Isotropic and anisotropic elements Let $q$ be a quadratic form on a vector space $V$ over a field $F$. A non-zero vector $v \in V$ is said to be _isotropic_ if $q(v) = 0$. Otherwise $v$ is said to be _anisotropic_. If $a,b \in V$ are anisotropic vectors, is it true then, that the sum $a+b$ and the difference $a-b$ are anisotropic, too?

It's false: consider the following as a quadratic form on $\mathbb{R}^2$: $$\Phi((x_1,x_2))=x_1^2-x_2^2$$

then $(1,0)$ and $(0,1)$ are both anisotropic but $\Phi((1,0)+(0,1))=0$