A king minus a man is a monarch. Add a woman and you get a queen. I don't see why this is so complicated. It is just like adding or removing adjectives from something.
This is nonsensical if you don't already have some existing notion of a relationship. These are vectors, not concepts. There a loads of operations that don't have any intuitive notion, yet are well defined in the context of vectors and most likely yield useless results.
Well isn't that true of all abstractions? They give us tools that we couldn't use before, and some of those tools don't make sense. I mean, in real physics it doesn't make sense to add different dimensions, eg time and distance, but math lets us do that, because they are 'just numbers'. We need to know that dimensions can only be multiplied and divided, and dimensional analysis needs to be done afterwards to check the unit of the result.
However, something like the dot-product here does make sense, since you can use it to determine similarities of vectors.
Stop thinking literally, and instead think of the concept. A king is a male monarch. So remove the male, and you have monarch. Add female and you have female monarch = queen.
Similarly with woman + person, the concept of woman is femininity. It makes sense if you think in terms of concepts.
