Plücker Coordinates for Lines 4

    A line l in 3-space spanned by vectors

    
has Plücker coordinates pi,j:= xi yj -  xj yi.

    These identify the line l with a point pl in 5-space

satisfying the quadratic Plücker equation
  =   0.
Conversely, every point in 5-space satisfying this Plücker equation is the Plücker coordinate of a line in 3-space.