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1) I know electric dipole is the strength of the polarization but I really wonder that what it means actually. For example;
When we consider the following situations.
The electric dipole moment magnitudes of the first situation and the second situation are the same but I wonder What this actually mean?
2) Why is the electric dipole moment a vector quantity? Why we need the electric dipole moment to be a vector quantity?
Dipoles can be understood as a vector quantity located in the middle point between the charges, therefore only useful when you place that vector there and forget about the charges, but that is only possible and useful if you are far away from that point. That is you set a $\vec{p}$ and observe from $\vec{r}$ with $\vec{p} = q \vec{d}$ with $|\vec{r}| \gg |\vec{d}|.$ Otherwise it is not a useful quantity. It is a vector since it points in one direction (from one charge to the other), has an associated magnitude and so on. In a way, when you consider a multipole expansion it makes sense, the monopole momentum is just the net charge (a scalar), the dipole momentum a vector and from then on matrix quantities.
