# Affine coordinate system

A rectilinear coordinate system in an affine space. An affine coordinate system on a plane is defined by an ordered pair of non-collinear vectors $\mathbf e _ {1}$ and $\mathbf e _ {2}$( an affine basis) and a point $O$( the coordinate origin). The straight lines passing through the point $O$ and parallel to the basis vectors are known as the coordinate axes. The vectors $\mathbf e _ {1}$ and $\mathbf e _ {2}$ define the positive direction on the coordinate axes. The axis parallel to the vector $\mathbf e _ {1}$ is called the abscissa (axis), while that parallel to the vector $\mathbf e _ {2}$ is called the ordinate (axis). The affine coordinates of a point $M$ are given by an ordered pair of numbers $(x, y)$ which are the coefficients of the decomposition of the vector $\overline{OM}\;$ by the basis vectors:
$$\overline{OM}\; = x \mathbf e _ {1} + y \mathbf e _ {2} .$$
The first number $x$ is called the abscissa, while the second number $y$ is called the ordinate of $M$.
An affine coordinate system in three-dimensional space is defined as an ordered triplet of linearly-independent vectors $\mathbf e _ {1} , \mathbf e _ {2} , \mathbf e _ {3}$ and a point $O$. As in the case of the plane, one defines the coordinate axes — abscissa, ordinate and applicate — and the coordinates of a point. Planes passing through pairs of coordinate axes are known as coordinate planes.