>>12915597Vectors are best understood as elements of a Vector Space.
You can't really define them otherwise.
What is a vector space?
https://en.wikipedia.org/wiki/Vector_space#DefinitionNotions of length and distance are not essential properties of Vector Spaces. Only Normed Vector Space have these notions.
> Is it a line?No. A line is a collection of vectors (a subspace of dimension 1).
> Is it direction?A direction can be viewed as the half-line {c v | c>0} if the underlying field is the Reals (or some other totally ordered Field I guess)
>Is it speed?The tangent (aka speed) of a curve in R^n at a vector p (better viewed as a point) is a vector v. That vector v is best viewed as an arrow starting at p and ending at p + v.
