Kalid,

thanks a lot for your answer!

Yes, I also thought that when you choose a frame where |b|=by the whole contribution comes from ay*by, and therefore it is reasonable to somehow combine ax*bx and ay*by, but I couldn’t figure out how come the combination turns out to be so simple (just summing the two components). There’s probably something simple I’m failing to see

Thanks for the hint on the law of cosines, it might indeed be the key to answer my question, although the circularity you already spotted out makes me feel suspicious about it. I’ll need to give it some deeper thought!

By the way, I love the way you reason in terms of analogies. Indeed a letter-by-letter comparison of “apple” and “manzana” is not the way to match the two terms, and their semantic association to the concept of apple is fundamental. However, I think the letter-by-letter comparison has also something to give, but I need to change the analogy in order to explain in what sense.

The way I look at math is as a wild island I have to explore. In the beginning, I have no map of the territory. As I proceed with the exploration and discover interesting places, I draw my own map by putting crosses (theorems or concepts) and tracing paths between them (proofs or reasoning lines), trying to find ways that connect all the interesting places.

There might be several ways to get to the same place from the same starting point, and it might not be obvious that some paths lead to the same place. Some might be longer and easier, some harder and shorter. Sometimes I feel surprised by ending up in the same place as I did by following another path; sometimes I’m surprised to end up somewhere different than I thought. That makes me realize I don’t know the territory.

Now given a good map with places and paths (theorems and formal proofs), I am normally (not always!) able to get from anywhere to anywhere, but that’s not enough to say I “know” the island. Instead, if I were able to orienteer well enough to be aware at any time where I am as I follow one path, and where I would be if I would be following an alternative one, that would make me feel confident that I know the territory, as if I could watch the whole island from above. Besides, that’s what I need if I want discover new places or find new ways of my own.

I like this analogy because I do not think mathematical concepts are conceived in the first place as “useful ideas to represent mathematically” (like “weighted parallelism”), and then given a mathematical formulation. I rather believe that they come up as people explore the island, and only after, when it’s clear that these places are frequently visited and somehow useful, they are given a name and (when possible) an interpretation. Like putting a cross and an intuitive label on the map.

But the intuitive label is a sign for tourists, and knowing where the useful places are and what paths lead to them is not enough for an explorer; I want to know where I am on the island as I follow each path, what obstacles I’m going to encounter, what deviations I will likely need to take, and so on. That makes me able to trace a step-by-step (letter-by-letter) comparison of any path.

Anyway, I’ll try to follow the path you pointed me to with the law of cosines and let you know if that will make my map richer! I look forward to your next article

Regards,

Andrea