Dear fellas, please have in mind that during this exposition I am thinking of simple 3D physical vectors: I know not all 3 numbers form a vector. In addition to being 3 numbers, they must transform correctly under a change in coordinate systems. I can superficially understand why this is true. For example, my age and the age of my two brothers are numbers, which could be the components of a vector, but they are not, since they don't transform correctly under rotation, they don't get "mixed up". (I don't get older, and my brother doesn't get younger when the coordinate system changes.) But I'm trying to understand this on a deeper level. I don't want to merely say "vectors are 3 numbers that transform correctly under Lorentz transformations", I want to be able to find a deeper principle, such as, for example, "vectors are 3 numbers, which, by themselves, represent a quantity which has direction", or "vectors are three numbers that have some form of dependence to space" or something like that. In summary: how can I state this idea (precisely) not in terms of "numbers that behave in a certain way under an operation" but in terms of "numbers that are such a way". I hope I expressed myself well enough to be understood. Thank you for your help!