Why can a boat move forward when it is close to the wind?

Answer

A sail can work in 2 ways. In position 8 on your figure, the wind blows at the back of the sail and then the force applied is in the direction of the wind. However, if a sail is approached at a small angle, it works like an airplane wing. An airplane flies horizontally, but the greatest force acting on the wing is in the vertical direction (lift force). This force is typically more than 10 times greater than the drag force on the wing. For a good understanding: I reason for a moment as if the aircraft is stationary and the air comes in at the speed of the aircraft from the flight direction: then the lift force is perpendicular to the direction of the oncoming air/wind (not perpendicular to the wing) and the aerodynamic drag force is simply in the direction of the oncoming air/wind.

For example, in situation 2 (close to the wind), the force exerted by the wind on the sail is mainly directed to the left (perpendicular to the direction from which the wind comes). A component of that force is therefore in the direction in which the boat is sailing and will therefore propel it in that direction. The faster the boat goes, the more resistance it will experience from the water and if that resistance force becomes the same as the driving force, the boat will continue to sail at that speed. The other component of the force is perpendicular to the direction the boat is traveling and so will cause the boat to drift sideways but this will be compensated by the force exerted by the boat’s sword.

PS. When the boat has a speed you have to look at how the wind is blowing relative to the boat and this determines the direction of the lift and the aerodynamic drag force. eg. in situation 8, if you go as fast as the wind, it becomes relatively calm. Then there is no longer any force that drives the boat and so the speed of the boat decreases due to the resistance of the water. So you can never go as fast as the wind in that direction. If you sail diagonally to the wind, the relative direction from which the wind is blowing changes. The faster you go, the more relative headwind you will get. However, as long as the relative wind direction makes an angle with the sailing direction you will get a ‘lift’ force on the sail which has a component that will propel the boat. The less resistance the boat has in the direction of travel, the faster you move forward.