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 v₁(x₁;y₁) and v₂(x₂;y₂) is the vector s(x₁+x₂;y₁+y₂).

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.