Does zero vector only exist in dim 0?

The zero vector is the additive identity; it exists in all Rⁿ, and is defined as [itex]<0, 0, ... , 0>[/itex].

EDIT: My statement is misleading. "The zero vector" does not exist in every Rⁿ; rather, for every n, there exists an element of Rⁿ which is the additive identity and is called the zero vector.

 

But in what sense is the vector "which is the additive identity and is called the zero vector" NOT the "zero vector"?

But, having written that, I do see your point. It is a mistake to think that there exists a single vector that is in all vector spaces. In, for example, R³, the "zero vector" is (0, 0, 0). But in the vector space of "all polynomials of degree 2 or less" the "zero vector" is the function f(x) such that f(x)= 0.

Every vector space contains a zero vector but it is misleading to talk about "the" zero vector as if all zero vectors were the same thing.

(But all vector spaces of dimension n are isomorphic and that isomorphism maps the zero vector in one to the zero vector in the other. A "natural isomorphism" from R³ to the space of polynomials of degree 2 or less takes (a, b, c) to ax^2+ bx+ c and so maps (0, 0, 0) to 0x^2+ 0x+ 0= 0 for all x.)