4.. Vetor dlm Rng Dmens Tg Seenrny pengertn etor pd dng dmens d sm hlny pengertn etor dlm rng dmens tg, etor pd sng mempny d omponen, m etor dlm rng mempny tg omponen. Yt ;,,,, Dmn merpn etor stn t etor ss pd rh etg sm, t etor stn serh sm- x, etor stn serh sm- dn etor stn serh dengn smz Pnng etor yt : Gmr 4. y dtnn oleh yng merpn rms r ser oordnt dmens tg dgmrn sepert Z X Y Gmr 4.. Vetor dlm Rng Dmens Tg,, dn,, m J deth d etor yng dset Hsl Kl Tt ddefnsn seg ert : Geometr dlm Rng, Vetor 47
Cos Contoh 4. : Tentn 5,, sdt ABC C dn Gmrn A,,,,4, 6 B dn Penyelesn 4. : J dgmr seg ert : A,, B,4, 6. Msln etor ngny d tt A dlh etor yng tt slny d tt B t etor BA dn etor dn tt dlh etor yng tt slny d tt B dn tt ngny d tt C t etor dpt dtentn seg ert. m etor dn BA ( ),( 4),( ( 6)), 6,. 9. BC ( 5 ),( 4),( ( 6)), 7, 8 Cos C 5,, BC, Geometr dlm Rng, Vetor 48
Cos Cos Cos Cos Cos, ( )() ( 6)( 7) (9)(8) ( ) ( 6) 9 ( 7) 8 4 7 6 8 9 49 64 8,86,45 9,988,95 4... Sdt dn Kosns Arh dn J deth st etor yt etor, sdt yng t nol ntr etor dengn etor stn yng serh dengn sm oordnt yt, dset sdt-sdt rh etor, sdtsdt terset dlmngn dengn, dn, etor dnytn seg, m sdt-sdt t dnytn seg, Cos dn Cos dmn ser rms dern : Cos Cos Cos Cos Berl g Cos Cos Cos sepert Gmr 4. Geometr dlm Rng, Vetor 49
Z X Y Gmr 4.. Sdt-Sdt Arh Vetor Contoh 4.4 : Deth etor etor 4 tentn sdt-sdt rh nt Penyelesn 4.4 : Deth etor Cos Cos Cos 4 9 9 4 9 4, m 4 9 6 9 Geometr dlm Rng, Vetor 5
4... Bdng Dent Oleh Vetor Unt melsn seh dng d eerp r, slh stny dengn menggnn hs etor t dengn menggnn ntn etor, msln n A, B, C dlh seh etor yng t nol dn P x, y z dlh tt tetp, oordnt, P x, y, z yng memenh persmn P P n dlh seh dng yng mell dn teg lrs n, sepert pd Gmr 4.4 P n A, B, C PP x x, y y z z, Px, y z P x y, z,, Bdng Gmr 4.4. Bdng Mell tt P ren etor P t : n A, B, C P n x x y y, z z teg lrs dengn etor, A, B, C x B y y C z z A x P P t Sehngg ser mm deth seh etor yng teg lrs pd seh dng d tt x, y z persmn dng dpt dtentn yt :, n A, B, C P, m A x x By y Cz z Geometr dlm Rng, Vetor 5
Contoh 4.5 : P,4, Tentn persmn dng yng mell tt n 4,, lrs dengn etor 6 dn teg Penyelesn 4.5 : Deth tt,4, z dn P sehngg ddpt nl x, y 4 sert etor n 4,, 6 sehngg ddpt nl C 6 A 4,, dn B,, ren rms nt menentn persmn dng A, m : x x B y y C z z dlh A x x By y Cz z 4x y 4 6z 4x 8 y 6z 8 4x y 6z 8 Contoh 4.6 : P dn teg Tentn persmn dng yng mell tt 4,, lrs dengn etor n 6 4 Penyelesn 4.6 : P sehngg ddpt nl x 4, dn z sert etor n 6 4 A, B 6, dn C 4 persmn dng dlh x x By y Cz z A x x By y Cz z x 4 6y 4z x 8 6y 4z x 6y 4z 6 Deth tt 4,, y, sehngg ddpt nl, ren rms nt menentn A, m : Geometr dlm Rng, Vetor 5
4... Jr Tt Ke Bdng J dern st tt Px, y z, dn seh dng yng mempny persmn Ax By Cz D, m L menytn st r dr tt tertent e st dng, m r t dnytn dengn rms : L Ax By A B Cz C D Pndng seh dng sepert Gmr 4.5 n A, B, C x, y z, L m x, y z, Msln tt x, y, z terlet pd dng dtr, ndn m x x, y y, z z dlh etor dr tt x, y, z e tt x, y z., n A, B, C dlh etor yng teg lrs terhdp dng yng dern, m lngn L dperoleh : L L m A m n os n x x By y Cz z Gmr 4.5. Jr Tt e Bdng A B dlh proyes etor m pd n, m C Geometr dlm Rng, Vetor 5
L L L Ax Ax Ax Ax By By By A B Cz By Cz C Ax By A B C Cz D Cz Cz ren tt dng, m A B C Ax By Cz D x, y z, terlet pd Contoh 4.7 : Tentn r tt 4,, P e dng x 4y 5z 9 Penyelesn 4.7 : A, 4 P 4,, deth nl 4 4,, Dr dng dtr x 4y 5z 9 deth nl C 5, dn 9 y, dn 4y 5z 9 D, dr tt z, m r tt x dlh : Ax By Cz D L A B C ()( 4) ( 4)() (5)( ) 9 L ( 4) 5 8 5 9 L 9 6 5 4 L 5 B, x, P e dng Geometr dlm Rng, Vetor 54
4..4. D Bdng Ser Deth d d h dng yng msng-msng mempny persmn A x B y Cz D dn A x B y Cz D ed dng dtn ser : Cttn : A A, B B P,,, C C dn D D St tt A x B y Cz D A B C D St tt A x B y Cz D A B C D P,, dtn terlet pd dng dtn terlet pd dng, Contoh 4.8 : Deth seh dng dengn persmn x 4y z 8, tentn seh dng yng mell tt P,, dn ser dengn dng x 4y z 8 Penyelesn 4.8 : x 4y z Deth dng dengn persmn 8, m dng yng ser dengn dng x 4y z 8 dlh x, ren dng yng ser dengn dng 4y z D 4y z 8 x 4y z D 4 D 6 8 4 D D x mell tt P,,, m dperoleh nl D yt Sehngg persmn dng yng ser dengn dng 4y z 8 x dn mell tt,, P dlh x 4y z Geometr dlm Rng, Vetor 55
Contoh 4.9 : 5x y 4z Deth d dng ser yt dng I :, dn dng II : 5x y 4z 4, erp r ed dng yng ser t Penyelesn 4.9 : J t lstrsn dengn gmr, seg ert : Bd I : 5x y 4z L Bd II P,, : 5x y 4z 4 Unt menentn r ed dng t t L, m t hrs, yng terlet pd dng II, menentn seh tt P, rny dlh : 5x y 4z 4 t er nl dperoleh : 5 4z 5 4z 8 4z 4z z 4 x dn y, m 4 4 4 Sehngg dperoleh tt yng terlet pd dng 5x y 4z 4 yt P,,, dn nt mengeth r ed dng t gnn r seh tt e dng, dlm hl n t tentn 5x y 4z dengn r tt,, menggnn rms : P e dng I yt L Ax By A B Cz C D Geometr dlm Rng, Vetor 56
Ax By Cz D L A B C 5 4 L 5 4 5 4 L 5 9 6 8 L 5 L 8 5 5 Sehngg dperoleh r dng I : x y 4z e dng II : 5 y 4z 4 x dlh 8 5 4..5. D Bdng Teg Lrs A x J deth d dng yt dng I B y Cz D dn A x B y C z dng II, d dng terset dtn teg lrs sepert Gmr 4.6 D m A, B C, A x B y C z D n A, C B, A x B y C z D Gmr 4.6 : Bdng Slng Teg Lrs Geometr dlm Rng, Vetor 57
A, C B, Dr Gmr 4.6 dpt deth, hw etor dlh etor yng teg lrs dng I A x B y Cz D, sedngn etor A n m A, B C dlh etor yng teg lrs, x B y C z D dng II, dng I dtn teg lrs B, dng II etor n A, C teg lrs etor m A, B, C, d h etor A, B, C m A, B C dtn teg lrs A A, B B C C n dn etor n m t Contoh 4. : Deth d h dng yt dng I dengn persmn x y z 7 dn dng II dengn persmn x y z 9 ph ed dng terset teg lrs? Penyelesn 4. : Deth dng I x y z 7 m etor yng teg lrs dng I dlh n,,, dng II y z 9 m,, yng teg lrs dng II dlh dtn slng teg lrs n m, m : n m A A B B C C 6 4 n m rtny etor,, Kren etor,, x m etor, d dng terset n slng teg lrs dengn m, tny dng I y z 7 dng II x y z 9 x teg lrs Geometr dlm Rng, Vetor 58
Contoh 4. : Deth d h dng yt dng I dengn persmn x 4y z 9 dn dng II dengn persmn x y z tentn nl gr ed dng t teg lrs Penyelesn 4. : Deth dng I x 4y z 9 m etor yng teg lrs dng I dlh n, 4, dn dng II x y z m m,, etor yng teg lrs dng II dlh, gr ed dng t teg lrs, m hrslh ed etor n, 4, dn m,, g hrs teg lrs, etor, 4, m,, n m etor n m A A B B C C 4 6 4 4 4 4 4 4y z x y z, m nl n teg lrs Sehngg gr dng I x 9 teg lrs dng II Geometr dlm Rng, Vetor 59
4..6. Sol-Sol Lthn A. Unt tp psngn tt sets rs grs errh dlm ent P,, P 4,5, P P P. dn. P,, dn. dn 4. P,, dn P P,, P P,4,,, 4,, dn P dwh n, ern dn emdn tls etorny B. Tentn sdt ntr etor m dn etor n d wh n. m 4,, dn n,, 5 m, 4, dn n,,. m,, dn n,,. 4. m 5 dn n C. Tentn Persmn Bdng yng mell tt P lrs etor n,,5 n P dn P dn n P dn n 4 P dn n..,,.,, 4.,, 5 dn teg D. Tentn Persmn Bdng yng Mell tt P dn Ser dengn dng Ax By Cz D.,,.,,. 4,, P dn dng x 4y z P dn dng x 4y z 6 P dn dng x y 4z E. Tentn Jr Tt P.,,.,6, e Bdng P dn dng x y z 7 P dn dng x y z 9 Ax By Cz D F. Tentn Jr Bdng-Bdng Ser Bert. x y z 9 dn dng 6x 4y z 9. 5x y z 5 dn dng 5x y z 7 Geometr dlm Rng, Vetor 6
Geometr dlm Rng, Vetor 6 4.. Hsl Kl Slng (Cross Prodt) Sdh t eth hw hsll tt dr d h etor,, dn etor,, dlh seh slr yt mell rms : Ln hlny dengn hsll slng (ross prodt) yng menghsln seh etor, deth d h etor,, dn,, m yng dset hsl l slng drmsn seg ert : Unt memperlnr pemhsn hsll slng, t ngt eml r menghtng Determnn, mslny :. Determnn x Msl deth determnn A dengn orde x yt : d d A. Determnn x Msln deth determnn A dengn orde x yt : A,, x
Geometr dlm Rng, Vetor 6 Sehngg t dentn dmn,, dn,, m : x,, x J t menr x mend x m omponen etor menempt rs e tg dn omponen etor menempt rs e d, yt : x,, x Msln,, dn,4, tentn dn 4 Contoh 4. : Penyelesn 4. :
4 4 ( )() (4)( ) ()() ( )( ) ()(4) ( )( ) 4 4 4 4 4 4 ( 4)( ) ( )() ( )( ) ()() ( )( ) ()(4) 4 4 4 4... Tfsrn Geometr x Art dr hsl l slng g perl dgmrn ser geometr nt memperels. Teorem A : Andn dn etor-etor dlm rng dmens tg dn sdt ntr mere, m : x x errt x teg lrs terhdp. dn., dn x mement st system tngn nn rngp tg. x sn Geometr dlm Rng, Vetor 6
Bt :, dn, Msln deth d h etor yt, m ses dengn rms hsll slng ddpt x,, sehngg dperoleh :, x, x Artny etor x teg lrs terhdp etor Teorem B : D etor dn dlm rng dmens tg dlh ser dn hny x = Penerpn dr hsl l slng ed etor slh stny dlh nt menentn persmn dng yng mell tg tt yng td segrs. Msln deth tg tt yt A,,,,, C,, B dn, dr etg tt terset dpt deth d h AB, etor yt etor AC Vetor,, AB AC. p q r, dn etor dlh etor yng mell tt A,, dn teg lrs dng yng memt tt,, B,, dn C,, A,,, B,, dn C,, mempny persmn : p x qy rz A,, m dng yng memt tg tt Geometr dlm Rng, Vetor 64
Contoh 4. : P,, Tentn persmn dng yng mell tg tt dn tt P,, P 4,,, tt Penyelesn 4. : Msln etor P P dn P P menentn etor dn etor yt : P P 4,, ( ) 5 sehngg t dpt, P P 5 P P Geometr dlm Rng, Vetor 65 4, ), ( ), 6, P P 6 4 Dperoleh : x 6 4 4 5 5 6 5 6 4 ( )() ( 4)(5) ( )() ( 6)(5) )( 4) 6 6 8 4 4 6 Sehngg dng yng mell tt 4,, 4 4 6 mempny persmn : A x x By y Cz z 4x 4 4y 6z 4x 56 4y 4 6z 4x 4y 6z 56 4 4x 4y 6z 44 4, ( ( 6)( ) P dengn norml
At dpt g dtentn dengn mengml tt,, P P ren tt g terlet pd dng dengn norml 4 4 6 mempny persmn : x x By y Cz z x 4y 6z A 4 4x 4 4y 48 6z 8 4x 4y 6z 4 48 8 4x 4y 6z 44 Contoh 4. : Perlhtn hw ls rn genng yng dent oleh etor dn seg ss erdmpngn dlh. x. Penyelesn 4. : J t gmrn ser geometr, m seg ert : Sn Kren ls rn genng t dlh Ls ls. x. tngg dmn ls. dn tngg Sn dlh :. sehngg Ls rn genng t L Sn dn ren x. Sn ls rn genng d ts dlh., m tert hw L. x. Geometr dlm Rng, Vetor 66
Contoh 4.4 : Seh rn genng yng dent dr d h etor dn etor 4 dlh Penyelesn 4.4 : Deth etor dn 4, m 4 4 4 6 9 4 6 8 4 4 Ls rn genng dlh L L 4 4 L 6 69 96 L 8 L Geometr dlm Rng, Vetor 67
4... Sol-Sol Lthn A. Dern, 4 dn w 4 tentn :. x. (xw). 4. x( w) x(xw) B. Tentn etor stn yng teg lrs terhdp dng yng dent oleh tg tt yt : A,, B 5,, C 4,,., dn., dn C, 4, A,,5 B,, C. Tentn Ls Jrn Genng yng dent dr etor A dn B seg d ss yng erdmpngn. A 4 dn B 4 5. A 5 dn B 6 D. Tentn Ls segtg yng dent dr tg tt A, B dn C yt :. A,,, B,4,6 dn C,,7. A,,, B,, dn C 4,5,6 E. Tentn persmn dng yng mell tg tt A, B dn C yt. A,5,6, B,, dn C 4,,6. A,,, B 4,, dn C 5,,6 Geometr dlm Rng, Vetor 68