The cross product of #< a_x,a_y,a_z > xx < b_x,b_y,b_z ># can be evaluated as:

#{(c_x=a_yb_z-b_ya_z),(c_y=a_zb_x-b_za_x),(c_z=a_xb_y-b_xa_y) :}#

#color(white)("XXX")#if you have trouble remembering the order of these combinations see below

Given

#{:(a_x,a_y,a_z),(2,5,4):}# and #{:(b_x,b_y,b_z),(4,3,6):}#

#c_x= 5xx6-3xx4=30-12=18#

#c_y=4xx4-6xx2=16-12=4#

#c_z=2xx3-4xx5=6-20=-14#

**This is the "below" mentioned above** (skip if not needed)

One way to remember the order of the cross product combinations is to treat the system as if it we like calculating a **determinant** for

something like:

#color(white)("XXX")| (c_x,c_y,c_z),(,=,),(a_x,a_y,a_z),(b_x,b_y,b_z) |#

to get something like:

#color(white)("XXX")c_x=+| (a_y,a_z),(b_x,b_z) |#

#color(white)("XXX")c_y = -| (a_x,a_z),(b_x,b_z) |#

#color(white)("XXX")c_z=+| (a_x,a_y),(b_x,b_y) |#

Don't forget to alternate the signs and remember this is just a memory aid not real determination evaluation!