Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 5A / b / Solution

The cross product of the plane normals is proportional to $(1,-1,-1)$, soThe point on the intersection perpendicular to $m$ iswhich satisfies both plane equations and $u\cdot m=0$. Hence the line is $r\times m=u\times m$.

The equally inclined line has direction $d=(1,1,1)/\sqrt{3}$. The distance between the two skew lines is