Begin2.DVI

It therefore remains to show that
v 1 =
v. The geometry of figure 6-14, provides an

aid in demonstrating that the vectors
r 1 and
r are related by the vector equation

r 1 =A
+
r,

where A
is a vector from the origin of one system to the origin of the other system

and lying along the axis of rotation and in the same direction as
ω.These results

demonstrate that ω×A
=
0 and

d
r 1

dt

=
v 1 =
ω×
r 1 =
ω×(A
+
r ) = 
ω×A
+
ω×
r =
ω×
r =
v =d
r

dt

,

Here the distributive law for cross products has been employed and the fact that

both
ωand A
have the same direction produced a cross product of zero.

Let B
 denote any vector connecting two fixed

points within a rigid body which is rotating about

a line with angular velocity
ω. Let
r 1 denote a

vector to the terminus of B
and let
r 2 denote a

vector to the origin of B
, as measured from some

origin on the axis of rotation. One can then write

d
r 1

dt

=ω×
r 1 and d
r^2

dt

=ω×
r 2

Observe that by vector addition
r 2 +B
=
r 1 so that

dB


dt =

d
r 1

dt −

d
r 2

dt =ω×
r^1 −ω×
r^2 =ω×(
r^1 −
r^2 ) = ω×

B

Therefore one can state that in general, if B
is any fixed vector lying within a rigid

body which is rotating, then with respect to any origin on the axis of rotation, one

can state that
dB

dt

=
ω×B
 (6 .59)

This is an important result used in the study of rotating bodies.

Two-Dimensional Curves

The graphical representation of a function y=f(x) in a rectangular cartesian

coordinate system can also be presented in a vector language. A graph of the

function y =f(x) can be represented by a position vector
r , measured from the