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 pointis 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,
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