This is a technical question coming out of my research.

Let $\angle(\cdot, \cdot)$ be the angle ($\in [0, \pi]$) between vectors. Consider two vectors $u, v$ in $\mathbb R^3$. Is it true that $$ \angle(u, v) \le \sum_{\Omega} \angle(u_{\Omega}, v_{\Omega})? $$ Here the $\Omega$ indexes length-two subvectors. My research problem only requires the version when $u, v \in \mathbb R_{++}^3$ (i.e., the positive orthant), but I suspect the general version might hold.

Also, does this generalize to $\mathbb R^n$, i.e., $\mathbb R^3$ is changed to $\mathbb R^n$, but we still only use length-two vectors on the right?