The vector orthogonal to 2 vectros in a plane is calculated with the determinant
#| (veci,vecj,veck), (d,e,f), (g,h,i) | #
where #〈d,e,f〉# and #〈g,h,i〉# are the 2 vectors
Here, we have #veca=〈0,20,31〉# and #vecb=〈32,-38,-12〉#
Therefore,
#| (veci,vecj,veck), (0,20,31), (32,-38,-12) | #
#=veci| (20,31), (-38,-12) | -vecj| (0,31), (32,-12) | +veck| (0,20), (32,-38) | #
#=veci(20*-12+38*31)-vecj(0*-12-31*32)+veck(0*-38-32*20)#
#=〈938,992,-640〉=vecc#
Verification by doing 2 dot products
#〈938,992,-640〉.〈0,20,31〉=938*0+992*20-640*31=0#
#〈938,992,-640〉.〈32,-38,-12〉=938*32-992*38+640*12=0#
So,
#vecc# is perpendicular to #veca# and #vecb#
The unit vector is
#hatc=vecc/||vecc||=(<938,992,-640>)/||<938,992,-640>||#
#=1/1507.8<938,992,-640>#
