## Vector equation (Challenging)

The lines $l_{1}$ and $l_{2}$ have vector equations
$r=(\lambda cos\theta)i+(\sqrt{2}\lambda sin\theta)j+4k$
$r=(\sqrt{2}+\frac{\mu}{\sqrt{2}}sin\theta)i+j+(3+\ mu cos \theta)k$
respectively, where $\theta$ is a constant such that $0\leq\theta<\pi$, and $\lambda$ and $\mu$ are real parameters.
The points P is on $l_{1}$ and Q on $l_{2}$ are such that PQ is perpendicular to both lines.

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

Thanks you.