Consider four planes with the Cartesian equations:
- Establish that two of them are parallel and the other two are perpendicular.
- For the two parallel planes, find the distance between them.
- The second and fourth planes are parallel since they have the same normal vector, .
To see that the first and third are perpendicular, we denote the normal vectors by and , respectively, and compute
Hence, they are perpendicular.
- Denoting the second and fourth planes by and , respectively we have Cartesian equations
Therefore, the distance between them is