mutual positions of vectors

In this entry, we work within a Euclidean space E.

  1. 1.

    Two non-zero Euclidean vectors a and b are said to be parallelMathworldPlanetmathPlanetmathPlanetmath, denoted by  ab,  iff there exists a real number k such that


    Since both a and b are non-zero, k0.  So is a binary relationMathworldPlanetmath on on  E{0}  and called the parallelism.  If  k>0,  then a and b are said to be in the same direction, and we denote this by  ab;  if  k<0,  then a and b are said to be in the opposite or contrary directions, and we denote this by  ab.


    • Actually, the parallelism is an equivalence relationMathworldPlanetmath on  E{0}.  If the zero vectorMathworldPlanetmath 0 were allowed along, then the relationMathworldPlanetmath were not symmetricMathworldPlanetmathPlanetmathPlanetmathPlanetmath (0=0b  but not necessarily  b=k0).

    • When two vectors a and b are not parallel to one another, written  ab,  they are said to be diverging.

  2. 2.

    Two Euclidean vectors a and b are perpendicularMathworldPlanetmathPlanetmath, denoted by  ab,  iff


    i.e. iff their scalar productMathworldPlanetmath vanishes.  Then a and b are normal vectors of each other.


    • We may say that 0 is perpendicular to all vectors, because its direction is and because  0b=0.

    • Perpendicularity is not an equivalence relation in the set of all vectors of the space in question, since it is neither reflexiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath nor transitiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

  3. 3.

    The angle θ between two non-zero vectors a and b is obtained from


    The angle is chosen so that  0θπ.

Title mutual positions of vectors
Canonical name MutualPositionsOfVectors
Date of creation 2013-03-22 14:36:24
Last modified on 2013-03-22 14:36:24
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 25
Author pahio (2872)
Entry type Definition
Classification msc 15A72
Related topic AngleBetweenTwoLines
Related topic DirectionCosines
Related topic OrthogonalVectors
Related topic PerpendicularityInEuclideanPlane
Related topic MedianOfTrapezoid
Related topic TriangleMidSegmentTheorem
Related topic CommonPointOfTriangleMedians
Related topic FluxOfVectorField
Related topic NormalOfPlane
Defines parallel
Defines parallelism
Defines perpendicular
Defines perpendicularity
Defines diverging
Defines normal vector