BAB III VEKTOR DALAM R 2 DAN R 3. Bab III Vektor dalam R 2 dan R 3

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1 Bb III Vetor dlm R dn R BAB III VEKTOR DALAM R DAN R Dlm bgn n n dbhs mslh eto-etor dlm rng berdmens dn berdmens, opers-opers rtmet pd etor g n ddefnsn dn beberp sft-sft dsr opers-opers tersebt... VEKTOR DALAM DIMENSI- dn DIMENSI - Pengntr Vetor Slr dlh sebh besrn yng td meml rh t st ntt yng hny mempny besr s. Sedngn etor dlh sebh besrn yng mempny rh. dr besrn etor dlh eceptn, perceptn, gy. Sedngn contoh nt besrn slr dlh wt, tempertr, mss, pnng, blngn rel. Vetor bs dtls secr geometrs sebg rs grs yng berrh dlm rng dmens- dn dmens-. Arh pnh mennn rh etor. contoh B bsny etor dsmpng dtls = AB A Eor dr pnh dts dsebt pngl etor dn ng pnh dsebt tt ng. Vetor bsny dnotsn dengn hrf ecl tebl ( mslny,,,,w dn x ) UNTUNG USADA (U) 0

2 Bb III Vetor dlm R dn R Opers pd etor D etor yng sm / elen D etor dtn sm b serh dn sm pnng. b Jmlh d etor J dn b dlh d etor sebrng, m mlh + b dlh etor yng dtentn sebg bert : Letn etor b sedemn sehngg tt pnglny berttn dengn tt ng etor. Vetor + b ddefnsn oleh pnh dr tt pngl e tt ng b. b b c b. b b b Selsh d etor J dn b dlh d etor sebrng, m selsh b dr ddefnsn sebg -b = + (-b) b -b t -b -b -b Unt mendptn selsh -b tnp menysn b, possn dn b sehngg tt-tt pnglny bermptn; Vetor dr ng b e tt ng dlh etor - b. b UNTUNG USADA (U)

3 Bb III Vetor dlm R dn R Vetor dlm dmens- Z,, dlh etor-etor stn msng- P(x,y,z) msng pd rh smb X, smb Y, sb. Z r,,. Vetor poss r dr O e P(x,y,z) dlh O Y r x y z dengn pnng X r x y z. Komponen-omponen st etor Z A OA P proyes A pd bdng XOY A (,, ) OA, OA OA PA, Y OP OA OA A P A OA OP PA X Sft-sft Opers Vetor J,, dn w dlh etor-etor dlm rng berdmens- dn rng berdmens- dn, l dlh slr, m. + = + b. ( + ) + w = + ( + w ) c. + 0 = 0 + = d. + (-) = 0 e. ( + ) = + f. ( + l ) = + l UNTUNG USADA (U)

4 Bb III Vetor dlm R dn R J = (,-,) dn b = (4,,) m A + b = ( 5,-,), = (,-6,4), -b = (-,-5,) SOAL-SOAL LATIHAN. Gmbr etor-etor bert dengn tt pngl dletn pd tt sl :. = (, 6) b. = (-4, -8) c. = (, 4, 5) d. 4 = (,, 0) e. 5 = (0, 0, ) f. 6 = (-, -, -4). Msln = (-,, ), = (4, 0, -8), dn w = (6, -, -4) cr omponen dr ). w b). 6 + c). + d). 5( 8w) e). -( w). J,,dn w dlh sebrng etor-etor, dptn, b, c sedemn sehngg + b + cw = (, 0, 4) 4. Dptn, b,dn c sedemn sehngg (-, 9, 6) + b(,, ) + c(0,, ) = (0, 0, 0).. NORMA SUATU VEKTOR dn HASIL KALI TITIK (Dot Prodct) DEFINISI NORMA SUATU VEKTOR Pnng st etor t norm ( ) ddefnsn = + (norm etor = (, ) dlm rng berdmens-) = + + ( norm etor = (,, ) dlm rng berdmens-) J P(x,y,z ) dn Q(x,y,z ) dlh d tt dlm rng berdmens-, m r ntr ed tt tersebt dlh norm etor PQ = x x + (y y ) + (z z ) Z Q(x,y,z ) P(x,y,z ) Y X UNTUNG USADA (U)

5 Bb III Vetor dlm R dn R. Norm Vetor = (-,,) dlh = ( ) + + = 4 b. Jr ntr tt P(,-,-5) dn Q(4,-,) dlh PQ = 4 + ( + ) + ( + 5) = 44 = HASIL KALI TITIK (Dot Prodct) Defns Hsl Kl Tt J dn dlh etor-etor dlm rng berdmens- t berdmens- dn dlh sdt ntr dn, m hsl l tt t hsl l dlm Eclden ddefnsn sebg = cos θ 0 dn 0 0 = 0 t = 0 Rms omponen nt hsl l tt Msl =(,, ) dn = (,, ) m = + + Sedngn nt mendptn sdt ntr d etor : cosθ = Msln (,-,) dn = (,,), dptn dn tentn sdt ntr dn Penyelesn = + + = ()() + (-)() + ()() = cosθ = = 6 6 = UNTUNG USADA (U) 4

6 Bb III Vetor dlm R dn R Sft-Sft Hsl Kl Tt J,, dn w dlh etor-etor dlm rng berdmens t berdmens dn dlh st slr, m :. = b. ( + w) = + w c. ( ) = () = () d. 0 0, dn = 0 = 0 PROYEKSI ORTOGONAL Vetor-etor yng teg lrs dsebt g etor-etor ortogonl. D etor dn ortogonl (teg lrs) dn hny = 0. w w w w Vetor dlh mlh dr w dn w, dmn w ser dengn dn w teg lrs dengn. Vetor w dsebt proyes ortogonl dr pd t omponen etor dr yng ser dengn. n dnytn dengn Proy = Sedngn etor w dsebt etor yng ortogonl terhdp. Kren w = -w, m etor n bs dtls sebg - Proy = - Msl = (,-,) dn = (4,-,). Dptn omponen etor dr yng ser etor dn etor yng ortogonl terhdp Penyelesn = ()(4) + (-)(-) + ()() = 5 = 4 + (-) + = Jd, omponen etor yng ser dlh Proy = 5 = 4,, = (0 7, 5 7, 0 7 ) Dn omponen etor yng ortogonl terhdp dlh - Proy = - = (,-,) (0 7, 5 7, 0 7 ) =( 6 7, 7, 7 ) UNTUNG USADA (U) 5

7 Bb III Vetor dlm R dn R Secr lstrs geometrs dpt dgmbr sepert dbwh n cos - cos 0 π π < θ π SOAL- SOAL LATIHAN. Dptn norm dr etor. = (4, -) b. (,, ) c. = (-7,, -) d. = ( 0, 6, 0). Dptn r ntr A dn B. A(, 4), B(5, 7) b. A(-, 6), B(-, -4) c. A(7, -5, ), B(-7, -, -) d. A(,, ), B(6, 0, ). J = (, -, ), = (, -, 4), dn w = (, 6, -4), m tnn :. + b. + c. + d. 5 + w 4. Dptn. = (, ), = (5, -7) b. = (-6,-), = (4, 0) b. = (, -5, 4), = (,, ) d. = (-,, ), = (, 7, -4) 5. Dptn proyes ortogonl dr terhdp. = (6, ), = (, -9) b. = (-, -), = (-, ) c. = (,, -7), = (, 0, 5) d. = (, 0, 0), = (4,, 8).. HASIL KALI SILANG (Cross Prodct) Defns Hsl Kl Slng J =(,, ) dn = (,, ) dlh etor-etor dlm rng berdmens, m hsl l slng x dlh etor yng ddefnsn sebg t dlm nots determnn x = ( -, -, - ) x =,, UNTUNG USADA (U) 6

8 Bb III Vetor dlm R dn R Dptn x, dmn = (,,-) dn = (,0,) Penyelesn x = 0,, 0 = (,-7,-6) Hbngn ntr Hsl l Tt dn Hsl Kl Slng J,, dn w dlh etor-etor dlm rng berdmens, m :. ( x ) = 0 b. ( x ) = 0 c. x = ( ) d. x ( x w) = ( w) - ( )w e. ( x ) x w = ( w) - ( w) Sft-Sft Artmet Hsl Kl Slng J,, dn w dlh etor-etor dlm rng berdmens dn dlh sebrng slr, m :. x = - ( x ) b. x ( x w) = ( x ) + ( x w) c. ( x ) x w = ( x w) + ( x w) d. ( x ) = () x = x () e. x 0 = 0 x = 0 f. x = 0 Vetor-etor yng mempny pnng st dn terlet dsmbh oordnt dsebt etor stn stndrt. Setp etor = (,, ) dlm rng berdmens dpt dnytn dlm bent,, dn = (,, ) = (,0,0) + (0,,0) + (0,0,) = + +. Msln = (, -, 4) = + 4 x z x = x = x = 0 x = ; x = ; x = (0,0,) x = - ; x = - ; x = - (0,,0) y (,0,0) Gmbr. Vetor stn stndrt UNTUNG USADA (U) 7

9 Bb III Vetor dlm R dn R UNTUNG USADA (U) 8 Interprets Geometrs dr Hsl Kl Slng Ls rn genng yng dbent dn, dlh : x L Ls segtg yng dbent dn : x L Volme blo mrng (Prleleppedm) dengn ss-ss,, w : w w w x w V. Dptn ls segtg yng tt-tt sdtny P(,, 5); Q(4,, -); R(, 6, 4) Penyelesn: PQ PR Ls segtg: PQ L x 6 x PR Dptn olme prlleleppedm yng ss-ssny, 4, w. Penyelesn: w x V.

10 Bb III Vetor dlm R dn R SOAL-SOAL LATIHAN. J = (,, -), = (0,, -), dn w = (, 6, 7). Dptn. x b. x ( x w) c. ( x ) x w d. ( x ) x ( x w) e. x( w) f. ( x ) w. Dptn ls segtg yng mempny tt-tt sdt bert: ). A(0,0,0); B(,,); C(,-,4) b). D(,0,0); E(0,,0); F(,,). Dptn ls rn genng yng dbent oleh d bh etor: ). dn b 4 b). dn b. 4. Dptn s prlleleppedm yng ss-ssny OA, OB, OC dmn A(,,); B(,,); C(,,). UNTUNG USADA (U) 9