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The following was taken from https://en.wikipedia.org/wiki/Affine_space:

The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after you've forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps"[1]). Imagine that Alice knows that a certain point is the actual origin, but Bob believes that another point — call it p — is the origin. Two vectors, a and b, are to be added. Bob draws an arrow from point p to point a and another arrow from point p to point b, and completes the parallelogram to find what Bob thinks is a + b, but Alice knows that he has actually computed

p + (a − p) + (b − p).

How did Alice arrive at this conclusion. I tried drawing a diagram for the above situation and it seems that Bob's a+b would appear as a+b+p to Alice? Anyway I would really appreciate a geometrical picture and some explanation. Thanks in advance.

Kemish Jimenez
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2 Answers2

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Suppose that, in Alice's coordinates we have $P(1,2)$, $A(2,3)$ and $B(3,2)$.

Bob thinks the coordinates for $A$ are $A-P=(2,3)-(1,2)=(1,1)$, and that the coordinates for $B$ are $B-P=(3,2)-(1,2)=(2,0)$. Bob adds the two vectors (in his coordinates) and gets a result of $(3,1)$. So when he draws the result he makes an arrow that starts at $P$ and ends three units to the right and one above $P$ -- that is, Bob's arrow ends at the point Alice calls $(4,3)$.

If Alice adds $A$ and $B$ herself, she of course gets $(5,5)$. So Bob's result of $(4,3)$ is $(5,5)-(1,2)$, or in symbols $A+B-P = P+(A-P)+(B-P)$.

hmakholm left over Monica
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Alice and Bob agree on which points are $a$ and $b$, but disagree about the correspondence between points and vectors. Bob draws an arrow from $p$ to $a$ and thinks, "That's vector $a$." But $a$ is the name of a point; how did it become the name of a vector? Bob associates each point with the vector from $p$ to that point, which is exactly what we mean by "taking $p$ as the origin". Alice, not making any such identification, sees the arrow from $p$ to $a$ and thinks, "That's vector $a-p$."

This subtraction notation might seem odd, but it's sensible when combined with the addition notation $p+v$ for "the point you end up at if you start at point $p$ and make the movement represented by vector $v$"; evidently $p+(a-p) = a$. You'll also note that it's the same convention as we follow on the number line: to get from (point) $3$ to (point) $7$, you have to add (vector) $4$, and $4=7-3$.

So anyway, Alice sees Bob draw the usual parallelogram for vector addition and sees an arrow from $p$ to $a$ (vector $a-p$), an arrow from $p$ to $b$ (vector $b-p$), and an arrow representing the sum of those vectors (vector $(a-p)+(b-p)$), and that sum arrow placed with its starting point at $p$, so its ending point at $p+((a-p)+(b-p))$.

Bob identifies (point) $p$ to be (vector) $0$, and so simplifies this to $a+b$ (which, being a sum of points, wouldn't make sense without some such identification).