In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a nonzero 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 pseudoEuclidean space.
A pseudoEuclidean vector space may be decomposed (nonuniquely) into orthogonal subspaces A and B, X = A + B, where q is positivedefinite on A and negativedefinite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres:
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Unit and Null vector

Zero vector  Null vector

NULL VECTOR
Transcription
Examples
The lightlike 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 Singularityfree solution of the MaxwellEinstein 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 0387908722.
 Shaw, Ronald (1982). Linear Algebra and Group Representations. 1. Academic Press. p. 151. ISBN 0126392013.
 Neville, E. H. (Eric Harold) (1922). Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions. Cambridge University Press. p. 204.