representatives for two distinct points in the plane (i.e. M and
N are linearly independent 3-vectors:
L=(a,b,c) be a the dual coordinates of a line (hyperplane) in the
plane. By the definition of a hyperplane, M lies on L if and only
if the dot product
vanishes, i.e. if and only if the
3-vector L is orthogonal to the 3-vector M. The line MN through
M and N must be represented by a 3-vector L orthogonal to both Mand N, and hence proportional to the cross product :
Another way to characterize the line MN is as the set of points
Dually, if L and L' are two lines defined by their dual coordinates, then is some other line through the intersection X of L and L' (since implies ). As varies, this line traces out the entire pencil of lines through X. By duality, .
One further way to approach lines is to recall that if M, N and Pare collinear, each point is a linear combination of the two others. In particular, the determinant |MNP| vanishes. If M and Nare fixed, this provides us with a linear constraint that P must satisfy if it is to lie on the line: .