directed segment

Let AB a line segmentMathworldPlanetmath. The directed segment AB¯ is to be taken the segment AB with a direction (similarMathworldPlanetmathPlanetmath to vectors). The defining property is then


(which relates to the property of vectors stating that v and -v have opposite direction and same modulus).

The additionPlanetmathPlanetmath of directed segments is done in a similar fashion of vectors, so the above relationMathworldPlanetmath is equivalentMathworldPlanetmathPlanetmathPlanetmath to


where 0 represents any segment of the form PP¯. If it is stated that we will work with directed segments, it’s customary to omit the overlining and to just write AB, convention we will follow now.


  • The definition does not says AB 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.

  • It does not make sense to compare signs of non-collinear segments. So if A,B,C are not on the same line (or parallel linesMathworldPlanetmath) we cannot relate the signs of AB,BC and CA.

It can be proved considering cases that no matter the relative position of three points A,B,C on a line, the following equality holds:


In the above picture AP+PB=AB. Notice that AB goes to the left since AB is the segment that starts at A and ends at B. Also, taking A=C gives AB+BA=AA=0 which is consistent with the earlier remarks.

Just like undirected segments in Euclidean geometryMathworldPlanetmath (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 A,B on a line, we can locate any other point P on the line considering the ratio AP/PB. In other words, P=Q if and only if AP/PB=AQ/QB. Moreover, to each point P corresponds an extended 11We use extended reals to avoid dealing with separate cases, allowing us to deal also with points at infinity. real r=AP/PB and to each extended real r corresponds a point P such that r=AP/PB.

Notice that AP/PB is positive when AP and PB have the same direction, which happens if and only if p is between A and B. If P lies outside AP, then AP and PB have negative signs and so the ratio will be negative.

Title directed segment
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 MidpointMathworldPlanetmathPlanetmathPlanetmath