Perpendicular vectors in $\Bbb R^3$

Question

Hi I am struggling with this simple question. Let $\vec{v}$ be a unit vector in $\Bbb R^3$.

How can I construct two periodic functions $\vec{x}(\theta)$ $\vec{y}(\theta)$

such that $\vec{v}$, $\vec{x}(\theta)$ and $\vec{y}(\theta)$ are all perpendicular to each other, for all $\theta\in R$ ?

pic: https://upload.wikimedia.org/wikipedia/commons/thumb/5/51/Euler_AxisAngle.png/800px-Euler_AxisAngle.png

Answer 1

Take $v=(v_1,v_2,v_3)^T$, take $x_0=(-v_2,v_1,0)$ and $y_0=v\times x_0$ and rotate \begin{align} x(\theta)&=R(\theta)x_0,\\ y(\theta)&=R(\theta)y_0 \end{align} with Rodrigues' rotation matrix.