The distance between the line $\vec{r}= 2\hat{i} - 2\hat{j} + 3\hat{k} + (\hat{i} - \hat{j} + 4\hat{k})$ and the plane $\vec{r}. (\hat{i} +5 \hat{j} + \hat{k})= 5, $ is

$\frac{10}{3\sqrt{3}}$

Clearly, given line passes through the point $(2\hat{i} - 2\hat{j} + 3\hat{k})$ and is parallel to the given place.

∴ Distance between the line and the plane

= Length of perpendicular from $2\hat{i} - 2\hat{j} + 3\hat{k}$ to the given plane.

$=\frac{|(2\hat{i} - 2\hat{j} + 3\hat{k}).(\hat{i} +5\hat{j} + \hat{k})-5|}{|\hat{i} +5\hat{j} + \hat{k}|}=\frac{10}{3\sqrt{3}}$