Let θ be the acute angle between
OP and OQ Now recall that the directed line
segments OP and OQ are vectors with components
a1, b1, c1 and a2, b2, c2, respectively Therefore, the
angle θ between them is given by
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
+
+
+
+
+
+ (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