Derive distance formula between two parallel planes

Question

Given two parallel planes: $$ ax + by + cz = d \\ ax + by + cz = e $$ How would I derive a formula for the distance between them?

Answer 1

The distance between two parallel planes is measured along a line perpendicular to both planes. Choose any point on the plane ax+ by+ cz= d, say, (0, 0, d/c). The line through that point perpendicular to the plane is x= at, y= bt, z= ct+ d/c. That line intersects the plane ax+ by+ cz= e where a(at)+ b(bt)+ c(ct+ d/c)= a^2t+ b^2t+ c^2t+ d= e so (a^2+ b^2+ c^2)t= e- d. That is, t= (e- d)/(a^2+ b^2+ c^2).
Put that into the equation of the line, x= a(e- d)/(a^2+ b^2+ c^2), y= b(e- d)/(a^2+ b^2+ c^2), z= c(e- d)/(a^2+ b^2+ c^2)+ d/c.

Finally, find the distance between point and (0, 0, d/c), sqrt(a^2+ b^2+ c^2)(e-d)/(a^2+ b^2+ c^2)= (e- d)/(sqrt{a^2+ b^2+ c^2}.