(1)
and
2
2
x
x
a
−
 =
2
2
2
2
y
y
z
z
b
c
−
−
= (2)
where, a1, b1, c1 and a2, b2, c2 are the direction ratios of the lines (1) and (2), respectively,
then
cos θ =
1 2
1 2
1 2
2
2
2
2
2
2
1
1
1
2
2
2
a a
b b
c c
a
b
c
a
b
c
+
+
+
+
+
+
Example 9 Find the angle between the pair of lines given by
rr  =
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
3
2
4
(
2
2 )
i
j
k
i
j
k
+
−
+ λ
+
+
and
rr  =
ˆ
ˆ
ˆ
ˆ
ˆ
5
2
(3
2
6 )
i
j
i
j
k
−
+ µ
+
+
© NCERT
not to be republished
THREE D IMENSIONAL G EOMETRY
473
Solution Here 1b
r
 = 
ˆ
ˆ
2ˆ
2
i
j
k
+
+
 and 
2b
r
 = 
ˆ
ˆ
ˆ
3
2
6
i
j
k
+
+
The angle θ between the two lines is given by
cos θ =
1
2
1
2
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
(
2
2 ) (3
2
6 )
1
4
4
9
4
36
b b
i
j
k
i
j
k
b b
⋅
+
+
⋅
+
+
=
+ +
+
+
r
r
r
r
=
3
4 12
19
3 7
21
+
+
=
×
Hence
θ = cos–1 
19
21






Example 10 Find the angle between the pair of lines
3
3
x +
 =
1
3
5
4
y
z
−
+
=
and
1
1
x +
 =
4
5
1
2
y
z
−
−
=
Solution The direction ratios of the first line are 3, 5, 4 and the direction ratios of the
second line are 1, 1, 2 If θ is the angle between them, then
cos θ =
2
2
2
2
2
2
3 1
5