Multiplication of Vectors . Although the rest are also important, here I'll discuss only the latter. The direction defined by the right-handed rule is customarily preferred to the other one. The late Isaac Isimov once suggested apprehensively that technological advances may lead to thesaurus changes that would eliminate such dear to heart notions as the clockwise and counterclockwise directions. Luckily, no technological progress could possibly affect the physical underpinning of the right-handed rule.
Definition Let a and b be two noncollinear vectors.
Cross product of collinear vectors is defined as 0. Obviously the product has no unit element. One the positive side, both the associative and distributive laws hold. For the latter, it's obvious from the geometric considerations. It's harder to prove the associative law.