not the complete set of points in P5; the Plücker coordinates of a line L satisfy the quadratic Plücker relation For proof, write this homogeneous polynomial as determinants and use Laplace expansion (in reverse). Since both 3×3 determinants have duplicate columns, the right hand side is identically zero. Another proof may be done like this: Since vector is perpendicular to vector (see above), the scalar product of d and m must be zero! q.e.d. Point equations Letting (x0:x1:x2:x3) be the point coordinates, four possible points on a line each have coordinates xi = pij, for j = 0...3. Some of these possible points may be inadmissible because all coordinates are zero, but since at least one Plücker coordinate is nonzero, at least two distinct points are guaranteed. Bijectivity If (q01:q02:q03:q23:q31:q12) are the homogeneous coordinates of a point in P5, without loss of generality assume that q01 is nonzero. Then the matrix has rank 2, and so its columns are distinct points defining a line L. When the P5 coordinates, qij, satisfy the quadratic Plücker relation, they are the Plücker coordinates of L. To see this, first normalize q01 to 1. Then we immediately have that for the Plücker coordinates computed from M, pij = qij, except for But if the qij satisfy the Plücker relation q23+q02q31+q03q12 = 0, then p23 = q23, completing the set of identities. Consequently, α is a surjection onto the algebraic variety consisting of the set of zeros of the quadratic polynomial And since α is also an injection, the lines in P3 are thus in bijective correspondence with the points of this quadric in P5, called the Plücker quadric or Klein quadric. Uses Plücker coordinates allow concise solutions to problems of line geometry in 3-dimensional space, especially those involving incidence. Line-line crossing Two lines in P3 are either skew or coplanar, and in the latter case they are either coincident or intersect in a unique point. If pij and p′ij are the Plücker coordinates of two lines, then they are coplanar precisely when d•m′+m•d′ = 0, as shown by When the lines are skew, the sign of the result indicates the sense of crossing: positive if a right-handed screw takes L into L′, else negative. The quadratic Plücker relation essentially states that a line is coplanar with itself. Line-line join In the event that two lines are coplanar but not parallel, their common plane has