How does one see a vector through the lens of set theory? As just the end point? Or the set of points on the segment between the starting point and ending point of the vector? Or just the start point and end point?
Example:
Let $$\vec v=\langle x_1,x_2,x_3\rangle$$ start at the origin and end at $p=(x_1,x_2,x_3)$.
Then $s=\{(x_1t,x_2t,x_3t)|t\in[0,1]\}$ is the segment between the origin and $p$.
And thus if $|s|$ denotes the length of $s$, then $|s|=|\vec v|$.
But in terms of a set, what would $\vec v$ be? Would there be use to this sort of definition?
From the point of view of set theory, a vector in a finite dimensional vector space is simply an ordered $n$-tuple: for example $$ \begin{split} \vec v=(x_1,x_2)\triangleq\{\{x_1\},\{x_1,x_2\}\}&\qquad n=2\\ \vec v=(x_1,x_2,x_3)\triangleq(x_1,(x_2,x_3))&\qquad n=3 \end{split} $$ and so on (i.e. proceeding by induction), one gets the set theoretical structure of a vector: everything else, from the multiplication by scalars to the concept of space points, comes if you add further "structure" to the set of vectors so obtained.