I am look for a way to determine the definiteness of a quadratic form on the orthogonal complement of a line. More specifically, I have the quadratic form $Q^-(x) = -x_0^2 + x_1^2 + \cdots + x_n^2$, which I know is indefinite. The text I am reading makes the claim that if you have a vector $x$ such that $Q^-(x) >0 $, then it must be true that $Q^-$ is indefinite on the orthogonal complement of $x$. I am not sure how one would prove this. Any suggestions on a general strategy to prove something like this would be appreciated as I also would like to then determine what would be true about the orthogonal complement when $Q^-(x) = 0$ or $Q^-(x) < 0$.
Note that the inner product being used here is $\langle x,y\rangle = -x_0y_0 + x_1y_1 + \cdots + x_ny_n$.