Properly appreciated, a line, plane, and 3-dimensional space are each geometric objects with no assigned coordinates, origin, or axes. A line, though, may be identified with the set of real numbers by choosing a point O as the origin (which has coordinate zero), a direction for the line, and a choice of unit length. Using the axioms of synthetic geometry (or Euclidean geometry) every point can then be uniquely assigned a coordinate. There is a unique point P such that the line segment OP has length 1 and $\overrightarrow{OP}$ is in the direction of the line; another point Q is assigned coordinate x when $\begin{array}{l}\overrightarrow{OQ}=x\overrightarrow{OP}\end{array}$.

Likewise, in a plane, an origin O is chosen (which has coordinates x = y = 0), and a first line in the plane is chosen as the x-axis, and a point on the line is assigned an x-coordinate as previously, and a zero y-coordinate. A second line, perpendicular to the first, can be constructed and each point of that line assigned a y-coordinate as previously (using the same unit length) and an x-coordinate of zero. For any point Q in the plane, let M and N be points on the x-axis and y-axis such that QM is parallel to the y-axis and QN is parallel to the x-axis. If M has coordinates (x,0) and N has coordinates (0,y) then Q is assigned coordinates (x,y).

This process can be extended to any finite-dimensional Euclidean space. Here the coordinates have been defined in such a way that the canonical basis is an orthonormal basis, but a basis can more generally be used to assign coordinates; the axes would not generally be perpendicular and standard formulae (for example, for the scalar product) would not be valid.