I understand flaps increase the angle of attack so it increases lift, but how does it increase drag also? Do the lift and drag have a direct relationship?
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As you say, flaps on aircraft wings increase lift at the expense of also increasing drag. This also means that the minimum speed to get lift is lower. These tradeoffs is why airplanes don't fly with flaps down all the time, only during takeoff and landing when the extended low speed range of lift is really useful.
To get a quick intuitive idea of why flaps also create more drag, put your hand out the window of a car traveling at highway speed. Start with the hand flat and parallel to the ground. In this configuration there will be no up/down force from the wind, just some backward push. Now rotate your hand a little so that the leading edge is higher than the trailing edge. You will feel the air pushing upwards, but there will also be more force pushing backwards. In affect, the larger area presented perpendicular to the wind creates more drag. If this area is oriented and shaped correctly, then it can also produce lift.
You can see the opposite by tilting the leading edge of your hand down a bit. You still get a bit more drag again due to the larger area facing the wind, but now the wind is also actively pushing down. This is basically negative lift.
In case that sounds silly, there are situations where this is useful too. This is exactly what a "spoiler" does on a race car. The wind is being used to provide a downward force (negative lift), which increases the sliding friction of the tires against the road, thereby allowing for more aggressive accelleration and cornering without slipping.