I have three vectors, $\vec{a},\vec{b},$ and $\vec{c}$ in $n$-dimensional space. I know the coordinates of all three vectors and their dot products. Both $\vec{a}$ and $\vec{b}$ are rotated away from $\vec{c}$ by an angle $\alpha$, in their own respective directions, obtaining $\vec{a}'$ and $\vec{b}'$. What is the angle between $\vec{a}'$ and $\vec{b}'$?
I have worked on finding the generalized rotation matrices in $n$-dimensions, calculating $\vec{a}'$ and $\vec{b}'$ using Clifford Algebra using the answers from the posts below. However, these methods require many matrix operations and are too slow. I am wondering whether there is a neater solution without requiring the calculation of the two rotated $\vec{a}'$ and $\vec{b}'$ vectors first.
Generalized rotation matrix in N dimensional space around N-2 unit vector Finding the rotation matrix in n-dimensions How can I calculate a $4\times 4$ rotation matrix to match a 4d direction vector?