When a vertical dipole antenna is used in conjunction with a loop antenna for direction finding then the field pattern obtained will be?

Loop antenna generate the same field as electrical dipole, but E and H are exchanged (a proof can be obtained this way: you align an electrical dipole with an axis, say z, and a loop antenna with z - you have to define surface orientation - then you apply duality theorem, calculate far field and you got it).

For instance, in spherical coordinates (r,phi,theta) you have

el. dipole {Etheta,Hphi}

loop ant. {Ephi,Htheta}

So, you can superpose the two fields and see that you have gained polarization independency (if an electrical dipole is aligned with z-axis and has its gap at z=0, it will receive very well Etheta, not at all Ephi (plane z=0), if you put a loop antenna "around" electrical dipole - like a ring around dipole's gap, lying on z=0 plane - it will receive very well Ephi (plane z=0) ).

This is the way to make triaxial sensor, which have to be polarization independent.