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In a french book : Linear Algebra (6th edition) from Joseph Grifone, who is expected to be the best book on the topic in french language, it is stated, in first chapter : "Remark: one does not add two vectors of different origin."

I never heard about this at school. In particular, what we learned is that if wish to make the sum of a vector $\vec{u}$ and a vector $\vec{v}$ (who are distinct in the space, without a ame origin), one just makes the geometrical construction of moving the $\vec{v}$ to the end of the $\vec{v}$ to deduce the vectorial sum.

My question is thus : it is true that one does not add two vectors of different origin ?

Or is it a specific context ?

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    $\begingroup$ What was being said at school sound reasonable. It always do for two vectors of different origins. $\endgroup$
    – Mikasa
    Feb 4 at 8:56
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    $\begingroup$ The author of your book would consider the vectors in your example to have the same origin. I think it's stupid to put a remark like that in there without context, and I think you can safely ignore that remark because once you get the context you'd never do what you're warmed about here anyway $\endgroup$
    – Bananach
    Feb 4 at 9:06
  • $\begingroup$ Thanks. But I'm really puzzled that he may have put a stupid comment, because the author is supposed to the "best teacher" on this topic. $\endgroup$ Feb 4 at 9:11
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    $\begingroup$ The mathematical answer is: vectors actually don't have origins, so there is no problem! $\endgroup$ Feb 4 at 10:44

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When you introduce examples of geometric vectors as arrows (an origin and an endpoint), and you don't want to introduce complicated things such as equivalence classes just yet, then you cannot add two vectors with different origins, because how would you decide where to put the origin of the sum ?

Also, in the specific example of forces acting on an object, two forces being applied on different points may not have the same effect as a single force acting from somewhere. Two opposing forces can rotate an object while their sum would a zero force and wouldn't do anything.

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