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

"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.

To express the component of the cross product vector C = A x B in terms of the Levi-Civita product vector and the components of A and B, we can use the following relation:

C_i = ε_ijk * A_j * B_k

Where ε_ijk is the Levi-Civita symbol, A_j is the j-th component of vector A, B_k is the k-th component of vector B, and C_i is the i-th component of the cross product vector C.

The Levi-Civita symbol ε_ijk is defined as follows:

ε_ijk = 0, if any of the indices (i, j, k) are the same
ε_ijk = 1, if (i, j, k) is an even permutation of (1, 2, 3)
ε_ijk = -1, if (i, j, k) is an odd permutation of (1, 2, 3)

Now, to show that A . (A x B) = 0, where "." denotes the dot product, we need to take the dot product of vector A with vector (A x B) and show that it equals zero.

Let's calculate the dot product:

A . (A x B) = A_i * (A x B)_i

Since the cross product of two vectors is orthogonal to both vectors, the dot product of vector A with the cross product (A x B) will be zero.

Therefore, A . (A x B) = 0.

In conclusion, we can express the component of the cross product vector using the Levi-Civita product vector and the components of A and B, and we have shown that the dot product of A with (A x B) is zero.