In what Structure point addition is not allowed and that makes points different from vectors.I mean in any Field or even Group i can add without problem but i have seen people saying point addition is not allowed and that is the main difference between vectors.Can someone show this to me in a more mathematical way and not just words .I mean rigorously with definitions as to what a point is?
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1An affine plane (or even just line) does not have a designated origin ... – Hagen von Eitzen Nov 02 '15 at 15:27
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1In some sense this is an arbitrary distinction. If we really want to make it impossible to add points, we can work in affine space instead of a vector space, where there is no origin. – Matt Samuel Nov 02 '15 at 15:27
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This is more of a geometrical understanding of points as objects. – Jean-François Gagnon Nov 02 '15 at 15:28
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Let’s first get an intuition, why adding points should not be possible. Consider a plane or a white piece of paper. If I mark two points $P$ and $Q$ on this paper where would the sum $P + Q$ be? It should not depend on the orientation or size of the paper: If I rotate the paper by $180^\circ$ or cut off some of the paper, the sum should remain where it is. This rules out constructions like “measure the distance from the bottom and right hand edges and add those”. In fact, the sum should not depend on any arbitrary choices, so you also can’t choose a point $O$ as the origin,then add the vectors from $O$ to $P$ and from $O$ to $Q$ and take as $P + Q$ the translation of $O$ by this vector, because choosing a different point as the origin will result in a different sum $P + Q$. (If you don’t believe me, just take a piece of paper and try an example.)
So adding points should not be possible. There is a slight “problem”, though: If we want to represent the points on the plane, basically the only way that allows us to compute with them is to choose a coordinate system and describe the points by their coordinates. But in this representation, we seemingly can add points: The sum of $P_1 = (x_1, y_1)$ and $P_2 = (x_2, y_2)$ is “obviously” $(x_1 + x_2, y_1 + y_2)$. This is because a coordinate system comes with a natural origin, the point $O = (0, 0)$. (And a coordinate system with a different origin gives a different sum, so maybe the sum isn’t so obvious, after all.)
The solution is to simply disallow the addition of points. There is nothing that stops you from adding coordinates, but the result is not meaningful, if your coordinates represent points.
(The difference of points is another beast: $P - Q$ is not a point but we can fruitfully interpret it as the vector from $Q$ to $P$. By having both points and vectors between them, you get affine space.)
Eike Schulte
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We cant allow addition but can allow subtraction ? i mean its the same as adding,. – Jam Nov 03 '15 at 12:24
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2No, it is not the same thing in general. To define subtraction in terms of addition (or vice versa), you need to be able to say what the negative of something is. (Just look at the formula: $x - y = x + (-y)$ – here we use the negative of $y$.) For points $P$, we don’t have a concept of $-P$, so we can’t do that. However, we can interpret $P - Q$ as the vector from $Q$ to $P$ – but note: the difference is a vector, not a point. – Eike Schulte Nov 03 '15 at 14:32
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In general, point addition does not have any meaning in topological spaces and manifolds. However, some very specific spaces under those descriptions, such as ordinary three-dimensional euclidean space, do admit an additive structure. In the case of 3D euclidean space, for example, we can choose an origin and then define addition on vectors from that origin to points in the space.
John Bentin
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