CHAPTER 5 Vectors and Vector Space

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1 HAPTE 5 Vetors d Vetor Spe

2 5. Alger d eometry of Vetors. Vetor A ordered trple,,, where,, re rel umers. Symol:, B,, A mgtude d dreto.. Norm of vetor,, Norm =,, = = mgtude. Slr multplto Produt of slr d vetor,,, slr,,,, 4. Sum of vetors Addto,,,,,, 5. Alger of Vetors ommuttve lw,,,,,,,, H H Assotve lw 0,0,0 0,0,0 0 Addtve detty 4 5 6

3 6. Norm of slr multplto 0 Iff 0 0 0,0, Prllelogrm Lw for Vetor Addto 8. Stdrd represetto of Vetor or rtes Vetor form where,0,0, 0,,0 d 0,0, re ut vetors, whh represet the dretos of x, y, z xes, respetvely. 5. Dot produt. Dot produt Slr produt, er produt The dot produt of vetors d s the slr.. Propertes of dot produt ommuttve lw H H Dstrutve lw :slr

4 4 5 0 ff 0. eometrl terpretto of dot produt,,,, Determe the gle etwee vetors Lw of ose os os os θ= gle etwee the two vetors = os Orthogol vetors A. Two ozero vetors d re orthogol ff 0 os0 0, 0 orthogol os 0, 90 B. 0 0 for every

5 zero vetor 0 orthogol to every vetor. Ex. Two les L : 4 t,6 t,t L L : p,7 p, 4 p L? Sol: pot of terseto of L & L t 0,6,0 t,7, p 0,7, p,9, 4,,,, L s ot L Equto of ple P x, y, z,, x y z N N N Ple P y vetor o ple s orthogol to ts orml vetor N P N 0 x y z 0 x y z 4. uhy-shwrz equlty

6 os os 5. The ross Produt. ross produt The ross produt of vetors d s the vetor. Propertes of ross produt At-ommuttvty orthogol to oth d pf. 0 0 s

7 os s ff // s 0, 0 s 0, 0 or // 5 H H Assotvty 6. Ple determed y d 0, 0, Norml vetor N Equto of ple N P 0 P Ex.5. ve: d: equto of ple Sol:,,, 4, N 5 4 Equto of ple N P,,5 x, y, z 0 x y 5 z 8 4. Are of prllelogrm Are ormul of geometry Are = h s

8 5. Volume of prllelepped V = re of se lttude = H os = 6. Slr trple produt H,, = H = Propertes: 5.4 Vetor spe H,, =, H, =,, H H,, = - H,,. -vetor A -vetor s ordered -tuple x, x x x, Where x s rel umer The set of ll -vetors s deoted : set of order pr x, y : set of ll -vetors 4 : set of ll 4-vetor u, v, w, s. Alger of vetor operto Addto of -vetors x, x, x x + y, y, y y = x y, x y x y Slr multplto α x, x, x x = x, x, x x. Propertes of -vetors H H 0 4 5

9 Beuse of these propertes of ddto of -vetors d multplto y slr, s lled vetor spe or rel vetor spe. 4. Norm of -vetor x, x x = x x x x 5. Dot produt of -vetors x, x, x x. y, y, y y = x y x y x y A. propertes of dot produt of -vetors H H ff 0 Notes: No prllegrm lw No hgher dmesol logue of the ross produt > No geerl verso of the lw of ose B. uhy-shwrz equlty Purely omputtol proof efer to P.4. Agle etwee -vetors d 0, 0 os 0 orthogol to ff 0 6. Stdrd represetto of vetors x, x,, x x e x e x e x e where,0,0,,0, 0,,0,,0,... e e

10 Ad 0,0,0,, e re mutully orthogol ut vetors whh defe the dretos of e e 0 7. Suspe of -spe A set s of vetors If 0 S s lled suspe of S whe & S S : rel umer Ex. ve: S =, d: Is S suspe of Sol: S S S S s ot suspe of 4 Ex. ve: =, -,4,,0 d: Is suspe of 4? Sol: 0 0 -,4,, ,4,,0 -,4,,0 -,4,,0 -,4,,0 -,4,,0 -,4,,0 s suspe of Ler Idepedee d Dmeso. Ler omto of -vetors

11 ... Here,,... slr EX:,,,7,4,,0 Ler omto?,4,,0,,,7 8,0,0,0 4 8,0,0,0. Ler Depedee of -vetors,..., re ler depedee f there re rel umer ot ll zero suh tht... 0 EX:, 6,0,0,0,,0,0,0 5 lerly depedet? Sol:Let Te ot ll zeros d re lerly depedee. Ler Idepedee of -vetors,..., re ler depedee ff... 0 hold oly f ll 0 EX:

12 ,0,0 0,,0 Sol:Let 0 0 d lerly depedet 4. odto for lerly depedet Solve ove system of equto to ot, If 0 the re, lerly depedet, otherwse lerly depedet., vetors re lerly depedet f I. No s zero vetor, d th II. If the ompoet of oe vetor s ts frst ozero ompoet, the ll other vetors hve ompoet equl to zero. Ex : 0,4,0,0, 0,0,6,0, 5 0,0,0, 4, Sol : lerly depedet Let,, e mutlly orthogol ozero vetors. The,, Ex re lerly depedet

13 4,0,0 0,, 0,, 0 0 lerly depedet Bss S s suspe of,, S form ss for S f, re lerly depedet. Every vetor S e wrtte s ler omto of, Ex rteso ut vetor,, Is,, ss for? Sol,0,0 0,,0 0,0, 0, 0,, lerly depedet,, form ss for y vetor x, y, z x, y, z x,0,0 0, y,0 0,0, z x,0,0 y 0,,0 z 0,0, x y z 6. Dmeso of suspe the umer of vetors y ss of suspe of. Ex. => Dmeso Bss:,, Ex => Dmeso

14 Bss: e, e, e, e Ex S:{ : x, y, z, 0, x-y, x+y, z 6 }. Is S suspe of 6. Bss of S. Dmeso of S Sol:. Suspe of 6 A, x = y = z = 0 = 0 S B, x, y,0, x y, x y, z x, y,0, x y, x y, z x x, y y,0, x x y y, x x y y, z z x, y,0, x y, x y, z S, = x, y, 0, x-y, x+y, z = x, y, z, 0, x-y, x+y, z S S s suspe of 6. Bss of S 6 x, y, z, 0, x-y, x+y, z = x,0,0,x,x,0 + 0,y,0,-y,y+0 + 0,0,0,0,0,z = x,0,0,,,0 + y0,,0,-,,0 + z0,0,0,0,0, xe ye z e e, e, e form of ss of S. Dmeso of S, D = 6

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