, any plane passing through the intersection of the planes
r n1
⋅r r  =
1
2
2
dand
r n
d
⋅
=
r r
has the equation
1
2
(
)
r
n
n
⋅
+ λ
r
r
r
= d1 + λd2 (1)
Cartesian form
In Cartesian system, let
1nr  =
1
2
1 ˆ
ˆ
ˆ
A
B
C
i
j
k
+
+
2nr  =
2
2
2 ˆ
ˆ
ˆ
A
B
C
i
j
k
+
+
and
rr  =
ˆ
ˆ
ˆ
xi
y j
z k
+
+
Then (1) becomes
x (A1 +  λA2) + y (B1 +  λB2) + z (C1 +  λC2) = d1 +  λd2
or
(A1x + B1y + C 1z – d1) +  λ(A2x + B2 y + C 2z – d2) = 0 (2)
which is the required Cartesian form of the equation of the plane passing through the
intersection of the given planes for each value of λ Example 20 Find the vector equation of the plane passing through the intersection of
the planes 
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
) 6 and
(2
3
4 )
5,
r
i
j
k
r
i
j
k
⋅
+ +
=
⋅
+
+
= −
r
r
and the point (1, 1, 1)