Generalizing to N dimensions is often seen as a pointless mathematical exercise because of …
… nothing interesting happens in the Nth dimension. Topologically, there is little difference between any spaces with more than 6 dimensions (because of h-cobordism, which shows that spaces with 6 or more dimensions can be “mapped” into each other). Fun fact: this has something to do with the (Abel-Ruffini)[https://en.wikipedia.org/wiki/Abel-Ruffini] theorem and the impossibility to find a close formula for a nth-degree polynomial, with n greater than 5. For spaces with 5 dimensions or lower, there is still some debate on how to work in those dimensions (see (low-dimensional topology)[https://en.wikipedia.org/wiki/Low-dimensional_topology]).
As far as adding a component to a vector, there are a lot of different things happening that make the exercise far from obvious. For example going from a 3-dimensional space to a (3+1)-dimensional space, you will build a (projective space)[https://en.wikipedia.org/wiki/Projective_space] which has its own set of rules, most notably the fact that there are no more parallel lines.
I do not think that reducing the problem to the tools (linear algebra) used to study the geometrical/topological properties of the space fully explains the mathematical misconceptions about this.
I’m okay with not fully correcting the misconceptions; I was only trying to give a taste of the complexity behind the mathematics I’ve seen. I was writing this for my high school self.
Blindly looking at h-cobordism, it says there is a sufficient condition which might not be met in this particular domain. I’ve seen domain experts use N dimensions and there’s even the problem of dimensionality reduction present; I’m inclined to believe that N dimensions are needed for the use cases I’ve seen.
Can you please update all the JS/Webfont links? My Chrome refuses to load HTTP content from HTTPS so all LaTeX expressions and everything don’t show up :(.