Editing Null vector (section)

{{Short description|Vector on which a quadratic form is zero}}
{{about|zeros of a quadratic form|the zero element in a vector space|Zero vector|null vectors in Minkowski space|Minkowski space#Causal structure}}
[[File:Conformalsphere.pdf|thumb|A null cone where <math>q(x,y,z) = x^2 + y^2 - z^2 .</math>]]

In [[mathematics]], given a [[vector space]] ''X'' with an associated [[quadratic form]] ''q'', written {{nowrap|(''X'', ''q'')}}, a '''null vector''' or '''isotropic vector''' is a non-zero element ''x'' of ''X'' for which {{nowrap|1=''q''(''x'') = 0}}.

In the theory of [[real number|real]] [[bilinear form]]s, [[definite quadratic form]]s and [[isotropic quadratic form]]s are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector.

A [[quadratic space]] {{nowrap|(''X'', ''q'')}} which has a null vector is called a [[pseudo-Euclidean space]]. The term ''isotropic vector v'' when ''q''(''v'') = 0 has been used in quadratic spaces,<ref>[[Emil Artin]] (1957) [[Geometric Algebra (book)|''Geometric Algebra'']], [https://archive.org/details/geometricalgebra033556mbp/page/n129/mode/2up?view=theater&q=isotropic isotropic]</ref> and '''anisotropic space''' for a quadratic space without null vectors.

A pseudo-Euclidean vector space may be decomposed (non-uniquely) into [[orthogonal subspaces]] ''A'' and ''B'', {{nowrap|1=''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:
<math display="block">\bigcup_{r \geq 0} \{x = a + b : q(a) = -q(b) = r, \ \ a, b \in B \}.</math>
The null cone is also the union of the [[isotropic line]]s through the origin.