Given two angles between three vectors in $\mathbb{R}^3$, detemine bounds on the third angle

Question

Consider three vectors $v_1, v_2, v_3$ in $\mathbb R^3$. Let $\theta_{12}$ be the angle formed by $v_1$ and $v_2$ (with equivalent definitions for $\theta_{13}$ and $\theta_{23}$ ). Now, suppose I give you the values of $\theta_{12}$ and $\theta_{13}$, but not the coordinates of any of the vectors. With this information, can you provide some information about $\theta_{23}$? It seems intuitive to me that $\theta_{13}$ cannot take any arbitrary value, but I do not see any rigorous way of finding upper and/or lower bounds for it. Moreover, it seems that the dimension of the problem is an important parameter, as in $\mathbb R^4$ this angle could take any value and in $\mathbb R^2$ it is trivial to see that $\theta_{23} = \theta_{13}-\theta_{12}$. Any help would be appreciated!

Answer 1

Since, we're talking about angles, then we can take all three vectors to be unit vectors. Since the actual vectors are not given, we can take $v_1$ to be

$v_1 = (0, 0, 1) $

and then using spherical coordinates, we can fix $v_2$ to lie in the $xz$ plane, as follows

$ v_2 = (\sin \theta_{12} , 0, \cos \theta_{12} ) $

And finally, we have the angle between $v_1$ and $v_3$ as $\theta_{13}$, therefore,

$ v_3 = (\sin \theta_{13} \cos \phi, \sin \theta_{13} \sin \phi, \cos \theta_{13} ) $

And immediately we can see that $\cos \theta_{23}$ is given by

$ \cos \theta_{23} = v_2 \cdot v_3 = \sin \theta_{12} \sin \theta_{13} \cos \phi + \cos \theta_{12} \cos \theta_{13} $

Hence, the minimum $\cos$ corresponding to the largest angle $\theta_{23}$ is

$ \cos_{Min} \theta_{23} = \cos \theta_{12} \cos \theta_{13} - \sin \theta_{12} \sin \theta_{13} = \cos( \theta_{12} + \theta_{13} ) $

Thus the maximum angle $\theta_{23}$ is

$ \text{Largest } \theta_{23} = \theta_{12} + \theta_{13} $

And similarly, the maximum $\cos$ corresponding to the smallest angle $\theta_{23}$ is

$ \cos_{Max} \theta_{23} = \cos \theta_{12} \cos \theta_{13} + \sin \theta_{12} \sin \theta_{13} = \cos( \theta_{12} - \theta_{13} ) $

Thus the minimum angle $\theta_{23}$ is

$ \text{Smallest } \theta_{23} = | \theta_{12} - \theta_{13} | $