# Sum of angles in $\mathbb{R}^n$

Given three vectors $v_1,v_2$ and $v_3$ in $\mathbb{R}^n$ with the standard scalar product the follwing is true $$\angle(v_1,v_2)+\angle(v_2,v_3)\geq \angle(v_1,v_3).$$

It tried to substitute $\angle(v_1,v_2) = cos^{-1}\frac{v_1 \cdot v_2}{\Vert v_1 \Vert \Vert v_2 \Vert}$ but I could not show the resulting inequality. What is the name of the inequality and do you know reference that one can cite in an article?

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You can reduce the problem to $\mathbb{R}^3$, and there wlog v2=(0,0,1) then one gets an easy to prove inequality if one writes everything in polar coordinates.