From the definition:
#bb A times bb B = abs bb A abs bb B sin alpha_("AB") \ bb hat n#
So #bb A times bb B# lies in the direction that is normal to both #bb A# and #bb B#, labelled #bb hat n#, and has magnitude #abs bb A abs bb B sin alpha_("AB")#, where #alpha_("AB")# is the angle between #bb A# and #bb B#
On the other hand:
#bb B times bb A = abs bb B abs bb A sin alpha_color(red)(bb "BA") \ bb hat n#
#= abs bb A abs bb B sin (-alpha_color(green)(bb "AB")) \ bb hat n#
#= bb color(red)(-) abs bb A abs bb B sin alpha_("AB") \ bb hat n#
IOW, these vectors have same magnitude, but are anti-parallel, so the angle between them is 180 degrees
It is therefore true to say that:
#bbA times bb B = - bb B times bb A#
And vice versa.
