Vectors. Vectors in Plane ( 2

Transcription

1 Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector s directed lie segmet The legth of tht lie segmet is the mgitude of the vector, the directio of tht lie segmet is prtilly represeted by the slope of the lie Fig Here it is importt wht is the strtig (iitil ) d wht is the edig (termil ) poit The rrow ottio used shows tht Q is the edig poit We ow represet vector with two coordites The first oe is the differece of x s of the poits, the secod the differece of the y s of the poits Precisely v x x, y y which is,4 for the vector i the figure which is compoet form of vector We lso write vector i mtrix form s colum mtrix,

2 v v v, v v Exmple Fig 7 5 v 7,,,5, 3 5, v 5 v Note: From ow o we shll use mtrix ottio which is more coveiet for the purpose of our course We lso use bold ottio for vectors d there is lso rrow covetio i prctice, ie, v, v mes the sme Every vector hs mgitude tht is clculted by the stdrd legth formul betwee two poits i the ple,

3 For exmple i the cse of the vector from the figure tht is v v v v v () v 4 Opertios with Vectors We defie dditio of vectors compoet-wise mer s u v d sclig ie, sclr multiplictio k u i the u v u v u v u v u k u k u k u where k i the secod defiitio is usully clled sclr The subtrctio would follow from these two There is geometric visuliztio of these opertio s o the figure 3, 4, d 5 below The first two visuliztios re ofte referred to s the rule of prllelogrm Fig 3 3

4 Fig 4 Fig 5 Note: I our textbook vectors re usully plced with iitil poit i coordite origi This is ofte referred to s the stdrd positio 4

5 Note: We c lso defie sclr multiplictio from the right u k s d it is obvious tht k u u k u uk k u u k Specil Vectors is referred to s zero or ull vector We lso hve the so clled uit vectors s follows, i, j Every vector c be decomposed or preseted s lier combitio of these uit vectors Lier combitio of vectors v, v,, v is vector tht c be writte s k v k v k v where k, k,, k re y rel umbers Exmple v i j See the figure 6 below 5

6 Fig 6 Vectors i -dimesiol Spce ( ) All the defiitios bove c be esily geerlized i cse of dimesios by simply hvig vector with directios (compoets) of these dimesios such s v v v v The opertios with vector we oted erlier hve the followig lgebric properties For ll u, v, w vectors i d ll sclrs k d l the followig holds: u v v u (commuttive property of vector dditio) u v w u v w (ssocitive property of vector dditio) 3 u u u (property of eutrl for the dditio) u u u u (iverse property of vector dditio) 4 6

7 k u v k u k v (distributive property of sclr multiplictio with respect to 5 vector dditio) k l u k u l u (distributive property of sclr multiplictio with respect to rel 6 umber dditio) k l u k l u (homogeeity property of sclr d rel mulptiplictio) 7 8 u uu (property of eutrl for sclr multiplictio) All of these properties re esy to verify We lso defie two vectors to be ideticl (equl) whe they hve the sme compoets, ie, u v u v u v u v u v u v Now we c see tht systems of lier equtios c be writte i vector formt, x x x b x x x b x x x b m m m m OR b b x x x b m m m m () Exmple 3 x x x 3 x 8x 8 3 5x 5x 3 becomes x x x

8 Sice mtrix of the system is mde of the vectors m m,, m we c write the system i the followig formt x x x b (3) with b b bm b I short we shll write (3) s mtrix multiplictio this cocept Ax b We shll soo expli You c esily covice yourself tht the followig is true, k k A x y Ax Ay A x Ax (4) where k is y rel umber (sclr) We eed oe more defiitio tht we shll mke use of lter We cll subset S of subspce sped by v, v,, v k if it is set of lier combitios of,,, k Sp v, v,, v k v v v We lbel this s For exmple, oe c view xy-ple i 3 s subspce sped by i d j The whole spce 3 is its ow subspce sped by i, j, d k It is esy to covice oeself tht subspce sped by vectors hs ll the properties from # to #8 If we were to defie vector spce by usig those properties (s it is ctully 8

9 doe xiomticlly) the subspce of vector spce is lso vector spce These ides will be clerer lter i the course Now otice tht (3) shows tht the system of lier equtios hs solutio whe b is the lier combitio of colums of system mtrix,,, ie, whe b belogs to Sp,,, the vector spce The ext theorem is simply cosequece from the Theorem from the previous lecture Theorem The system Ax b hs solutio for every b i m if d oly if system mtrix A (mtrix of coefficiets) hs pivot positio i every row This lso mes tht colum vectors of A sp m Note A is ot ugmeted mtrix of the system Homework: Check olie 9