Firstly, the horizontal forces acting on a boat aren’t as big as you might think. Certainly less than the weight of the vessel.
Then consider the shape of an anchor. In all cases they are designed so that if you pull them horizontally they dig into the ground. The further you pull the deeper they dig. So suddenly you aren’t just trying to pull the anchor, you are trying to pull all of the ground that it is under. You can imagine that an anchor doesn’t have to be buried very deep before the amount of ground you have to pull far exceeds any force you can apply.
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