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Vector nature of Angular momentum; vector cross products

I have intentionally not paid much attention to the vector quality of angular momentum and the rotational variables. If angular momentum is to become a versatile tool for us we must do this.

For a rotating object the angular momentum points along the axis of rotation. This is the only choice we have since it is the only direction which stays constant as the object rotates. I have a choice as to which direction to choose as the positive direction though clearly if I switch the direction of rotation then the direction of angular momentum should also switch to reflect this change. To be definite we have once and for all chosen to have the direction of vectors associated with rotation oriented according to the right handed rule. For a rotating object this means that the direction of angular momentum (and the vectors tex2html_wrap_inline73 and tex2html_wrap_inline75 for that matter) is found by curling your right hand around the rotation direction. The direction in which your thumb points is the direction of these vectors. This is simply an arbitrary choice of convention complete with a simple way to remember the convention.

How do I transfer this convention to our new and more general definition of angular momentum. I consider a general particle displaced by a vector, tex2html_wrap_inline77 from our reference point and moving with linear momentum, tex2html_wrap_inline79. As in simple rotation the direction of angular momentum is normal to the plane spanned by tex2html_wrap_inline77 and tex2html_wrap_inline79 and it points in the direction given by the right hand rule namely that if we let our right hand fingers sweep through the smallest angle from tex2html_wrap_inline77 to tex2html_wrap_inline79 then our thumb points in the direction of angular momentum.

What is happening here is that we are constructing a vector, tex2html_wrap_inline61 from two vectors, tex2html_wrap_inline77 and tex2html_wrap_inline79. The particular construction is denoted a vector cross product:
equation38
note that not only does the vector cross product produce a vector normal to the plane spanned by tex2html_wrap_inline77 and tex2html_wrap_inline79 but the length contains the factor, tex2html_wrap_inline99:
equation45
where tex2html_wrap_inline69 is the angle between tex2html_wrap_inline77 and tex2html_wrap_inline79. Please consult your favorite math book or the discussion in Fishbane et al about the more details about the cross product. We have mentioned what is most important for you to know but there is also an important results on how to get the coordinates of the cross product given the coordinates of the constituent vectors. You should also check out a link on the course WEB page to an interactive JAVA applet which allows you to view how the cross product works out in terms of direction and magnitude in real time as you use your mouse to adjust the factors in the cross product. Note that the cross product is not commutative and specifically tex2html_wrap_inline107.


next up previous
Next: About this document Up: Angular Momentum Previous: Generalization of Angular momentum

Collin Broholm
Wed Oct 22 13:19:47 EDT 1997