The first kind of product is the scalar product or the dot product which is defined as,
A . B =
It is clear from the definition that, the dot product of two vectors yields a scalar.
Another kind of product we encounter is the vector product or cross product which is defined as,
A X B =
In case of B X A, rotate the screw from B to A.
Thus, one more conclusion to draw from this is that the vectors A X B is oppositely directed to B X A but, both have the same magnitude.
Thus, vector cross product is anti-commutative.
A X B = - B X A
Alternative and more general definitions given below.
Alternative and more general definitions:
The dot product of the 2 vectors is defined to be the real or complex number given by :
The cross product is defined to be equal to the determinant :
These definitions hold not only in any real or complex finite dimensional vector space, but also in real or complex finite and infinite dimensional metric spaces, normed spaces, Banach spaces, Inner-product spaces and Hilbert spaces. :)