However, \\(\mathbb{N}^n\\), which consists of all n-tuples of natural numbers, definitely exists, but it is not considered to be a vector space.

That's because in a vector space, we expect certain operations to always be defined: we should be able to add two vectors to get another, take the negative of vector to get its 'opposite', and scale a vector by a number to get another vector.

Let's choose a 3-tuple in \\(\mathbb{N}^3\\), say x = (4,2,3).

If we try to take the negative of x, that would be the 3-tuple (-4,-2,-3). But that doesn't belong to \\(\mathbb{N}^3\\), where components must be non-negative.