Vector indexes.

Question

For a vector of two dimensions I can use $\vec x = (x_a, x_b)$ or $\vec x = (x_x, x_y)$, if no letters are more appropriate.

What if I need to name more dimensions than letters in the alphabet? What's better, $\vec x = (x_0, x_1, x_2, ... x_{29})$, or $\vec x = (x_1, x_2, x_3, ... x_{30})$? If none is better than the other, which one's most used, and why?

Answer 1

Usually the following is adopted for $\vec x\in \mathbb R^n$

$$\vec x = (x_1, x_2, x_3, \ldots, x_{n})$$

but you can use also other notations declaring that in order to avoid confusions.

Answer 2

We usually don't care whether indices start at $0$ or $1$ (except in the sense we'd rather start with our favourite if it doesn't matter, and the old joke is that set theorists start at $0$ and non-Peano number theorists start at $1$). I say usually, because there are a few cases where we do:

• If you're working in a Riemannian geometry with one dimension having the wrong sign to make the usual Euclidean contribution to the squared norm on vectors, it makes sense to put that Cartesian coordinate at the start, and conventionally its index is $0$. While sci-fi often calls time the fourth dimension, physicists tend to think of it as the zeroth. (That being said, occasional texts make time $x_4$ rather than $x_0$, or perhaps that should be $x^0$, but that's another story.)
• If the $i$th entry of a vector, or $ij$ entry of a matrix etc., is to be written as a function of the index or indices, use a convention that simplifies things. For example, the Thue-Morse sequence should start at $0$ (actually, that's a good idea with sequences in general), as should ladder-operator matrices.
• If there's programming involved, use the same indexing for your mathematical exposition as in the code itself, which varies by language. For example, Python starts at $0$, whereas R starts at $1$.