A long straight wire carries a current along the ‘z’ axis. One can find two points in the plane perpendicular to the conductor such that
i) The magnetic fields are equal
ii) The direction of the magnetic fields are perpendicular
iii) Magnitudes of the magnetic fields are equal
iv) The field at one point is opposite to that at the other point.

At every point equidistant from the wire, the magnitude of magnetic field is same and its direction is always along the tangent of the circle at that point.

Hence, magnetic fields cannot be same around the wire in a plane. 

At two diametrically opposite points, magnetic fields are opposite.

At every quarter of the circle, the magnetic field becomes perpendicular.