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 
and 
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, 
from our reference point
and moving with linear momentum, 
. As in simple rotation
the direction of
angular momentum is normal to the plane spanned by
and 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 to
then our thumb points in the direction of angular momentum.
What is happening here is that we are constructing a
vector, 
from two vectors, and .
The particular construction is denoted a vector cross
product:
note that not only does the vector cross product produce a
vector normal to the plane spanned by and but
the length contains the factor, 
:
where 
is the angle between and .
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 
.