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I have a general question about vectors and Euclidean space:

Suppose we are working in $\mathbb{R}^3$. In this space we can identify points. Choose an arbitrary point and label it $O$. Then we can consider the vector space $V_{O}$ of displacement vectors. Then consider $$E = \bigcup_{A \in \mathbb{R}^3} V_A$$

Note that $E$ is the set of free vectors which we can write as the union of bound vectors over points $A \in \mathbb{R}^3$.

Is $E$ just a union of vector spaces? More specifically:

  • In looking at $\mathbb{R}^3$ we transition from points to displacement vectors with some fixed origin $O$. Consider a vector $\textbf{v}$ and its parallel copies with respect to $O$. Then this is an equivalence class. Is this where the identification ends? Or should we also consider other equivalence classes with different origins? The set of all of the equivalence classes is forms a quotient space?
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I don't understand what it means to take this union. – Qiaochu Yuan Jul 7 '11 at 21:47
@Qiaochu Yuan: $E$ is the set of free vectors which is the union of bound vectors. – Damien Jul 7 '11 at 21:50
@Damien: the union of all these vectors can actually be made into a manifold; the collection of vectors based at a fixed point O is actually a vector space ( of dimension n; the same as that of $R^n$)usually called the tangent space of $R^n$, based at O; the union of all these spaces also has a name, and . Look up Tangent Bundle. BTW, tangent spaces based at any two different points are naturally isomorphic to each other; basically by translating vectors to the origin and then back. – gary Jul 7 '11 at 21:54

In some sense, it is just a union of vector spaces. However, as you point out, there is a natural correspondence between vectors in one vector space and vectors in another ("vectors that point in the same way and are of the same magnitude are the same"). If you want to regard them as just a union of vector spaces or as parametrized by the underlying space ($\mathbb{R}^3$) is entirely up to you! But in differential geometry you probably will want to consider vectors that "look the same, but with different origins" to be related.

The concept you are looking for is vector bundle. Another, perhaps more exciting example of a vector bundle would be the tangent vectors of a sphere (but in this case it is somewhat trickier, since the underlying space is not Euclidean).

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