Consider the line $L_1:x=y=z$ and the line $L_2:2x+y+z-1=0=3x+y+2z-2$ then:

perpendicular distance of origin from plane containing line $L_2$ and parallel to line $L_1$ is $\frac{1}{\sqrt{2}}$

the shortest distance between the two lines is $\frac{1}{\sqrt{2}}$

Plane containing the line and parallel to line $L_1$ is $z – x + 1 = 0$

The shortest distance between the two lines is $\sqrt{2}$