I read the message about dimensions of dot products, but what about cross products. when you have two vectors in the xy-plane, the result is a vector in the z direction. so why is the units cm^2, or am I doing something wrong?
Everything above looks ok, it is true that the cross-product of two vectors in the x-y plane will be a vector in the z-direction. It is also true that if the two vectors are displacement vectors with units of cm, that the cross product will have units of cm^2. In the case that you describe above, the cross product vector is just (AxBy - AyBx)zhat -- Ax, By, Ay, and Bx all have units of cm and the unit vector is dimensionless, so the final units are cm^2. This discussion is relevant for prob.20 of HW#2, for example, in which you should conclude that the cross product of two displacement vectors is not a displacement vector.