Some quantities cannot be represented with only a value. For example, velocity, can be represented with a value that states the speed you are going at but you also need to show in which direction you are going.

Below is a table showing some examples of scalar quantities and vector quantities:

Your speed is a scalar quantity, which means it only is composed of a number, with a unit, while your velocity is a vector quantity, which means it has a magnitude (the size of it, a number) with a unit but it also has a direction. There are a few quantities that can have a “sibling” such speed and velocity. For example, your mass is a scalar quantity as it is just a number, but your weight is a vector quantity that points downwards as it is the result of a force that it is exerted on you by the centre of Earth.

A very important difference to be aware of is the one between distance and displacement. Distance is a scalar quantity that refers to how much ground an object has covered during its motion. Displacement is a vector quantity that refers to how far out of place an object is, or more easily, it is the object’s overall change in position.

A vector is, in fact a arrow which has a size, called magnitude and a direction. You are required to be able to add vectors.

The rule to add vectors is that they must be added tip to tail.

Let’s suppose we have the two vectors, A and B like shown in figure a below, in order to add them we translate the tail (the bottom, the one without the arrow) of B just after the tip of A.

figure a

By adding them we create the diagram shown in figure b.

figure b

In fact, in this way, the arrows are now tip to tail, and then the resultant of the vector, is the one that connects the two other extremes of the vectors.

This method will also work on sum of more than two vectors. The other way of resolving them is using trigonometry. All vectors can be broken down into two components, the y-component and the x-component. The following image shows a vector being split into its horizontal, $A_{x}$ and vertical, $A_{y}$ components:

figure c

The useful thing about the vector components is that they form a right angled triangle with the vector, so various trigonometric identities may be used. From the triangle in figure c, we can state the following relationships:

$A_{x} = A \times \cos \theta$

$A_{y} = A \times \sin \theta$

A_{y} = A_{x} \times \tan \theta \$

This way of solving vectors is very useful when dealing with inclined planes and fields (e.g. magnetic and electric) which exert forces on objects.

Page written by Luca Quinci – Thank you!