angle and plane of rotation in 3D geometric algebra

Question

Suppose that we perform a rotation of an orthonormal basis $\textbf{e}_1$, $\textbf{e}_2$ and $\textbf{e}_3$ in an Euclidean vector space. We want to rotate a certain angle $\varphi$ about a certain plane defined by the bivector B. As far as I know, individual rotation of single vector can be accomplished by sandwich operation of rotors, e.g. $R=e^{\frac{\theta}{2}\hat{B}}$. Now suppose we know the final position of the rotated basis, i.e., $\textbf{e}^{\prime}_1$, $\textbf{e}^{\prime}_2$ and $\textbf{e}^{\prime}_3$. How can we recover $\varphi$ and B if we know the initial and the rotated basis?

Answer 1

I am simply going to quote the result from Doran & Lasenby (2003) Geometric Algebra for Physicists p. 103 (with minor adaptations of symbols to your problem statement). Note use of the summation convention on indices.

$$R = \frac{1+e'_k e_k}{|1+e'_k e_k|} = \frac{\psi}{\sqrt{\psi\tilde{\psi}}}$$

where $\psi = 1+e'_k e_k$. The case of $\psi=0$ (180-degree rotation) must be handled separately, essentially by inspection.

$\frac{\mathbf{B}}{2}$ must then be recovered as the logarithm of $R$. If we wish, $\mathbf{B}=\varphi\hat{B}$ can be resolved into $\varphi = |\mathbf{B}|$ and $\hat{B}$, with $|\hat{B}|=1$.