A line in projective 3-space through the points
(x0:x1:x2:x3) and
(y0:y1:y2:y3)
has Plücker coordinates
pij = xiyj - xjyi.
In this way, lines in P3 correspond to points in P5 satisfying the equation
p01p23-p02p13
+p03p12=0.
The equation for a general line p to meet a fixed line q is linear in these coordinates:
p01q23-p02q13
+p03q12
+p12q03
-p13q02
+p23q01=0.
The equations for a line to be tangent to a quadric is
.