Question
Find the Cartesian as well as the vector equation of the planes passing through the intersection of the planes 
which are at unit distance from the origin.
Solution
The equation of first plane is
or
The equation of second plane is
or
or
...(2)
The equation of any plane through the intersection of planes (1) and (2) is
(x + 3 y + 6) + k (3 x – y + 4 z) = 0
or (3 k + l).x + (–k + 3)y + 4 kz + 6 = 0 ...(3)
From given condition,
perpendicular distance of origin (0, 0, 0) from plane (3) = 1
Taking k = 1, from (3), we get,
Taking k = – 1, from (3), we get,
The required cartesian equations are
2x + y + 2z + 3 = 0, x – 2y + 2z — 3 = 0
and vector equations are
or
The equation of second plane is
or
or
...(2)The equation of any plane through the intersection of planes (1) and (2) is
(x + 3 y + 6) + k (3 x – y + 4 z) = 0
or (3 k + l).x + (–k + 3)y + 4 kz + 6 = 0 ...(3)
From given condition,
perpendicular distance of origin (0, 0, 0) from plane (3) = 1
Taking k = 1, from (3), we get,
Taking k = – 1, from (3), we get,
The required cartesian equations are
2x + y + 2z + 3 = 0, x – 2y + 2z — 3 = 0
and vector equations are
