In Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry, by Jürgen Richter-Gebert we are asked to consider the three-dimensional vector space over an arbitrary field $\mathbb{K}$, and treat the one-dimensional subspaces as as points and two-dimensional subspaces as lines. He then goes on the speak of the condition in which the scalar product is zero. But he has not specified a scalar product.
Last I checked, the axioms of a vector space do not specify an inner product. In the case of $\mathbb{R}^n,$ I would just assume the author was was being presumptive, and the standard scalar product was intended. But I don't even know what a "standard" scalar product over $\mathbb{H}^3$ (quaternions) would be. Is there a general definition for a standard scalar product on a vector space over a field?
You can’t define an inner product over an arbitrary field, as has been discussed here: https://mathoverflow.net/q/129413. For example the condition $\left<v,v \right> > 0$ requires an ordering on your field. But if that‘s out of the way the statement in the text follows from the scalar product axioms without specifying the scalar product explicitly:
$\left<v,v \right> = 0 \Rightarrow \left<\lambda_1 v,\lambda_2 v \right>=\lambda_1\bar{\lambda_2} \left<v,v \right> =0$
Where I just used linearity in the first argument and conjugate symmetry. (The latter means that you need an automorphism on your field that plays the role of complex conjugation)