On page 4 of these notes by John Dusel (http://math.ucr.edu/~jmd/Root_Systems.pdf) it reminds us that if we have a symmetric positive bilinear form (pageg 2) we can define an angle between vectors $\alpha$ and $\beta$ by asking that it satisfies $(\alpha ,\beta) = \lVert \alpha \rVert \lVert \beta \rVert \cos{\theta}$. However, it goes on to say that $\cos{^2\theta} = 1$ iff $\alpha = \pm \beta$. I'm not sure how this can be true though, as I think $\cos{^2\theta} = 1$ iff $\alpha = \pm \beta$ implies that $(\alpha ,\alpha) = (\beta ,\beta)$.
I'm guessing I'm missing something here?