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## Main Question or Discussion Point

*[Moderator's Note: Spun off from previous thread due to increase in discussion level to "A" and going well beyond the original thread's topic.]*

A vector space has no origin to begin with ;-)).An affine space is basically a vector space without an origin.

An affine space is a set of points and a vector space ##(M,V)##. Then you have a set of axioms which boils down to what you know from Euclidean geometry, i.e., to a pair of points ##A,B \in M## there's a vector ##\overrightarrow{AB}## (an arrow connecting ##A## with ##B##). If you have two other points such that the arrow ##\overrightarrow{CD}## is parallel to ##\overrightarrow{AB}## these to vectors are by definition the same.

If you have an arbitrary vector ##\vec{v} \in V## and a point ##A \in M## there's a unique point ##B## such that ##\overrightarrow{AB}=\vec{v}##.

Then you can add vectors ##\vec{v}_1,\vec{v}_2## and the meaning in connection with the arrows connecting points is ##\vec{v}_1+\vec{v}_2=\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}##. Here we have chosen ##\overrightarrow{AB}=\vec{v}_1## and ##\overrightarrow{BC}=\vec{v}_2##.

The dimension of the vector space is by definition the dimension of the affine space.

If your vector space has in addition a scalar product, you can measure distances and angles between points in the usual way using this scalar product. That's then an Euclidean affine space. E.g., space as described by Euclidean geometry is simply a 3D (or if you are restricted to a plane 2D) Euclidean affine.

In special relativity you have a 4D pseudo-Euclidean affine space to describe spacetime. Here the Minkowski product is not positive definite and thus not a proper scalar product, but it has the signature (1,3) (or (3,1) depending on the convention of your book). Otherwise it's pretty much analoguous to a Euclidean affine space.

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