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

$\frac{4}{3}$

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

$\begin{vmatrix}(4-3)×\frac{2}{3}+(6-4) × (-\frac{2}{3})+ (3-5) ×\frac{1}{3}\end{vmatrix}= \begin{vmatrix}\frac{2}{3}-\frac{4}{3}-\frac{2}{3}\end{vmatrix}=\frac{4}{3}$