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relation with velocity

Now remember we learned that the velocity at a point on the object tex2html_wrap_inline660 , where tex2html_wrap_inline662 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 tex2html_wrap_inline664 , measuring the moving point in relation to an origin that is on the axis, we'd like to say tex2html_wrap_inline666 , 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 tex2html_wrap_inline668 pointing up, and a vector r pointing onto some point on the object as pictured below


tex2html_wrap_inline662 in this picture is tex2html_wrap_inline674 . tex2html_wrap_inline676 is the angle between tex2html_wrap_inline664 and tex2html_wrap_inline668 . so


What about it's direction. It's not hard to see that tex2html_wrap_inline682 is perpendicular to both bom and tex2html_wrap_inline664 .

This is what's called a cross product or vector product. If you have two vectors tex2html_wrap_inline688 and tex2html_wrap_inline690 , then we can define a vector tex2html_wrap_inline692 . It has a magnitude tex2html_wrap_inline694 , with tex2html_wrap_inline676 being the angle between tex2html_wrap_inline688 and tex2html_wrap_inline690 . And it's perpendicular to both tex2html_wrap_inline688 and tex2html_wrap_inline690 .

So we see that tex2html_wrap_inline706 . Could I have said tex2html_wrap_inline708 ? 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, tex2html_wrap_inline688 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 tex2html_wrap_inline712 !

Joshua Deutsch
Sun Feb 23 15:54:50 PST 1997