© NCERT
not to be republished
 MATHEMATI CS
488
Let a1, b1, c1 and a2, b2, c2 be the direction ratios of 1b
r
and 
2b
r
, respectively Then
2
1
2
1
2
1 ˆ
ˆ
ˆ
AB (
)
(
)
(
)
x
x i
y
y
j
z
z k
= − + − + −
uuur
1
1
1
1
2
2
2
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
and
b
a i
b j
c k
b
a i
b j
c k
= + +
= + +
r
r
The given lines are coplanar if and only if 
(
)
1
2
AB
0
⋅b b
×
=
uuur
r
r In the cartesian form,
it can be expressed as
2
1
2
1
2
1
1
1
1
2
2
2
0
x
x
y
y
z
z
a
b
c
a
b
c
−
−
− = (4)
Example 21 Show that the lines
+3
1
5
–3
1
5
x
y
z
−
−
=
=
 and  
+1
2
5
–1
2
5
x
y
z
−
−
=
=
 are coplanar