Exponential of a Bivector & Rotor Sandwich

1. General bivector exponentiation

• Let $B\in {\mathrm{\Lambda }}^2({\mathbb{R}}^3)$ with magnitude $|B|$.
• Power‐series definition:$$e^{B\theta }=\sum _{n=0}^{\mathrm{\infty }}\frac{(B\theta {)}^n}{n!}.$$
• Since $B^2=-|B{|}^2$, group even/odd terms to get$$e^{B\theta }=\cos (|B|\theta )+\frac{B}{|B|}\sin (|B|\theta ).$$
• Unit bivector case ($|B|=1$) recovers
  $e^{B\theta }=\cos \theta +B\sin \theta$.

2. Rotor & sandwich formula

• Define the rotor
  ${\displaystyle R=e^{B\theta }}$,
  whose reverse is $\tilde{R}=e^{-B\theta }$.
• Rotation of any vector $a\in {\mathbb{R}}^3$ by angle $2|B|\theta$ in the plane of $B$:$$a\mapsto Ra\tilde{R}=e^{-B\theta }a{e}^{B\theta }.$$
• This “sandwiching” composes two reflections across planes orthogonal to the bivector axis.

Key points:

• Non‐unit bivectors scale the rotation‐angle by $|B|$.
• Rotors live in the even subalgebra $\mathbb{R}\oplus {\mathrm{\Lambda }}^2$.
• Sandwiching preserves vector length and implements proper rotations.