directed segment
Let a line segment. The directed segment is to be taken the segment with a direction (similar to vectors). The defining property is then
(which relates to the property of vectors stating that and have opposite direction and same modulus).
The addition of directed segments is done in a similar fashion of vectors, so the above relation is equivalent to
where represents any segment of the form . If it is stated that we will work with directed segments, it’s customary to omit the overlining and to just write , convention we will follow now.
Notes.
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The definition does not says is either positive or negative (and it does not really matters doing so). It merely states that traveling the segment on different directions give different signs.
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It does not make sense to compare signs of non-collinear segments. So if are not on the same line (or parallel lines) we cannot relate the signs of and .
It can be proved considering cases that no matter the relative position of three points on a line, the following equality holds:
In the above picture . Notice that goes to the left since is the segment that starts at and ends at . Also, taking gives which is consistent with the earlier remarks.
Just like undirected segments in Euclidean geometry (and unlike vectors), directed segments can be divided to obtain a ratio. Such ratio is the number obtained dividing the undirected segments, but taking signs int oaccount (ratio of two segments with the same direction is positive, and negative otherwise).
Given two points on a line, we can locate any other point on the line considering the ratio . In other words, if and only if . Moreover, to each point corresponds an extended 11We use extended reals to avoid dealing with separate cases, allowing us to deal also with points at infinity. real and to each extended real corresponds a point such that .
Notice that is positive when and have the same direction, which happens if and only if is between and . If lies outside , then and have negative signs and so the ratio will be negative.
Title | directed segment |
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Canonical name | DirectedSegment |
Date of creation | 2013-03-22 14:56:59 |
Last modified on | 2013-03-22 14:56:59 |
Owner | drini (3) |
Last modified by | drini (3) |
Numerical id | 9 |
Author | drini (3) |
Entry type | Definition |
Classification | msc 51F99 |
Classification | msc 51M25 |
Related topic | CevasTheorem |
Related topic | Midpoint |