Let and let . Let be the plane determined by the Cartesian equation .
- Determine if the two planes are parallel.
- If we define to be the plane with Cartesian equation find two points on .
- We know for any point , there is a unique plane parallel to containing . We pick a point on that is not on and show that the unique plane parallel to containing is, in fact, .
The point is on since
It is not on since
has no solution . So, the unique plane parallel to containing is
Then, we obtain the Cartesian equation of this plane. We have
This gives us . Then, which implies
Hence, this has the Cartesian equation . But this is the plane ; hence, is parallel to .
- The points and are both in the intersection since they both satisfy the equations