Plucker Coodinates provide a convenient representation for directed lines in affine 3-Dimensional space, R3[1,2]. This section reviews relevant notation for, and some useful properties of, Plucker coordinates.
Each ordered pair of distinct points p = (px, py, pz) and q = (qx, qy, qz) defines a directed line in 3-Dimensions. This line corresponds to the following set of six coefficients called the Plucker coordinates of the line:
L = (L, L, L, L, L, L)
Each Plucker coordinate is the determinant of a 2x2 minor of the matrix:
px py pz 1
qx qy qz 1
We adopt the following convention relating these minors and the Plucker coordinates L[i]:
L = px qy - qx
L = px qz - qx pz
L = px - qx
L = py qz - qy pz
L = pz - qz
L = qy - py
If a and b are two directed lines, and a[i] and b[i] their corresponding Plucker mappings, a relation side(a,b) can be defined as the permuted inner product
side(a,b) = a*b + a*b + a*b + a*b + a*b + a*b
which is zero whenever a and b intersect or
are parallel, and non-zero otherwise.
However, although every directed line in R3 maps to a point in the Plucker coordinates, not every six-tuple X of P5 corresponds to a real line. Only the points satisfying the quadratic relation:
side(X,X) = 0 and X, X and X not all equal to zero
correspond to real lines in R3. If
then it corresponds to a line at infinity. The remaining points do not correspond to lines in P3.
The plucker coordinates are
determined only to within a scale factor. That is if points p, q and r
lie on a line L, plucker(p,q) and plucker(p,r) are identical up to a scale,
that is there is some constant c (not zero) such that plucker(p,q) = c*plucker(p.r).
Note that this property is made use of in converting
from the plucker coordinates to the 3-Dimensions.
 Jorge Stolfi. Primitives for computational geometry. Technical Report 36, DEC SRC, 1989
 Seth Teller & Michael Hohmeyer, Determining the Lines Through Four Lines, Journal of Graphic Tools.