There are no $3$ linearly independent lightlike vectors such that $u+v+w = 0$.

Question

Consider the Lorentz-Minkowski space $E^n_1$, also known as $\mathbb{L}^n$. I want to prove that there are not lightlike linearly independent vectors $u, v, w \in E^n_1$ such that $u + v + w = 0$. How to do it? I'm still unfamiliar with the intuition behind such space.

Answer 1

Hint Suppose there were. Expand $$0 = [{\bf u} + {\bf v} + {\bf w}] \cdot [{\bf u} - ({\bf v} + {\bf w})]$$ to conclude that ${\bf v} \cdot {\bf w} = 0$.

Additional hint What is the matrix representation of the bilinear form $\cdot$ with respect to the basis $({\bf u}, {\bf v}, {\bf w})$?