Null vector

In mathematics, given a vector space "X" with an associated quadratic form "q", written , a null vector or isotropic vector is a non-zero element "x" of "X" for which .
In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.
A quadratic space which has a null vector is called a pseudo-Euclidean space.
A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces "A" and "B", , where "q" is positive-definite on "A" and negative-definite on "B". The null cone, or isotropic cone, of "X" consists of the union of balanced spheres:
The null cone is also the union of the isotropic lines through the origin.
Examples.
The light-like vectors of Minkowski space are null vectors.
The four linearly independent biquaternions , , , and are null vectors and can serve as a basis for the subspace used to represent spacetime. Null vectors are also used in the Newman–Penrose formalism approach to spacetime manifolds.
A composition algebra "splits" when it has a null vector; otherwise it is a division algebra.
In the Verma module of a Lie algebra there are null vectors.