The line $$x + y - z = 0 = x - 2y + 3z - 5$$ is parallel to the vector

$$\overrightarrow b = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 1 & 1 & { - 1} \cr 1 & { - 2} & 3 \cr } } \right| = (1,4, - 3)$$

Equation of the line through $$P(1,2,4)$$ and parallel to $$\overrightarrow b $$

$${{x - 1} \over 1} = {{y - 2} \over { - 4}} = {{z - 4} \over { - 3}}$$

Let $$N \equiv (\lambda + 1, - 4\lambda + 2, - 3\lambda + 4)$$

$$\overrightarrow {QN} = (\lambda , - 4\lambda + 4, - 3\lambda - 1)$$

$$\overrightarrow {QN} $$ is perpendicular to $$\overrightarrow {b} $$

$$ \Rightarrow (\lambda , - 4\lambda + 4, - 3\lambda - 1)\,.\,(1,4, - 3) = 0$$

$$ \Rightarrow \lambda = {1 \over 2}$$

Hence $$\overrightarrow {QN} = \left( {{1 \over 2},2,{{ - 5} \over 2}} \right)$$ and $$\left| {\overrightarrow {QN} } \right| = \sqrt {{{21} \over 2}} $$