Definition of Vectors
Definition
A sequence of numbers is called a vector.
Description
In the general curriculum, a vector is learned as a ‘geometric object with magnitude and direction’. Since it’s the concept you first come across in physics, you inevitably become familiar with vectors of $3$ dimensions or less.
$$ (3,4) = \begin{bmatrix} 3 \\ 4 \end{bmatrix} $$ $$ (x,y,z) = \begin{bmatrix} x \\ y \\ z \end{bmatrix} $$
However, vectors can actually be generalized to more coordinates. It’s sufficient to just list more numbers below, for example, a $4$-dimensional vector considering time $t$ can be denoted as follows.
$$ (t,x,y,z) = \begin{bmatrix} t \\ x \\ y \\ z \end{bmatrix} $$
What does it mean to have vectors of more than $4$ dimensions? For instance, if you want to represent the position $(x,y,z)$ at time $t$ and the thermal energy $E$ for each oxygen molecule, you can extend it to $5$ dimensions as follows.
$$ (t,x,y,z,E) = \begin{bmatrix} t \\ x \\ y \\ z \\ E \end{bmatrix} $$
The point is, there’s no need to be overly afraid of the length of the vector, that is, the increase in dimensions. In the endless world of mathematics given under a specific format, such expansion of dimensions is natural and obvious. In the same manner, it’s possible to think of vectors generalized up to $n$ dimensions, usually denoted by boldface $\mathbf{x}$.
$$ \mathbf{x} = \left( x_{1}, \cdots , x_{n} \right) = \begin{bmatrix} x_{1} \\ \vdots \\ x_{n} \end{bmatrix} $$
From this simple definition, a $n$-dimensional vector is indistinguishable from a $n$-tuple. The farther you get from physics and closer to mathematics, the less you use expressions like $\vec{x}$ with arrows, and as you enter into abstract and general mathematics, you get precise and strict definitions without terms like ‘coordinates’ or ‘sequence’.