Question

express the component of a cross product vector C= A x B in terms of Levi civita product vector and the component of a and B, using this relation show that A . ( AxB ) = 0

Answer 1

"Product vector" is not the usual name used for this object in the scientific literature. It is called a tensor (actually a pseudo-tensor in this case)

If we denote this by e_{i,j,k}, we have

C_i = e_{i,j,k} A_{j} B_{k}

If you take the inner product with A, you get:

C_{i}A_{i} =e_{i,j,k} A_{i} A_{j} B_{k}

Now, e_{i,j,k} is anti-symmetric w.r.t. interchanging any two indices, but
A_{i} A_{j} is oviously symmetric w.r.t. interchanging i and j, therefore the summation vanishes.