In the plane euclidean geometry,

- given two distinct points A and B,
- given the straight line (which is unique) through A and B,
- given the direction from A to B so that we can say that A precedes B,

the segment AB is the set of the points P such that A is equal to or precedes P and P is equal to or precedes B: A≤P≤B.

Usually in the plane euclidean geometry the direction of a segment is not
considered: the only relevant property of a segment is its length, that is its **magnitude**,
which doesn't depend upon its orientation.

But many properties of the natural phenomena are characterized not only by their
**magnitude**, but also by their **line of action** and by the **direction**
on this line.

For example, to describe the displacement of an airplane on a geographical chart, it isn't sufficient to say that the plane has covered 1000 miles: we need to specificate the line of the fly, for instance W-E, and the direction of the fly, for instance from E to W.

Displacements and other properties with the same attributes are said **vectorial
quantities**. To efficiently represent such quantities we can use oriented segments
drawn as arrows, in which, given a suitable scale, the length is proportional to the
magnitude, the line is the line of action and the head gives the direction.
These arrows are said **geometric vectors**. In this note, *vector* means always
*geometric vector*.

Usually vectors are denoted by boldface latin letters or by latin letters with an arrow written on top, so the following notations are equivalent

**a**, **b**, **c,...**

For example, to represent on a geographical map, such that 1cm corresponds to 1km, the trip of a car which has moved 4km toward W, we'll draw a 4cm long horizontal arrow with the head on the leftmost end. The other end is the 'tail' of the vector.

It is important to emphasize that **the displacement doesn't depend from the
starting or ending point**. If a car starts from Paris and another car starts from London
and both move toward W covering 100km, at the end of the trip they we'll be in
different places, but they we'll have the same displacement.

In general, we can say that **parallel, equally oriented and equally long vectors
are equivalent**. Therefore **a vector can be freely translated in the space.**