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A non-zero vector that is orthogonal to itself. Let  $  E $
be a [[Vector space|vector space]] over the field of real or complex numbers and let  $  \Phi $
be a non-degenerate [[Bilinear form|bilinear form]] of [[Signature|signature]]  $  ( p , q ) $, 
$  p \neq 0 $, 
$  q \neq 0 $, 
on  $  E \times E $. 
Then an isotropic vector is a non-zero vector  $  x \in E $
for which  $  \Phi ( x , x ) = 0 $. 
One sometimes says that an isotropic vector has zero length (or norm). The set of all isotropic vectors is called the isotropic cone. A subspace  $  V \subset  E $
is called isotropic if there exists a non-zero vector  $  z \in V $
orthogonal to  $  V $(
that is, the restriction of  $  \Phi $
to  $  V \times V $
is degenerate:  $  V \cap V  ^  \perp  \neq \{ 0 \} $). 
A vector subspace  $  V $
is said to be totally isotropic if all its vectors are isotropic vectors.

In the relativistic interpretation of the Universe, [[Space-time|space-time]] is locally regarded as a four-dimensional vector space with a form of signature  $  ( 3 , 1 ) $, 
the trajectories of photons are isotropic lines, while the isotropic cone is called the light cone.

{{SourceattributionEOM|Isotropic vector}}
[[Category:Mathematics]]