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In greek, we learn that a vector in 3D space has three basic characteristics, namely,

  • μήκος --> magnitude (length)
  • διεύθυνση --> a line the vector is on
  • φορά --> the way the vector faces on that line

So, a vector is a line segment having direction. (Wikipedia)

It can be seen that the only thing that distinguishes a vector from a line segment is the way the vector faces on the line that it is on.

In greek referring to both last two characteristics, we say 'κατεύθυνση' which translates to direction. It seems though, all three words translate to direction, so, I was wondering what to call the line the vector is on and the way the vector faces on that line if needed to separate them.

To be clear, the way the vector faces given a line can be one of two things. If we arbitrarily pick a plus and minus side of the line, the vector faces the negative side when to draw it we start at a point on the line and draw a line segment of some length towards the negative side.

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  • $\begingroup$ I would say a vector is parallel to a line, not that it is on the line. Thinking that a vector has a particular location leads to confusion later when you need to understand that the vector from $(2,0)$ to $(3,2)$ and the vector from $(0,0)$ to $(1,2)$ are the same vector. $\endgroup$ – David K Feb 12 '19 at 3:55
  • $\begingroup$ Thank you for your comment! When mentioning the line the vector is on I was not thinking about location. I was thinking about orientation. I would point it out in my answer and question. $\endgroup$ – giannisl9 Feb 12 '19 at 4:26
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    $\begingroup$ "Parallel to" conveys the idea of orientation without implying anything about location. That's why I like those particular words. $\endgroup$ – David K Feb 12 '19 at 4:28
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A vector in 3D space can be fully described by three characteristics.

Magnitude which comes from ordering vectors according to size.

Orientation which comes from rotating the axes of a reference frame.

Sense which comes from associating the vector with a set of half lines.

Orientation and sense fully describe direction.

We actually care for the orientation of the line the vector is on and determining its sense (the set of all half lines who have the same orientation and by pairs belong to the same part of the half plane they define) is equivalent to the way the vector faces on that line.

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