vectora    = {ax, ay, az}
vectorb    = {bx, by, bz}
scalara    = Norm[vectora]
directiona = vectora/scalara

Cross[vectorb, directiona]

{(az by)/Sqrt[Abs[ax]^2 + Abs[ay]^2 + Abs[az]^2]
- (ay bz)/Sqrt[Abs[ax]^2 + Abs[ay]^2 + Abs[az]^2],
-((az bx)/Sqrt[Abs[ax]^2 + Abs[ay]^2 + Abs[az]^2])
+ (ax bz)/Sqrt[Abs[ax]^2 + Abs[ay]^2 + Abs[az]^2],
  (ay bx)/Sqrt[Abs[ax]^2 + Abs[ay]^2 + Abs[az]^2]
- (ax by)/Sqrt[Abs[ax]^2 + Abs[ay]^2 + Abs[az]^2]}

{(az by)/scalara - (ay bz)/scalara, -(ax bx)/scalara + (ax bz)/scalara, (ay bx)/scalara - (ax by)/scalara}