The simplification of the plane equations to one tensor equation proceeds from the similarity of the three-vector equations for

,

,

, and

. These equations can be rewritten together using the antisymmetric permutation tensor

; however, the three-vectors

,

,

for points in

and the origin

need to be rewritten as four-vectors so that they have a compatible dimension. The first three components of the vectors

,

,

, and

are equal to the coordinates of a point in

and the fourth component of these vectors is

. Similarly, the first three components of the vector

are equal to the free variables

,

, and

, while the fourth component of this vector is

. The matrix

allows the first, second, and third components of a four-vector to be selected for summation.