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 Example 30 Find the coordinates of the point where the line through the points
A (3, 4, 1) and B(5, 1, 6) crosses the XY-plane Solution The vector equation of the line through the points A and B is
 rr  =
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
3
4
[ (5
3)
(1
4)
(6
1)
]
i
j
k
i
j
k
+
+
+ λ
−
+
−
+
−
i