The projection of the line joining the points $( 3, 4, 5)$ and $( 4, 6, 3)$ on the line joining the points $(- 1, 2, 4)$ and $(1, 0, 5)$, is

The direction ratios of the line joining $P (-1, 2, 4)$ and $Q (1, 0, 5)$ are proportional to $2, - 2, 1$

Its direction cosines are  $\frac{2}{3},-\frac{2}{3},\frac{1}{3}$

Thus, the projection of the line joining $A (3, 4,5)$ and $B(4, 6, 3)$ on $PQ$ is given by 

$\left|(4-3)\times \frac{2}{3}+(6-4)\times \left(-\frac{2}{3}\right)+(3-5)\times \frac{1}{3}\right|=\left|\frac{2}{3}-\frac{4}{3}-\frac{2}{3}\right|=\frac{4}{3}$