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单词 MutualPositionsOfVectors
释义

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

    a=kb.

    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.

    Remarks

    • 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

    ab=0,

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

    Remarks

    • 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

    cosθ=ab|a||b|.

    The angle is chosen so that  0θπ.

Titlemutual positions of vectors
Canonical nameMutualPositionsOfVectors
Date of creation2013-03-22 14:36:24
Last modified on2013-03-22 14:36:24
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id25
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 15A72
Related topicAngleBetweenTwoLines
Related topicDirectionCosines
Related topicOrthogonalVectors
Related topicPerpendicularityInEuclideanPlane
Related topicMedianOfTrapezoid
Related topicTriangleMidSegmentTheorem
Related topicCommonPointOfTriangleMedians
Related topicFluxOfVectorField
Related topicNormalOfPlane
Definesparallel
Definesparallelism
Definesperpendicular
Definesperpendicularity
Definesdiverging
Definesnormal vector
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更新时间:2025/5/4 6:13:28