21Kinematics
© 2010 Taylor & Francis Group, LLC
If this is followed by a second innitesimal rotation δθ
2
about a different axis
through O, the nal
radius vector r
12
of the particle for δθ
1
followed by δθ
2
is
rrr
rr
rr
12 12
121
12
=+
=+ ×
=+ ×+ ×
δ
δθ
δθ δθ
()
()(
rr
+×
=+ ×+ ×+
δθ
δθ δθ δθ δθ
1
12
.
We can write r
12
as
rrr
12 12
=+δ
(1.38)
with
δδθδθδθδθ
rr
12 12
=×+×+× ×
. (1.39)
If the orders of rotations were reversed, that is, if δθ
2
was followed by δθ
1
, then we would have
rrr
21 21
=+δ
(1.40)
with
δδθδθδθδθ
rr
21 21112
=×+×+× ×
. (1.41)
For innitesimal rotations, we can neglect the higher-order terms; then Equations 1.39 and 1.41
become identical, and
. We get the same result: the two innitesima ...