Statements such as "any two lines in a plane meet" are called incidence propositions.
Historically, projective geometry was developed in order to make the propositions of incidence true without exceptions, such as those caused by the existence of parallels.
This is most significant for projective planes due to the universal validity of Desargues' theorem in higher dimensions.
In contrast, the analytic approach is to define projective space based on linear algebra and utilizing homogeneous co-ordinates.
However these computations can be naturally extended to higher-dimensional projective spaces, and the field may be replaced by a division ring (or skewfield) provided that one pays attention to the fact that multiplication is not commutative in that case.
With respect to this basis, the solution space of a single linear equation {(x, y, z) | ax + by + cz = 0} is a two-dimensional subspace of V, and hence a line of P(V).
Point coordinates may be written as column vectors, (x, y, z)T, with colons, (x : y : z), or with a subscript, (x, y, z)P. Correspondingly, line coordinates may be written as row vectors, (a, b, c), with colons, [a : b : c] or with a subscript, (a, b, c)L. Other variations are also possible.