Question

does null vector exist

Answer 1

Yes null vector exist

Answer 2

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 Xfor 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 there exists a nonzero null vector. Where such a vector exists, (X, q) 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:

⋃r≥0{x=a+b:q(a)=−q(b)=r,a∈A,b∈B}.

The null cone is also the union of the isotropic lines through the origin.

So, in short, it does exist.