Taking the example of projective space of dimension three, there will be homogeneous co-ordinates (*x*:*y*:*z*:*w*). The **plane at infinity** is usually identified with the set of points with *w* = 0. Away from this plane we can use (*x/w*, *y/w*, *z/w*) as an ordinary Cartesian system; therefore the affine space complementary to the plane at infinity is co-ordinatised in a familiar way, with a basis corresponding to (1:0:0:1), (0:1:0:1), (0:0:1:1).

If we try to intersect the two planes defined by equations *x = w* and *x = 2w* then we clearly will derive first *w = 0* and then *x = 0*. That tells us that the intersection is contained in the plane at infinity, and consists of all points with co-ordinates (0:*y*:*z*;0). It is a line, and in fact the line joining (0:1:0:0) and (0:0:1:0). It cannot be given by a single equation in the co-ordinates. In fact a line in three-dimensional projective space corresponds to a two-dimensional subspace of the underlying four-dimensional vector space, therefore given by two linear conditions.