There are several meanings of "null vector" in mathematics. 1. The most common meaning of null vector is the n-dimensional vector 0 of length 0. i.e., the vector with n components, each of which is 0. 2. When applied to a matrix A, a null vector is a nonzero vector x with the property that A x = 0. 3. When applied to a vector space X with an associated quadratic form q, a null vector is a nonzero element x of X for which q(x) = 0. 4. When applied to a geometric product satisfying the contraction rule a^2 = ϵ_a ( left bracketing bar a right bracketing bar )^2 for a an element of an n-vector space, a null vector is a value of a such that a!=0 but left bracketing bar a right bracketing bar = 0.