Exponential Map, Cross Product

Question

Let $G=\mathbb{R}^3$, $0\neq x\in G$, and $X$ be the linear transformation of $p\mapsto (x\times p)$ of $G$ (cross product). Choose a right-handed orthogonal basis $(e_{1},e_{2},e_{3})$ for G, with $e_{3}$ a unit vector parallel to $x$. I want to show that $$exp(X)e_{1}=cos||x||e_{1}+sin||x||e_{2},$$ $$exp(X)e_{2}=-sin||x||e_{1}+cos||x||e_{2},$$ and $$exp(X)e_{3}=e_{3}.$$ I really don't know what to do with this one. I have read the theory about the exponential map and Lie groups/algebras, but I really struggle finding it. Any help is really appreciated.

Answer 1

The case $x=0$ is easy. Suppose $x\ne0$. If $x=\|x\|e_3$, then \begin{cases} Xe_1=x\times e_1=\|x\|(e_3\times e_1)=\|x\|e_2,\\ Xe_2=x\times e_2=\|x\|(e_3\times e_2)=-\|x\|e_1,\\ Xe_3=x\times e_3=0. \end{cases} Now you may use the Taylor series for the exponential, since and cosine functions to finish the proof.