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If you take as a concrete example the affine space $\mathbb{R^2}$, the vector subspace $V = \{(x,y)|x=y$ and $x,y \in \mathbb{R} \}$ and the affine point $(2,3)$, you get the affine subspace $\{(x+2,y)|x=y$ and $x,y \in \mathbb{R} \}$.

I used this example to imagine the sum of an affine point and a vector as an affine point and am comfortable with this. However, I am still having troubles trying to imagine the difference of two affine points as a vector. Could someone illustrate this with my example? I illustrated the sum of a vector and a point as translating the point in $\mathbb{R^2}$ using all the vectors of the vector subspace.

Note: I don't want a proof or anything, just a visualisation or some argument as to why this is logical, I know it follows out of $x \in Y = p + W \Rightarrow x = p+w => w = x-p$.

Thanks in advance!

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up vote 2 down vote accepted

An affine space can be thought of as a vector space in which you've forgotten which point is the origin. Say it is proposed to add two points. If you knew which point is the origin, you'd draw arrows from the origin to each of the two points to be added, then complete the parallelogram and find the sum. But no point has any better claim to being the origin than any other, so you can't do that. But you can still draw an arrow from one point to another, and that's what it means to say you add a vector to a point in the affine space to get another point in the affine space. But it's also what is meant by saying one point in the affine space minus another point in the affine space is a vector. The arrow you draw from a point $A$ to a point $B$ is the vector you need to add to $A$ to get $B$; hence it is the vector you get when you subtract $A$ from $B$. The vectors belong to a space in which you do know which point is the origin: the origin is the vector of length $0$ that you draw from any point in the affine space to itself.

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It is also of interest to notice that although you cannot in geneneral take a linear combination of points in an affine space, because no particular point is the origin, you can still take an affine combination. An affine combination is a linear combination in which the sum of the coefficients is $1$. Let's say Adam thinks one point is the origin and Even thinks the other point is, and they compute their linear combinations accordingly. They will get different results, unless the sum of the coefficients is $1$. In that case, they will agree. – Michael Hardy Oct 1 '11 at 16:53
Great, I got it now, thanks :) – user12205 Oct 1 '11 at 20:41

How much does the position of the point $q$ differ from the position of the point $q$? Well, by the vector $\overrightarrow{pq}$ of course! Hence it makes sense that the difference $q-p$ is $\overrightarrow{pq}$.

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