Derive an expression (in vector form) for electric field of a dipole at a point on the equatorial plane of the dipole. How does the field vary at large distances?

We are considering +q and -q charges on line. They are separated by a distance 2d. The equatorial line of a dipole is the line which is perpendicular to the axial line and passing through a mid-way between the charges

Electricity intensity at an axial point is twice the electric intensity on the equatorial line.