A slab is cut along the plane 2x - y + 2z = 3, with coordinates in metres…

A slab is cut along the plane $2x - y + 2z = 3$, with coordinates in metres. A guide rail passes through the point $(1, 0, 2)$ in the direction $(1, 1, 1)$, so its equation is $x - 1 = y = z - 2$. A technician must fix sensors on the rail at exactly $2$ metres from the slab. She notes that the distance of a point of the rail from the slab changes at a steady rate along the rail, so there are two places for the sensors. One is on each side of the slab. Find the values of the parameter $t$ of the points $A + t\vec{b}$ at which this happens. Find the $x$-coordinates of the two points, and the distance between the two points.