Orthonormal Basis - Angle of Rotation with respect to Standard Orthonormal Basis I have an orthonormal basis ${\bf{b}}_1$ and ${\bf{b}}_2$ in $\mathbb{R}^2$. I want to find out the angle of rotation. I added a little picture here. I essentially want to find $\theta$

I know that I can compute the angle between two vectors but then there are $4$ combinations here

*

*$\theta_{11} = \arccos({\bf{b}}_1^\top{\bf{e}}_1)$

*$\theta_{12} = \arccos({\bf{b}}_1^\top{\bf{e}}_2)$

*$\theta_{21} = \arccos({\bf{b}}_2^\top{\bf{e}}_1)$

*$\theta_{22} = \arccos({\bf{b}}_2^\top{\bf{e}}_2)$
How would one know which angle is correct? Importantly, here I used the labels $b_1$, $b_2$ in the same order as $e_1$ and $e_2$ but that's not necessarily the same order geometically!

The simplest way is to construct orthogonal matrix with column vectors $B=[b_1 \ \ b_2 \ \ b_1 \times b_2]$.
(I assume here that $b_1$ and $b_2$ are normalized to unit length)
and to use trace of such matrix $\text{tr}(B)$.
Then we have immediately  $\text{tr}(B)=1+2\cos(\theta)$.
See also https://en.wikipedia.org/wiki/Rotation_matrix#Determining_the_angle
