A point on the given line $\overrightarrow r=(2\hat i-2\hat j+3\hat k)+\lambda(\hat i-\hat j+4\hat k)$ is $A(2,-2,3)$
$\perp$ distance of $P$ from the plane $x+5y+z-5=0$ is $\bigg|\large\frac{2-10+3-5}{\sqrt {1+25+1}}\bigg|$
$=\large\frac{10}{3\sqrt 3}$