The important thing to understand is that a wing generates a force (which we call lift) at right angles to the flow of air past it. An aircraft is moving forward through the air, so the wing sees the air moving backward, and the force generated is upwards.
If a boat is at right angles to the wind then the force generated by the sail will be at right angles to that which is forward.
If a boat is pointing 45 degrees into the wind (so halfway between across the wind and straight into it) then the sail will generate a force at right angles to that, which is forward and sideways (45 degrees off the bow).
So why doesn’t the boat go forward and sideways? Well there is another wing underwater called the keel, and it generates another force which is sideways *in the other direction*. This cancels out the sideways force from the sail and all that is left is a forward force.
The closer you point the boat into the wind the more the resulting force from the sail is pointing sideways, and the less it is pointing forwards. At some point the forwards force won’t be big enough to counteract all the drag acting on the boat, the sail and the keel, and you will stop moving forward.
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