Will the banner of this airplane be always in the proper direction if the airplane flies in any direction on a windy day?
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Other than User58220's answer, I'm reading a lot of nonsense here.
When you fly an airplane (I and many other people on this site do), when you are cruising in the air, you center the rudder.
The plane has no awareness of the movement of the air mass over the ground (wind). The plane has a vertical stabilizer (tail) which causes it to point into its relative wind, which is entirely unrelated to the motion of the air mass over the ground. If it is towing a banner, that also flows in a line with the airplane.
If you deflect the rudder pedals, that pushes the tail of the plane left or right, causing the relative wind to blow more against one side of the airplane. You don't do that to compensate for wind over the ground (except during landing and takeoff, when you don't have a banner).
The rudder is used in turns to compensate for adverse aileron yaw, during takeoff to compensate for asymmetric propeller thrust, and during landing to keep the nose lined up with the runway in a crosswind.
If you do anything with the rudder in cruise, you are not doing it to compensate for air movement over the ground.
EDIT: adding some educational videos:
Better late than never: Suppose the air mass is simply not moving, but the ground underneath is moving (with respect to the air). The plane just flies in the still air, banner and all, irrespective of what the ground's doing.
However, if it is needed to fly from airport A to airport B, as User58220 pointed out, that's a navigation problem, not an aircraft-operation problem. It is necessary to turn the plane to a direction such that it will arrive at the place where airport B will be when both the plane and the airport get there. This is called the wind triangle problem, and there is a simple tool to help solve it, called an E6B.
Further edit: I almost forgot that when I first started using the Microsoft Flight Simulator, the plane wouldn't go where I pointed it, which made it very hard to land. It would always drift off center. Later I learned a fundamental difference between cars and planes. Cars go where they are pointed. Planes go where they are carried. There's always some sort of wind, so the way you get where you're going is by looking at the ground to see where you're being carried. If you are going to the right of where you want to go, you adjust your heading to the left, and vice versa. You never do it by pointing the plane at your destination, except in a general sense. I only mention this because not having that understanding could have led to the OP's question.
If a plane is flying without any rudder input, then the banner will always fly straight behind the plane, with nose-tail-banner in a straight line, no matter what the speed or direction of the plane and/or wind. The only thing that affects the plane and banner is the flow of air over the control surfaces. How would the banner know that there was wind blowing?
Forget about taxiing planes and crabbing into the wind on final approach. These all change the problem by adding the frame of reference of the ground.
A pilot doesn't need or have a wind speed meter. A pilot doesn't need one, only an air speed indicator. The navigator would love to have a wind speed meter, because the navigator is the one that has to worry about ground speed and course made good...
The banner is basically a wind vane while the aircraft can resist torque by using its rudder. Even without using the rudder the torque on the craft due to angled wind speed will be lower than the torque on the banner due to the difference in masses.
Both @user58220 and @Gregsan give valid responses but with relevance to the question I think it is fair to say the banner cannot possibly be always straight regardless of flight conditions. The slant angle no matter how small can be measured according to the formula given by Gregsan, and of course the picture above says a lot.