Why is it so that we consider multiplying a vector by a negative scalar as reversing the direction of it. I know that negative numbers reside in the opposite direction of positive numbers in the number line. I want a solid argument regarding what might the intuition be behind reversing the direction of a vector when multiplied by a negative scalar in any dimension (1, 2 or 3)
 A: This happens simply because $-v$ is the additive inverse of $v$: $-v$ is the only vector $∈ V$ such that $v +(-v)=0∈V$
A: I think at first at least it's best to motivate the algebra by thinking geometrically. From a geometric perspective, we start by thinking of directed line segments in a reasonably simple space like $2$- or $3$-dimensional Euclidean space. Vector addition then is just "placing end-to-end:" to add two vectors, we translate one so that its tail rests at the head of the other, and then connect the tail of the first to the head of the second.
This gives us an obvious notion of the "additive inverse" of a vector $v$: it's just the same vector but with head and tail switched (the "backwards" version of the original vector). Call this "$v_{back}$." The point, then, is the following:

In order to satisfy the basic arithmetic properties we want, it needs to be the case that $$(-1)v=v_{back}.$$

This amounts exactly to the idea that multiplying by a negative reverses direction.
So how do we argue this? Well, we first observe that $$v+(-1)v=(1)v+(-1)v=(1-1)v=0v=0.$$ But now since we know geometrically that $v+v_{back}=0$ as well, we have $$v+(-1)v=v+v_{back},$$ and so by subtracting $v$ from both sides of the above (or if you prefer, adding - say - $v_{back}$ to both sides) we get $$(-1)v=v_{back}$$ as desired. All the above reasoning relies on only very basic assumptions about how vectors and scalars interact; essentially, we have that "multiplying by a negative reverses direction" is the only possible way to wind up with the basic algebraic properties we want.
