Angle between sum of vectors

Question

Let $u,v$ and $w$ be vectors in $\mathbb{R}^n$ and let $\theta(u,w), \theta(v,w)$ and $\theta(u+v,w)$ represent the angle between each listed pair of vectors. Does it hold that one of the following two statements must be true: $$\theta(u,w) \geq \theta(u+v,w)$$ Or: $$\theta(v,w) \geq \theta(u+v,w)$$ I feel this must be true and that it follows from some simple property that I'm forgetting. If it is true, does it generalize to any finite list of vectors?

Answer 1

I don't think this is true. Take the vectors $u=\langle-1, 1\rangle$, $v=\langle 1, 1\rangle$ and $w=\langle 0, -1\rangle$ in $\mathbb{R}^2$. Then $u+v=\langle 0, 2\rangle$, and $\theta(u+v, w)=\pi$, while both $\theta(u,w)$ and $\theta(v,w)=\frac{3\pi}{4}$.