Suppose I have three hermitian operators $\sigma _x,\sigma _y,\sigma _z$ that I don't explicitly say they are Pauli matrices but still use the similar notation $\sigma_j$. They only need to satisfy the following commutation relations $$ \left[ \sigma _x,\sigma _y \right] =2i\sigma _z,\left[ \sigma _y,\sigma _z \right] =2i\sigma _x,\left[ \sigma _z,\sigma _x \right] =2i\sigma _y \tag 1. $$
My question is, how can I( or can I?) directly get the rotation meaning of $e^{-i\frac{\theta}{2}\sigma _{\vec{n}}}$? Explicitly, we can have the following $$ e^{-i\frac{\theta}{2}\sigma _{\vec{n}}}\vec{a}\cdot \vec{\sigma}e^{i\frac{\theta}{2}\sigma _{\vec{n}}}=\vec{b}\cdot \vec{\sigma} \tag 2 $$ where $\vec a$ and $\vec b$ are two 3 dimentional vectors connected by a $SO(3)$ element $O$ such that $\vec b = O \vec a$. How can I show that the $O$ here is a rotation around $\vec n$ axis for $\theta$ angle? I may tediously show this with the following formula, but is there a better way to do so? The formula: $$ e^{\xi A} B e^{-\xi A}=B+\xi[A, B]+\frac{\xi^2}{2 !}[A,[A, B]]+\cdots \tag 3$$
