This is a very basic question about dot products or scalar products:-
If I want to move a block and I apply a force parallel to displacement, the block will move and some work will be done. So in the formula will be $W= F\cdot S$, here we won't calculate the force mg of the block but the force we applied (the parallel force).
Now let's say that the force is not parallel and is at some angle from the horizontal
So in this case the work done will be the projection of the force $F_1$ on the $x$ axis, because that is how dot products are defined (As Projections). But can't we say that the block moved due to its horizontal component $F_1\cos(\theta)$ and the answer would be same. And obviously we won't count $F_1\sin(\theta)$ as the work is not done by it.
So why do we say projection and not component in dot or vector product ?