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Is a line segment by definition directed? Does directed mean it is in movement? If its a segment of a line, doesn't it neccessarily follow the rest of the line?

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    $\begingroup$ You can think of it as a line segment marked with an imaginary arrowhead pointing toward one end or the other. In general one shouldn't think of this as "movement". $\endgroup$ Apr 30 '16 at 12:42
  • $\begingroup$ So not moving, but if you were to move, you could only go one direction? $\endgroup$
    – Anon
    Apr 30 '16 at 12:57
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    $\begingroup$ For most purposes, you shouldn't think of this as "movement", you should think of it as a label. If you know what the orientation of a triangle is, this is the 1-dim. analog of that. (In fact, an orientation of a triangle determines orientations, i.e., "imaginary arrowheads", of each of its sides.) $\endgroup$ Apr 30 '16 at 12:58
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    $\begingroup$ Think of it as a vector based at one endpoint and "directed" toward the other endpoint. It is analogous to the difference between an unordered pair (the endpoints of a line segment) v.s an ordered pair (of endpoints of a directed line segment). $\endgroup$
    – hardmath
    Apr 30 '16 at 13:07
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Given two points $a$, $b$ in a real vector space $V$ the segment $[a,b]$ is the convex hull of these points, in other words, the set $$\sigma:=\{(1-t) a+tb\>|\>0\leq t\leq1\}\ .$$ If one of $a$,$b$ is declared initial point, and the other endpoint of $\sigma$ then the segment becomes directed. There is not more to it.

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