The vector cross product of two vectors #veca" "#&#" "vecb" "#
is defined as
#vecaxxvecb=|veca||vecb|sinthetahatvecn#
where #theta" "#is the angle between the vectors & #" "hatvecn" "isa unit vector mutually perpendicular to
#veca" "#&#" "vecb" "#
If the vectors are in component form, in particular:
#veca=((a_1),(a_2),(a_3))#and #vecb=((b_1),(b_2),(b_3))#
we can evaluate the cross product by the use of a determinate
#vecaxxvecb=|(hatveci,hatvecj,hatveck),(a_1,a_2, a_3),(b_1,b_2,b_3)|#
In this case we have
#veca=((-1),(1),(0))##" "vecb=((2),(1),(3))#
#vecaxxvecb=|(hatveci,hatvecj,hatveck),(-1,1,0),(2,1,3)|#
expanding the determinant as usual
#vecaxxvecb=+hatveci|(1,0),(1,3)|-hatvecj|(-1,0),(2,3)|+hatveck|(-1,1),(2,1)|#
#vecaxxvecb=+3hatveci+3hatvecj-3hatveck#
By definition this is perpendicular tot eh original two vectors. A quick check using the dot product will confirm this.
#veca.((3),(3),(-3))=((-1),(1),(0)).((3),(3),(-3))#
#=-3+3+0=0" "#perpendicular
#vecb.((3),(3),(-3))=((2),(1),(3)).((3),(3),(-3))#
#=6+3-3=0" "#perpendicular