Calculate angle between two vectors, given their rotation w.r.t. a third vector.

I have three vectors, $\overrightarrow{a},\overrightarrow{b},$ and $\overrightarrow{c}$ in n-dimensional space. I know the coordinates of all three vectors and their dot products. Both $\overrightarrow{a}$ and $\overrightarrow{b}$ are rotated away from $\overrightarrow{c}$ by an angle $\alpha $, in their own respective directions, obtaining ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ . What is the angle between ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ ?

I have three vectors, $\overrightarrow{a},\overrightarrow{b},$ and $\overrightarrow{c}$ in n-dimensional space. I know the coordinates of all three vectors and their dot products. Both $\overrightarrow{a}$ and $\overrightarrow{b}$ are rotated away from $\overrightarrow{c}$ by an angle $\alpha $, in their own respective directions, obtaining ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ . What is the angle between ${\overrightarrow{a}}^{\prime}$ and ${\overrightarrow{b}}^{\prime}$ ?