parallelogram principle

  • A starting for learning vectors is to think that they are directed line segments. Thus a vector u has a direction and a length (magnitude) and nothing else. Therefore, if two vectors u and v have a same direction and a same length, one can consider them identical (and denote  u=v). So the location of a certain vector in the plane (or in the space) is insignificant; in fact one may also think that this vector consists of all possible directed line segments having a common direction and a common length.

  • However, a vector u as an infinite setMathworldPlanetmath of directed segments is quite uncomfortable to handle, and one can choose from all possible representants of u one individual directed segment AB, i.e. a line segmentMathworldPlanetmath directed from a certain point A (the initial point) to another certain point B (the terminal point). Although AB is only a representant of u, one may write  AB=u  or  u=AB.

  • For describing a vector u, it’s convenient to know its position in the coordinate systemMathworldPlanetmath of the plane (or the space); there one can say e.g. how great a displacement u means from left to right (i.e. in the direction of x-axis) and how great from below upwards (i.e. in the direction of the y-axis); those displacements may be expressed with two numbers.  One may for example write

    u=(+5-1), (1)

    where the first (upper) number +5 tells that the vector leads 5 length-units to the right and the second (lower) number -1 that it leads 1 length-unit downwards.

Addition of vectors

Since the vector may be interpreted as a of a horizontal displacement and a vertical displacement, it’s meaningful that by the additionPlanetmathPlanetmath of two vectors the horizontal displacements are summed and likewise the vertical displacements.  Accordingly, if we have

v=(+1-3), (2)

then the sum of the vectors (1) and (2) is


which result means a vector leading 6 length-units to the right and 4 down.


When we set the vectors u and v one after the other, as in the above picture, and take the sum vector from the initial point of the first addend to the terminal point of the second addend, then both the horizontal and the vertical displacements are respectively added.  The addition rule as a formulaMathworldPlanetmathPlanetmath using the points is


Note, that the sum vector u+v can be also obtained as the diagonal vector of the parallelogramMathworldPlanetmath with one pair of opposite sides equal to u and the other pair of opposite sides equal to v.  The parallelogram picture illustrates also that the vector addition is commutative, i.e. that  u+v=v+u.


If we think the second (dashed in the third picture) diagonal of the parallelogram, it is halved by the first (blue) diagonal, since the diagonals of any parallelogram bisect each other (see parallelogram theorems); as well the (blue) diagonal representing the sum u+v is halved into two equal vectors (better: directed segments) m=12(u+v).  In the triangleMathworldPlanetmath ABC, the vectors u, v, m may be called two side vectors and a median vector, all having the common initial point A.  Thus we can write the

Theorem.  In a triangle, the median vector emanating from a certain vertex is the arithmetic meanMathworldPlanetmath of the side vectors emanating from the same vertex.

Title parallelogram principle
Canonical name ParallelogramPrinciple
Date of creation 2013-03-22 17:47:04
Last modified on 2013-03-22 17:47:04
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 16
Author pahio (2872)
Entry type Topic
Classification msc 53A45
Synonym addition of vectors
Synonym vector addition
Synonym sum vector
Related topic CommonPointOfTriangleMedians
Related topic ProvingThalesTheoremWithVectors
Defines directed line segment
Defines sum of vectors
Defines side vector
Defines median vector
Defines diagonal vector