How many vectors are their which are perpendicular to acos (∂) i-cap+b sin (∂) j-cap

their can be many vectors but i will give you one example that is bsin(∂) i-cap – acos(∂) j-cap.......now just take the dot product which is surely zero , so that confirms that the given example is perpendicular to the vector in question now similarly you can find others ….............

hope this helps...............