Question

Question #b4ef9

Answer 1

The first kind of product is the scalar product or the dot product which is defined as,

A . B = `AB Cos theta` where `A` and `B` are magnitudes of the two vectors respectively.

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 = `AB sin theta` n, where n is a unit vector normal to the plane of A and B. It's direction is specified by the right handed screw rule : Take a right handed screw and place it at the point where the tails of the two vectors join. Rotate the screw from A towards B, the direction of advancement gives the direction of n.

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

Answer 2

Alternative and more general definitions given below.

Explanation:

Alternative and more general definitions:

Suppose `vecA= (A_1, A_2, ..... , A_n) and vecB=(B_1, B_2, ..... , B_n)` are two n-dimensional vectors in a real or complex n-dimensional vector space.
Then :

The dot product of the 2 vectors is defined to be the real or complex number given by :
`vecA*vecB= A_1B_1+A_2B_2+......+A_nB_n`

The cross product is defined to be equal to the determinant :

`|(hatn_1,hatn_2,....,hatn_n),(A_1,A_2,...,A_n),(B_1,B_2,...,B_n)|`,

where `hatn_i` are the orthogonal unit vectors normal to each dimension.

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. :)