The paper proposes generalization of geometric notion of vectors concerning dimensionality of the configuration space. Trivial mapping between an algebraic vector space and Euclidean space is possible as the Euclidean space is able to configure all elements of the algebraic vector space. Such configuration relies on the notion of globally valid directions those satisfy the vector axioms upon their direct product with lengths. We prove that, certain type of ordered direction exists in each number of Euclidean dimensions along which elements of vector spaces can be interpreted. We show that such general ordered directions equivalently exist at each point in Euclidean space and there exists a special metric for each kind of the ordered direction. An algebraic structure of addition and scaling exists for the direct product of such directions and path lengths along such directions. The path length is in terms of the special metric that comes with each dimension. We further show that this consideration satisfies the vector axioms and leads to the complete normed space within the Euclidean space. A mathematical framework is built with 3 lemmas, 8 theorems and a conjecture. Application of the framework to locally 3+1 dimensional universe leads to four fundamental versions as which a vector can exist geometrically. Thus any physical quantity in the universe should come in four versions of vectors as long as the underlying structure of spheres exists for the ordered directions.