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
=
+
+
) (3)
and
sin θ =
(
)
2
2
2
1
2
2
1
1
2
2
1
1 2
2 1
(
)
(
)
l m
l m
m n
m n
n l
n l
−
−
−
+
− (4)
Two lines with direction ratios a1, b1, c1 and a2, b2, c2 are
(i)
perpendicular i