1
2
AB (
)
b
b
×
uuur r
r
 = 0 or 
2
1
1
2
(
) (
)
a
a
b
b
−
⋅
×
r
r
r
r
 = 0
Cartesian form
Let (x1, y1, z1) and (x2, y2, z2) be the coordinates of the points A and B respectively © 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