Axis angle rotation as a differential equation

Question

I am trying to solve the equation

$\frac{d \vec{x}(\theta)}{d\theta} = \vec{n} \times \vec{x}(\theta)$

where $\vec{x}(\theta)$ is rotated vector $\vec{x}$ by $\theta$ about (normalized) axis $\vec{n}$ and $\vec{n} \cdot \vec{x} = 0$.

By sketching how the rotation proceeds I am able to show that $\vec{x}(\theta) = \vec{x}\cos(\theta) + \vec{n} \times \vec{x} \sin(\theta)$ which indeed solves the equation (it is a special case of the Rodrigues rotation formula)

How would I obtain the solution algebraically from the equation?

Thanks a lot!

Answer 1

Use $n\times(\vec n\times \vec x) = (\vec n\cdot \vec x) \vec n - (\vec n\cdot \vec n) \vec x$.