## 1. Plane vectors.

The elements of
(the cartesian square
)
are the ordered pairs of real numbers **v***(x;y)*.

The elements **v** of
can be represented in a
cartesian
plane Oxy by geometric vectors
applied to the origin O. So they can be named
**plane vectors**.
Vectors are usually denoted by latin bold uppercase letters or by latin uppercase letters
surmounted by an arrow.

The elements *x* and *y* of the pair are the coordinates (abscissa and ordinate)
of the geometric vector **v**.

The sum of the vectors **v**_{1}*(x*_{1};y_{1}) and
**v**_{2}*(x*_{2};y_{2}) is the vector
**s***(x*_{1}+x_{2};y_{1}+y_{2}).

The vector **z**(0;0) is such that its sum with any other vector
**v***(x;y)* is the vector **v** itself. The vector **z** is the neutral (or identity)
element of the sum. In a cartesian plane, this vector is represented by the origin.

For every vector **v***(x;y)* there is a vector **w***(-x;-y)* such that
the sum
**v**+**w** gives **z**. **w** is said to be the opposite of **v** and it is
usually written **-v**.

The sum between vectors of
gives this set the structure of
abelian group.