Hence, the equation 
1
2
1
2
(
)
r
n
n
d
d
⋅
+ λ
=
+λ
r
r
r
represents a plane π3 which is such
that if any vector rr satisfies both the equations π1 and π2, it also satisfies the equation
π3 i e , 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