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 !