Suppose we have three vectors in 3-D: $\vec{A},\vec{B},\vec{C}$. The angle between $\vec{A}$ and $\vec{B}$ is $\theta_{AB}$.
This is straightforward to find if we know the vectors explicitly: $\vec{A} \cdot \vec{B}=\lvert A\rvert \lvert B\rvert \cos \theta_{AB}$.
How would you define the angles with the third vector, C, i.e. $\theta_{BC}$ and $\theta_{AC}$ as functions of $\theta_{AB}$?
For some context, I'm looking at three abstract vectors, and am trying to study their behaviour with respect to their angles between one another. I know the angles are not independent and I know:
1) In 2-D the functions are very straightforward, just a substraction
2) I know what would happen if these vectors corresponded to points on a sphere
But since I'm looking at these completely analytically, without any restriction in 3-D, I don't know how to relate all three angles.
Any thoughts?
There is no function for the other two angles in terms of one angle. Think of it like this: in this case, we have $\theta_{AB} \quad\text{and} \quad \theta_{AC}$. Given just the angles, we have two cones around $\overrightarrow A$, one for all the possible $\overrightarrow B$ and one for all the possible $\overrightarrow C$. Any line on a cone is valid for the known angle. The only ways out of this situation are at least as complicated as using the dot product formula. They require you to know more angles or use the vectors themselves.
In short, there is no explicit formula.