Now remember we learned that the velocity at a point on the
object 
, where 
is the distance between the
point and the axis. It is the distance measured perpendicularly
from the axis to the point. Now lets try to express this relation
as a vector relation.
If we define a vector 
, measuring
the moving point in relation to an origin that is on the axis,
we'd like to say 
,
except that makes no sense since we haven't defined how
to multiply to vectors and get a vector! Well it's about
time we do that. So we have a vector 
pointing up,
and a vector r pointing onto some point on the object
as pictured below
in this picture is 
. 
is the
angle between and . so
What about it's direction. It's not hard to see that
is perpendicular to both bom and .
This is what's called a cross product or vector
product. If you have two vectors 
and 
, then
we can define a vector 
. It has
a magnitude 
, with being the
angle between and . And it's perpendicular to
both and .
So we see that 
. Could I have
said 
? The answer is no!
We have to
be careful about getting our signs right, and
again we use a right hand rule to make that clear.
If the first vector being multiplied, is represented by the
stretched fingers, and the second, by the bent ones then the
thumb gives the direction of the resulting cross product.
Notice that if the order was reversed, so would the directions
so that 
!
