In mathematics, given a vector space *X* with an associated quadratic form *q*, written (*X*, *q*), a **null vector** or **isotropic vector** is a non-zero element *x* of *X* for which *q*(*x*) = 0.

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 (*X*, *q*) 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*, *X* = *A* + *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 *l* = 1 + *hi*, *n* = 1 + *hj*, *m* = 1 + *hk*, and *m*^{∗} = 1 – *hk* are null vectors and { *l*, *n*, *m*, *m*^{∗} } 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.^{[1]}

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.

## References

**^**Patrick Dolan (1968) A Singularity-free solution of the Maxwell-Einstein Equations, Communications in Mathematical Physics 9(2):161–8, especially 166, link from Project Euclid

- Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. (1984).
*Modern Geometry: Methods and Applications*. Translated by Burns, Robert G. Springer. p. 50. ISBN 0-387-90872-2. - Shaw, Ronald (1982).
*Linear Algebra and Group Representations*.**1**. Academic Press. p. 151. ISBN 0-12-639201-3. - Neville, E. H. (Eric Harold) (1922).
*Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions*. Cambridge University Press. p. 204.