Vector Diagram

The distance from London to Paris is around \( 350\,\mathrm{km} \) if you flew straight there in a plane. However, if you flew the same distance in a different direction then you would end up somewhere else, of course. Amsterdam is also around \( 350\,\mathrm{km} \) from London. To get from one point to another you must know not just the distance but also the direction. In physics, quantities with both a size and a direction are called vectors. In this article, we will learn the different ways of drawing vectors on vector diagrams and how these diagrams can be used.

What is a vector diagram?

Vector diagrams are extremely important in physics and you will come across them throughout your studies. They are defined as follows:

As expected from the name, a vector diagram is a way to visualise vectors. It is important to understand what a vector quantity is exactly:

The reason for giving the definition of a vector quantity is to distinguish it from a scalar quantity, which only has a magnitude and does not have a direction. Some examples of different scalar and vector quantities are given in the table below.

What are vector diagrams used for?

Now that we have defined a vector diagram, let us take an example of one to see what these diagrams represent. Figure 1 shows the forces acting on a skydiver. They feel a downward force due to gravity and an upward force due to air resistance. In the diagram, the length of the arrows shows the relative magnitudes of the forces.

Vector diagrams are useful in physics because they can be used to find the magnitude and direction of the overall vector due to multiple vectors added together, which is called the resultant vector.

Fig. 2 - Vectors must be added together from tip to tail to find their resultant vector.

Vector scale diagram

Vector diagrams normally use a scale to help represent the vectors' magnitudes. If you are given several vectors of different magnitudes, then in order to find a scale to represent them, you could take the ratio of the largest to the smallest magnitude and use this value as the unit length for representing the rest of the vectors' magnitudes. This is just an example - there are different ways of arriving at a scale for representing vectors.

How to draw a vector diagram

Drawing a vector diagram is relatively simple. First, you must define a scale for the magnitudes of the vectors, as discussed above. If you were using a ruler and drawing on a piece of paper, then it might be suitable to use the \( \mathrm{cm} \) markings on your ruler for the scale. For example, you could set \( 1\,\mathrm{cm} \) to \( 1\,\mathrm N \) when dealing with forces, or maybe \( 1\,\mathrm{cm} \) to \( 1\,\mathrm m/\mathrm s \) when working with velocities. Of course, the scale depends on the magnitudes of the vectors being used.

Then you need to decide on a reference direction. Once you have these two decisions made, you can adapt the real length of your vectors to the scale and represent them by taking into account their direction with respect to the reference direction. Importantly, you always have to remember to draw the vector as an arrow, with the arrowhead pointing in the direction of the vector.

There are different types of vector diagrams. Sometimes it is more useful to draw the vectors from tip to tail, so that the resultant vector can be calculated easily, such as for displacements and velocities. In other situations, it is better to draw the vectors coming from one point. For example, when drawing vector diagrams for forces, the vectors are drawn from where they act on the object so that their effects on the object can be visualized.

Force vector diagram

The most common types of vector diagrams involve forces. These diagrams are called free-body diagrams.

Fig. 4 - A free-body diagram shows the forces acting on an object.

A general free-body diagram is shown in figure 4. The arrows on a free-body diagram originate from the centre of mass, as this is the point where all forces can be presumed to act.

Free-body diagrams can be used to find the acceleration of an object due to the forces acting on it. Newton's second law states that

$$\vec F=m\vec a$$

where \( \vec F \) is the resultant force acting on an object in \( \mathrm N \), \( m \) is the mass of the object in \( \mathrm{kg} \) and \( \vec a \) is the acceleration of the object in \( \mathrm m/\mathrm{s^2} \). This means that if the resultant force acting on a body is known, then its acceleration can be found by rearranging Newton's second law to

$$\vec a=\frac{\vec F}{m}.$$

If the direction of the vector is less important, such as when considering an object's free-fall, then we can use the vector magnitude form of the equation.

$$a=\frac{F}{m}$$

Vector diagram examples

Vector diagrams can be useful when answering practice questions.

Vectors can be split up into components. In three dimensions, three perpendicular axes can be drawn and the resultant velocity vector can be decomposed into three velocity vectors in each direction. This is shown in figure 6. Work through the following example to gain a better understanding of this concept.

Fig. 6 - Vectors in three dimensions can be split up into components in three perpendicular directions.