TITLE: Perpendicular vectors in $\Bbb R^3$
QUESTION [0 upvotes]: 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

REPLY [0 votes]: 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.
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