[jampekka]

Maybe the books assume that the geometrical interpretations of the dot product are already known by the reader? I think they (both the projection interpretation and relation to angle between vectors) were taught in high school at the latest. There's also a lot of interpretations and uses for the dot product, some of which aren't necessarily that useful for classical mechanics.

But in general, literature using and/or teaching mathematics does tend to be too algebraic/mechanistic. Languge models can be a very good aide here!