ˆ
ˆ
ˆ
3
4
3
34
i
j
k
−
+
=
17
34
3
Alternatively, find the equation of the plane passing through A, B and C and then
compute the distance of the point P from the plane Example 29 Show that the lines
x
a
d
−
+
α − δ
 = y
a
z
a
d
−
−
−
=
α
α + δ
and
x
b
c
−
+
β − γ
 = y
b
z
b
c
−
−
−
=
β
β + γ
 are coplanar Solution
Here
x1 = a – d
x2 = b – c
y1 = a
y2 = b
z1 = a + d
z2 = b + c
a1 = α – δ
a2 = β – γ
b1 = α
b2 = β
c1 = α + δ
c2 = β + γ
Now consider the determinant
2
1
2
1
2
1
1
1
1
2
2
2
x
x
y
y
z
z
a
b
c
a
b
c
−
−
−
 = 
b
c
a
d
b
a
b
c
a
d
− −
+
−
+
−
−
α − δ
α
α + δ
β − γ
β
β + γ
© NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
497
Adding third column to the first column, we get
2 
b
a
b
a
b
c
a
d
−
−
+
−
−
α
α
α + δ
β
β
β + γ
 = 0
Since the first and second columns are identical Hence, the given two lines are
coplanar