vector

An ordered list of numbers, viewable as an arrow, a list, or a point.

A vector $\mathbf{v}$ in $\mathbb{R}^n$ is an ordered list of real numbers, $\mathbf v = (v_1, v_2, \dots, v_n)$ , which can be pictured three equally valid ways: as an arrow from the origin to $(v_1,\dots,v_n)$ (a directed segment whose length and direction matter but whose drawn location doesn't), as the list of coordinates itself (the form every algebraic rule in this course is defined on), or as a point, the single location those coordinates name. None of the three is the "real" vector — they're one object.