Let be the plane which is parallel to
and which passes through the intersection of the planes
Find a Cartesian equation for .
First, we find the intersection of the two planes. This is the set of points which simultaneously satisfy
From the first equation we have
Plugging this into the second equation we have
This then gives us . So the line is the set of points
where
is arbitrary. Thus,
The plane must then contain this line and be parallel to the vector . Since it is parallel to
its normal must be perpendicular to
, thus,
and we must have
Therefore,
This has the Cartesian equation