(1)
The angle between the lines in terms of sin θ is given by
sin θ =
2
1
cos
− θ
=
(
)(
)
2
1 2
1 2
1 2
2
2
2
2
2
2
1
1
1
2
2
2
(
)
1
a a
bb
c c
a
b
c
a
b
c
+
+
−
+
+
+
+
=
(
)(
) (
)
(
) (
)
2
2
2
2
2
2
2
1
1
1
2
2
2
1 2
1 2
1 2
2
2
2
2
2
2
1
1
1
2
2
2
a
b
c
a
b
c
a a
bb
c c
a
b
c
a
b
c
+
+
+
+
−
+
+
+
+
+
+
=
2
2
2
1
2
2
1
1
2
2
1
1
2
2
1
2
2
2
2
2
2
1
1
1
2
2
2
(
)
(
)
(
)
−
+
−
+
−
+ +
+ +
a b
a b
b c
b c
c a
c a
a
b
c
a
b
c (2)
�Note   In case the lines L1 and L2 do not pass through the origin, we may take
lines 
1
′L andL2
′ which are parallel to L1 and L2 respectively and pass through
the origin Fig 11 6
© NCERT
not to be republished
 MATHEMATI CS
472
If instead of direction ratios for the lines L1 and L2, direction cosines, namely,
l1, m 1, n1 for L1 and l2, m 2, n2 for L2 are given, then (1) and (2) takes the following form:
cos θ = |l1 l2 + m 1m 2 + n1n2|    (as 
2
2
2
1
1
1
1
l
m
n
+
+
=
2
2
2
2
2
2
l
m
n
=
+
+
)