(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 © NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
487
Taking
ˆ
ˆ
ˆ
r
xi
y j
r= + +z k
, we get
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
) [(1 2 )
(1 3 )
(1 4 ) ] 6 5
xi
y j
z k
i
j
k
+
+
⋅
+ λ
+
+ λ
+
+ λ
= − λ
or
(1 + 2λ ) x + (1 + 3λ) y + (1 + 4λ) z = 6 – 5λ
or
(x + y + z – 6 ) + λ  (2x + 3y + 4 z + 5) = 0