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Vectorquantitiesalsosatisfytwodistinctoperations,vectoradditionandmultiplicationofavectorbyascalarWebeginbyreviewingthebasicalgebraic operationsbetweenvectorsinthree-dim-ensionalspaceR3;see[10]fordetails.Theconceptofavectorinthreedimensionsisnotmateriallydifferentfromthatof avectorintwodimensionsThesequantitiesarecalledvectorquantitiesTheonlychangeisthatvectorshavethreecomponentsTheprecisemathematical statementisthat:Geometricdefinitionofvectors:AvectorisadirectedlinesegmentThelengthofavectorvissometimescalleditsmagnitudeorthenormofv OpenStaxLocatepointsinspaceusingcoordinatesWritethedistanceformulainthreedimensionsThissuggeststhatwestartthestudyingofCalculuswithd geometry,beginningwiththecoordinatesystemandvectorsThree-dimensionalvectorsThreedimensionalCartesiancoordinatesystemLearningObjectivesThis suggeststhatwestartThedeterminantofthematrix(a,b,c),thatiswhosecolumnsarethevectorsa,bandc,isdefinedtobethesignedvolumeofthe parallelepipedwhoseverticesare0,a,b,c,Vectorsinthree-dimensionalspacevectorsformanabeliangroupunderaddition;theidentityvectorisandtheinverse ofxis x.Itisstillaquantitywithmagnitudeanddirection,exceptnowthereisonemoredimensionFindaformulaforthesetofpoints(w,x,y,z)in4–dimensional spacethatareatadistanceofunitsfromthepoint(5,3,–2,1)Weshallusecolumnvectornotationv=v1v2v3=(v1,v2,v3)T∈RThestandardbasisvectorsof R3aree1=i=,e=j=,e=k=()3-dgeometryisintimatelyconnectedtoCalculusandCalculustechniquescanbeusedtounderstandcertainproperties ofdimensionalobjectsAswehavelearned,thetwo-dimensionalrectangularcoordinatesystemcontainstwoperpendicularaxes:thehorizontalx-axisTHREE DIMENSIONALGEOMETRYHence,from(1),thedc’softhelineare,,abclmnabcabcabc=±=±=±++++++where,dependingonthe1 VectorsinTwoandThreeDimensionsInthesenotes,vectorswillbedenotedbyanunderlinedletterasfollows:v:Intheliteratureothernotationsareused,suchas v,v, v,Vectorsinthreedimensions.TheUseofVectorsinPhysics.Writetheequationsforsimpleplanesandspheres.VectorsareusefultoolsforsolvingtwodimensionalproblemsInthreedimensions,asintwo,vectorsarecommonlyexpressedincomponentform,\(v=x,y,z\),orintermsofthestandardunitvectors,\ (xi+yj+zk\)PropertiesofvectorsinspaceareanaturalextensionofthepropertiesforvectorsinaplaneThecombinationsofiandj(ori,j,k)produceallvectors vintheThree-DimensionalCoordinateSystemsPerformvectoroperationsin\(\mathbb{R}^{3}\)Describethree-dimensionalspacemathematicallyFromthe lastsectionwehavethreeimportantideasaboutvectors,(1)vectorscanexistatanypointPinspace,(2)vectorshavedirection3-dgeometryisintimately connectedtoCalculusandCalculustechniquescanbeusedtounderstandcertainpropertiesofdimensionalobjectsThedistance||xy||betweenxandyinRNotice howeasilywemovedintothreedimensions!Let\(v=x1,y1,z1\)and\(w=x2,y2,z2\)bevectors,andlet\(k\)beascalarThreenumbersareneededto representthemagnitudeanddirectionofavectorquantityinathreedimensionalspace.(Thissetis"4–sphere"withradiusandcenter(5,3,–2,1).)Solution:We wantthedistancefrom(w,x,y,z)to(5,3,–2,1),(w–5)2+(x–3)2+(y+2)2+(z–1)2,tobesoIfyouarehikingandsaythatyouaremiNNWofyourcampyouare specifyingavector