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    $\begingroup$ Well...in principle, you could define a vector of, say, R^2 to be an equivalence class of "directed line segments". $\endgroup$ Commented May 11, 2010 at 12:26
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    $\begingroup$ That's verbatim how I learned the definition of vector. But the "equivalence class" part of it changes everything (and did not go over too well with many of the other students; it was junior-high after all...) $\endgroup$ Commented Aug 18, 2010 at 22:06
  • $\begingroup$ This was (more or less) the definition I heard when I was 7 or 8. I think it's great for a seven or eight-year-old, but probably not so great for an undergraduate mathematics major. :) $\endgroup$ Commented Oct 17, 2014 at 19:41
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    $\begingroup$ Could you say in more detail what's wrong with this one? In an affine space, a directed line segment is indeed the same thing as a tangent vector. And there's no need for equivalence classes—line segments based at different points live in different tangent spaces, so they shouldn't be identified (although all the tangent spaces are canonically isomorphic through translation). I certainly agree that it's harmful to give the impression that all vectors are directed line segments, but I think it's very true and useful to point out that all directed line segments are vectors. $\endgroup$ Commented Feb 15, 2015 at 4:06