A×B = (Ax i + Ay j + Az k) × (Bx i + By j + Bz k )
= (Ax i + Ay j + Az k) × Bx i + (Ax i + Ay j + Az k) × By j + (Ax i + Ay j + Az k) × Bz k
= Ax Bx i × i + Ay Bx j × i + Az Bx k × i
+ Ax By i × j + Ay By j × j + Az By k × j
+ Ax Bz i × k + Ay Bz j × k + Az Bz k × k
= 0 - Ay Bx k + Az Bx j + Ax By k + 0 - Az By i - Ax Bz j + Ay Bz i + 0
= (Ay Bz - Az By) i + (Az Bx - Ax Bz) j + (Ax By - Ay Bx) k
= (Ay Bz - Az By, Az Bx - Ax Bz, Ax By - Ay Bx)