Vector equation (Challenging)

The lines $\displaystyle l_{1}$ and $\displaystyle l_{2}$ have vector equations

$\displaystyle r=(\lambda cos\theta)i+(\sqrt{2}\lambda sin\theta)j+4k$

$\displaystyle r=(\sqrt{2}+\frac{\mu}{\sqrt{2}}sin\theta)i+j+(3+\ mu cos \theta)k$

respectively, where $\displaystyle \theta$ is a constant such that $\displaystyle 0\leq\theta<\pi$, and $\displaystyle \lambda$ and $\displaystyle \mu$ are real parameters.

The points P is on $\displaystyle l_{1}$ and Q on $\displaystyle l_{2}$ are such that PQ is perpendicular to both lines.

Show that for all values of $\displaystyle \theta$, $\displaystyle PQ\leq\sqrt{2}$

Thanks you.