The cross product of 2 vectors 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=〈-3,5,8〉# and #vecb=〈6,-2,7〉#
Therefore,
#| (veci,vecj,veck), (-3,5,8), (6,-2,7) | #
#=veci| (5,8), (-2,7) | -vecj| (-3,8), (6,7) | +veck| (-3,5), (6,-2) | #
#=veci(5*7+2*8)-vecj(-3*7-6*8)+veck(3*2-5*6)#
#=〈51,69,-24〉=vecc#
Verification by doing 2 dot products
#〈51,69,-24〉.〈-3,5,8〉=-51*3+69*5-24*8=0#
#〈51,69,-24〉.〈6,-2,7〉=51*6-69*2-24*7=0#
So,
#vecc# is perpendicular to #veca# and #vecb#