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Looking for quadrupole moment of spherical shell with surface charge (sigma)=(sigma_o)*cos(theta) and (sigma_o) => constant

Sphere is at the origin with radius=a.

2.

Well, I think I am using the right equation, by integration of the quadrapole moment taken from the quadrupole term, but I am questioning my [(r')^2] factor.

3.

I use vector product I get a zero. I guess a vector crossed with itself is always zero because of no span. I have already found the monopole and dipole (monopole =0), and I think I take the vector product of r', and not the dot product, because the dot product is a scalar... and I found by another theorem that whether it is shell or sphere, the quadrapole and higher terms are zero. So is this where the quadrapole zeroes out?? from the [(r')^2] factor?