A particle is traveling in the direction of the positive $z$ axis, until eventually, it is deflected. The new direction is given by an azimuthal angle (w.r.t. the positive $z$ axis) $\theta$ and a polar angle (within the $x$-$y$ plane) $\phi$. The new direction is given by $\vec u = (\cos \phi \sin \theta, \sin \phi \sin \theta, \cos \theta)$, having length 1.
I am looking for an orthogonal direction $\vec v$ to $\vec u$, having the same length 1.
Is there a direct expression in $\theta$ and $\phi$ that yields a normalized $\vec v$, without having to compute some intermediate vector and diving all entries of said vector by its norm?