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What is the difference between Euclidean space and $\mathbb R^3$?

I have found in some books that they are the same, but in other references like Wikipedia, it says that a vector in $\mathbb R^3$ is a point in the Euclidean space, and the difference of two vectors in $\mathbb R^3$ is a vector in the Euclidean space.

user172798
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  • Its not an algebraic geometry question. It is a projective/affine geometry question. – Troy Woo Aug 30 '14 at 22:12
  • @TroyWoo : I've changed the tags. – Michael Hardy Aug 30 '14 at 22:13
  • Roughly speaking, in Euclidean space, you don't add two points; with the push of a vector, you move from one point to another. But if you choose a particular point as the "origin", the Euclidean space looks like a vector space: a vector starting from the origin corresponds uniquely to its end-point. – Troy Woo Aug 30 '14 at 22:16
  • And Euclidean space also has a metric, whereas $\mathbb R^3$ may have a natural metric only on each line separately. – Michael Hardy Aug 30 '14 at 22:18

2 Answers2

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Some books will tell you that they are the same, and some will speak of Euclidean spaces in higher dimensions. It's a matter of convention.

However, the space $\mathbb R^3$, when not assigned an inner product, is only a vector space, so that one cannot speak of angles and distances as one would in Euclidean space. And the space $\mathbb R^3$ has an origin, whereas in Euclidean geometry one does not single out a particular point to play a special role different from the roles of all other points.

Michael Hardy
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A concise mathematical term to describe the relationship between the Euclidean space $X = \mathbb E^n$ and the real vector space $V = \mathbb R^n$ is to say that $X$ is a principal homogeneous space (or ''torsor'') for $V$. This is a way of saying that they are definitely not the same objects, but they very much are related to each other.

In particular:

  • The objects $\mathbb E^n$ and $\mathbb R^n$ are exactly the same as sets of elements -- they both correspond bijectively to $n$-tuples of real numbers.

However (as mentioned by OP),

  • in the vector space $\mathbb R^n$ we are allowed to add any two vectors (using the ''tip to tail'' visualization), whereas in Euclidean space $\mathbb E^n$ there is no natural way to describe the process of ''adding'' two points. Instead, given two points $P,Q$ in $\mathbb E^n$ we can naturally define their difference $\vec{v} = P-Q$, which is a vector in $ \mathbb R^n$. This vector tells us how to get from point $Q$ to $P$.

  • in $\mathbb R^n$ there is a special ''zero vector'' $\vec{0} = (0,\ldots,0)$ which satisfies the additive property $\vec{0} + \vec{v} = \vec{v}$ for any $\vec{v}\in \mathbb R^n$, while in $\mathbb E^n$ there is no point that is somehow more special that the other ones -- i.e. the space is ''homogeneous'' meaning it looks the same around every point.

In the general context, we say that $X$ is a torsor for a group $G$ if $X$ is a copy of $G$ which ''forgets'' where the identity element is. This means the underlying sets are bijectively the same, and gives the group $G$ a natural action on the set $X$ in the same way that $\mathbb R^n$ acts on $\mathbb E^n$ (i.e. the action is free and transitive).

This blog post of John Baez discusses a few natural examples of torsors motivated by physics, and gives definitions in more detail if you are interested.

Harry Richman
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