isotropic vector
Over the reals a positive-definite form measures honest length, so only the zero vector has zero ‘length’. But indefinite forms — like the spacetime interval — allow nonzero vectors to measure zero. Such a self-orthogonal direction is called isotropic; in relativity these are the light-like directions traced by rays of light, which have zero spacetime length yet are far from being the zero vector.
Formally, a vector v ≠ 0 in a quadratic space (V, Q) is isotropic if Q(v) = 0. A subspace is totally isotropic if Q vanishes on all of it. A quadratic space that has an isotropic vector is called isotropic; one with none (every nonzero vector has Q ≠ 0) is anisotropic. Positive- or negative-definite forms are anisotropic; indefinite real forms always carry isotropic vectors.
Isotropic vectors govern the structure theory of forms. A nondegenerate form has an isotropic vector if and only if it contains a hyperbolic plane, so isotropy is the seed of hyperbolic splitting. Witt decomposition writes any form as an orthogonal sum of hyperbolic planes and an anisotropic core; the maximal dimension of a totally isotropic subspace is the Witt index. Over algebraically closed fields every form of dimension at least 2 is isotropic; over R isotropy is exactly indefiniteness.
For Q(x, y) = x^2 − y^2 on R^2, the vector (1, 1) is isotropic since Q(1, 1) = 1 − 1 = 0, as is (1, −1). These two span a hyperbolic plane, and the whole space is isotropic with Witt index 1.
Isotropic directions of the form x^2 − y^2.