(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) Solution Here, 
1
ˆ
ˆ
ˆ
n
i
j
k
=
+ +
r
 and 
2
nr  = 
ˆ
ˆ
ˆ
2
3
4 ;
i
j
k
+
+
and
d1 = 6 and d2 = –5
Hence, using the relation 
1
2
1
2
(
)
r
n
n
d
d
⋅
+ λ
=
+λ
r
r
r
, we get
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
[
(2
3
4 )]
r
i
j
k
i
j
k
⋅
+ +
+λ
+
+
r
 = 6 5
− λ
or
ˆ
ˆ
ˆ
[(1 2 )
(1 3 )
(1 4 ) ]
r
i
j
k
⋅
+ λ
+
+ λ
+
+ λ
r
 = 6 5
− λ                                … (1)
where, λ is some real number