, i.e., its DNA. In physics, data has its own DNA, which is not composed of four nitrogenous bases (ATCG, adenine, thymine, cytosine, and guanine) as for living beings, but of one or three fundamental pieces of information:

• module
• direction

### Example.

Let’s take a physical quantity like the volume of a blue whale, the largest animal found on our planet. Volume is a physical quantity that, in the International System, is measured in cubic metres.
To measure the volume of the whale, we immerse it in a salt water pool for its safety and for our ease of measurement: the whale remains still because, as an intelligent animal, it is also curious to know the result.
Let’s assume that the pool has the shape of a cube whose side is known to us and that it is almost full of water; we completely immerse the whale, always still and collaborative, we observe the rise in the water level, we thank Archimedes and measure the volume of water from the previous level without the whale to the new level with the whale immersed. We get a Volume of, let’s try, 230 m3.

## The measurement

Do we need to know anything else? No, the measurement is complete: we have a numerical value, the modulus, equal to 230, and a unit of measurement, the 3 m3, and nothing else is needed. We communicate this to the whale, who is pleased and offers to continue working with us but who asks, rightly, to be released into the open sea.

We release the whale into the open sea, then we ask ourselves the question of evaluating its speed expressed, in SI, in meters per second (��sm​) and which can also be expressed in one of its derived and most common units of measurement, such as kilometers per hour (��ℎhkm​).
To observe the speed of the whale, free to move in the ocean, and to describe it, it is not enough for us to know how many km/h it is moving, i.e. the speed module is not enough, but we need to have two additional pieces of information:

• the direction, i.e. the imaginary straight line along which the whale is moving, moment by moment
• the direction, i.e. knowing whether the whale, along this straight line, is going forwards or backwards.

We therefore associate, moment by moment, the modulus of the whale’s speed with its direction and its direction, and we more properly indicate the speed.

## How a piece of data is made

The data are not all the same: some just need a number and a unit of measurement, while others need something else: direction and direction!
We are talking about scalars and vectors.

Let’s now find out which quantities can be considered scalars and which can be considered vector!

The volume (ININ) is a scalar quantity because to define it, all you need is a number and a unit of measurement (3 m3) or multiples and submultiples. There is no need to establish other data.

The speed (in⃗in), however, requires a module , a direction and to be defined. Velocity is a vector quantity .

## What are vectors and what are they for?

• Those that can be expressed simply through a number, referring to a unit of measurement (for example, length, time, temperature, energy)
• Those for which a number and a unit of measurement are not enough (for example, displacement, speed, acceleration)

The former are called scalar quantities , the latter vector quantities .