vectors-matrics-coordinate transformations .pdf

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1、S.Widnall16.07DynamicsFall2009LecturenotesbasedonJ.PeraireVersion2.0LectureL3-Vectors,MatricesandCoordinateTransformationsByusingvectorsanddefiningappropriateoperationsbetweenthem,physicallawscanoftenbewritteninasimpleform.SincewewillmakingextensiveuseofvectorsinDynamics,wewillsummarize

2、someoftheirimportantproperties.VectorsForourpurposeswewillthinkofavectorasamathematicalrepresentationofaphysicalentitywhichhasbothmagnitudeanddirectionina3Dspace.Examplesofphysicalvectorsareforces,moments,andvelocities.Geometrically,avectorcanberepresentedasarrows.Thelengthofthearrowre

3、presentsitsmagnitude.Unlessindicatedotherwise,weshallassumethatparalleltranslationdoesnotchangeavector,andweshallcallthevectorssatisfyingthisproperty,freevectors.Thus,twovectorsareequalifandonlyiftheyareparallel,pointinthesamedirection,andhaveequallength.Vectorsareusuallytypedinboldfac

4、eandscalarquantitiesappearinlightfaceitalictype,e.g.thevectorquantityAhasmagnitude,ormodulus,A=

5、A

6、.Inhandwrittentext,vectorsareoftenexpressedusingthe−→arrow,orunderbarnotation,e.g.A,A.VectorAlgebraHere,weintroduceafewusefuloperationswhicharedefinedforfreevectors.MultiplicationbyascalarI

7、fwemultiplyavectorAbyascalarα,theresultisavectorB=αA,whichhasmagnitudeB=

8、α

9、A.ThevectorB,isparalleltoAandpointsinthesamedirectionifα>0.Forα<0,thevectorBisparalleltoAbutpointsintheoppositedirection(antiparallel).Ifwemultiplyanarbitraryvector,A,bytheinverseofitsmagnitude,(1/A),weobtainaun

10、itvectorwhichisparalleltoA.Thereexistseveralcommonnotationstodenoteaunitvector,e.g.Aˆ,eA,etc.Thus,wehavethatAˆ=A/A=A/

11、A

12、,andA=AAˆ,

13、Aˆ

14、=1.1VectoradditionVectoradditionhasaverysimplegeometricalinterpretation.ToaddvectorBtovectorA,wesimplyplacethetailofBattheheadofA.ThesumisavectorCfromth

15、etailofAtotheheadofB.Thus,wewriteC=A+B.ThesameresultisobtainediftherolesofAarereversedB.Thatis,C=A+B=B+A.Thiscommutativepropertyisillustratedbelowwiththeparallelogramconstruction.Sincetheresultofaddingtwovectorsisalsoavector,wecanconsiderthesumofmultiplevectors.Itcaneasilybeverifiedth