Must vectors in $\mathbb{R}^n$ have their "tail" at origin? I was looking the definition for an $n$-sphere centered at origin with radius $r$: $$\mathbb{S}^n = \{v \in \mathbb{R}^{n+1} : ||v|| = r \}$$
Although I understand that the $||v|| = r$ condition refers to all vectors $v$ with their initial point at origin for this definition to make sense.
But I always thought that $||v|| = r$ means the vector $v$ has length $r$. Any vector, even those far away from the original, can have lenght $r$, right? E.g. when $n = 2$, the vector $u = (0,10) - (0,9)$ is such that $||u|| = 1$, but it is not part of the unit circle $\mathbb{S}^1$ centered at origin.
Question: what is the reason for my confusion? Is it because vectors in euclidean $n$-dimensional vector spaces should always have initial point at the origin? (edit: just realized this is rubbish since we can have vectors of the same length which are orthogonal)
 A: In one sense, yes; the tails of all vectors in a vector space are at the same point.
There is a related notion of an affine space, which is more or less the same thing as a vector space, but vectors can have tails at any point. We get a vector space by picking a point (it can be any point) and call it the "origin" -- then, the set of all vectors whose tail is at the origin becomes a vector space.
In another sense, no; we can think of vectors as being "unpinned" (not a technical term). A vector represents an abstract displacement that is not tied down to any particular point -- however, if we pick a particular point and call it the "origin", then we can produce a visual representation of the vector by drawing it as if it had a tail at the origin. And in this visualization, the zero vector always winds up simply pointing from the origin to itself.
(caution: the phrase "affine space" is often used to mean several different things)
A: Vectors, on their own, can be thought as starting anywhere in space. 
But usually we want to do more than that. We want to algebraic operations with them: we want to think of $\mathbb R^n$ as a vector space, even a normed space, or even an inner product space. These notions are defined naturally when we think of $\mathbb R^n$ as the set of vectors with start at the origin and end at a fixed point. 
