Геометрия7_9
.pdf8 KJiac |
150 |
,zumaT xy TaK, HK noKaaaHO Ha MaJIIOHKY 225. ~a TQKHM B~6opoM CHCTeMH KOOp,ll;HHaT KOOPl_l;HHaTaMH_ BeKTopa _a 6y,11;yTh lal i 0, a Koop,11;HHaTaMH eeKTopa b 6y,11;yT1> lb! cos <pi lbl sin <p. CKaJIHPHHH
,11;06yTOK
ab = lal Ibl COS<p + 0 Ibl sin <p = lal Ibl cos <p. TeopeMy ,11;oee,11;eHo.
3 TeopeMH 10.3 BHnJIHBae, w;o JCOJ£u BeJCTopu nepneniJuJCyJ£sipni, To ix cJCa1uipnuii. iJo6yTo1C iJopiBnHJe ny.n.HJ. I HaenaKH, RJCUfO cJCa-
.n.sipnuii. iJo6yTo1C BiiJ1w.innux BiiJ "Y"" BeJCTopiB iJopiBHHJe Hy.n.HJ, TO BelCTopu nepneHiJuxy.n.apHi.
3 a ,11; a q a (38). ,D;oee,11;iTL, w;o cyMa Kea,11;paTiB ,11;iar0Ha- 0ne:H: napaJieJiorpaMa ,11;opiBHIOe cyMi KBa,11;paTiB :H:oro cropiH.
Po a e'Ha a H H a. Hexaii 'IOTHPHKYTHHKABCD - napaJieJiorpa.M (MaJI. 226). MaeMo eeKTOpHi pieHocTi:
AB+ AD= AC,
AB-AD= DB.
IU.,11;HeceMo 06H,11;ei qacTHHH yTeopeHHX piBHOCTeii ,11;0 KBa,11;paTa. ,D;icTaHeMO:
AB 2 + 2AB · AD + AD2 = AC2,
AB2 - 2AB · AD + AD2 = DB 2•
[
8
A
MaJI. 226
,D;o,11;aMO IlO'!JieHHO~ piBHOCTi. ,D;icTaHeMO:
2AB 2 + 2AD2 = AC 2 + DB 2•
OcKiJILKH y napaJieJiorpaMi npoTHJiemHi cTopoHH pieHi, TO IJ,H pieHi.CTL oaHa'Iae,w;o cyMa Kea,11;paTiB ,11;iar0Haneii napaJienorpaMa ,11;opieH106 cyMi KBa,11;paTiB :H:oro cTopiH, w;o ii Tpe6a 6yJIO ,IJ;OBeCTH.
99. P03KJIMABB.H BEKTOPA
3A KOOP,lJ;HBATBHMH OC.HMH
BeKTOp Ha3HBa6TLC.R oauHU'lHUM, .RKlll;O HOrO a6COJllOTHa BeJIH- 'IHHa,11;opiBHI06 O,IJ;HHH~. 0,IJ;HHH'!HiBeKTOpH, HKi MalOTb Hanp.RMH ,IJ;O)J;QTHHX KOOp,IJ;HHQTHHX nieoceii, HaaHBQIOTbCH Koopi}unarHUMU 8€KTOpaMU a6o opraMU. IloaHa'IaTHMeMOIX e1(l; 0) Ha oci xi e2(0; 1) Ha oci y (MaJI. 227).
§ 10. BeKTOpH |
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151 |
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OcKi.J:rbKH |
Koop,n;HHaTai BeKTOpH |
Bi,n;- |
y |
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Miaai |
ei,n; aym!_ i HeKoniaeapai, TO |
6y,n;L- |
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HKHH |
eeKTop |
a(a 1; a 2 ) M0i1ma aanHcaTH |
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y BHrnx,n;i: |
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a= |
(*) |
3Haii,n;eMo Koe<l>i~ieHTH A. i |
µ ~boro |
p03KJia,n;y. IloMHOatHMO piBHiCTL (*) Ha BeKTop ej . OcKiJILKH
Ci{a1; a 2)e1 = a1, e1 . e1 = 1, e1 . e2 =
= 0, TO a1 = "-·
AHanori'IHo,noMHOatHBWH o6H,n;Bi qacTHHH piBHOCTi (•) Ha BeKTOp e2, ,n;icTaHeMo: a 2 = µ.
TaKHM 'IHHOM,,IJ;JUI 6y,n;L-.SIKOro BeKTOpa |
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- |
- |
- |
a = |
a1e1 |
+ a 2e2. |
e, x
0 '
MaJI. 227
a(a ,; a 2) Ma6MO:
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KOHTPO~hHI3AITHTAHHH |
• |
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1.IIJ;o TaKe eeKTop? HK noaHa'laIOTLCHeeKTopH?
2.HKi BeKTOpH H83HBaIOTLCH o,n;HaKoeo Harrp.si:MJieHHMH (npoTHnemHo H81IpHMJieHHMH)?
3.IIJ;o TaKe a6coJIIOTHa eeJIH'lHHaeeKTopa?
4.IIJ;o TaKe HYJILOBHii BeKTOP?
5.HKi eeKTOPH Ha3HBaIOTLCH piBHHMH?
6.,D;oBe,n;iTb, ~o piBHi BeKTOpH o,n;HaKOBO HaIIpHMJieHi ii piBHi aa a6coJIIOTHOIO BeJIH'lHHOIO. I HaerraKH, BeKTOpH, o,n;HaKoeo Harrp.si:MJieHi ii pieHi aa a6coJIIOTHOIO BeJIH'lHHOIO,piBHi Mim co6010.
7.,D;oBe,n;iTb, ~o Bi,n; 6y,n;L-HKOl TO'IKHMOatHa Bi,D;KJiaCTH BeKTOp, HKHii ,n;opiBHIOe ,n;aHOMY BeKTopy, i TiJILKH o,n;HH.
8.IIJ;o TaKe KOOp,IJ;HHaTH BeKTOpa? "lfoMy ,n;opiBHI06 a6coJIIOTHa BeJIH"IHHa BeKTOpa 3 KOOp,D;HHaTaMH ai, a2?
9.,D;oee,n;iTL, ~o piBHi BeKTOPH MaIOTb Bi,n;rroBi,IJ;HO piBHi KOOp,n;H- HaTH, a BeKTOpH 3 Bi,D;IIOBi,IJ;HO piBHHMH KOOPAHHaTaMH piBHi.
10.,D;aiiTe 03Ha"leHHH ,n;o,n;aBaHHH BeKTopiB.
11.J];oBe,ll;iTL, ~O ,ll;JIH 6y,ll;b-HKHX B0KTOpiB aTa b:_a _T _b _ b+a.
12.,D;o_!le,n;iT~, ~OJl;JIH ~y,n;L-HKHX TPLOX BeKTopiB a, b, c: a+ (b +
+c.
13.,D;oBe,n;iTL BeKTOPHY piBHiCTL AB+ BC= AC.
14.,D;oBe,n;iTb, ~o ,D;JIH 3.!laxo,n;meHHH CYMH B~KTopiB aTa b Tpe6_!t Bi,n; KiHI~H BeKTopa a Bi,ll;KJiaCTH BeKTOp b'' HKHii ,u;opiBHI06 ~· To,u;i BeKTop, no"laToK .si:Koro a6i~aeTLCH a no"laTKOM _!leKTQPa a, a KiHe~L - 3 KiH~eM BeKTOpa b', ,u;opiBHIOBaTHMe a+ b.
15.Cct>opMyJIIOiiTe •rrpaBHJIO napaJieJiorpaMa• ,u;o,u;aeaHHH BeKTOpiB.
16.,D;aii:Te 03Ha"leHHH pi3HH~ BeKTOpiB.
17.,D;aii:Te 03Ha'leHHHMHOateHHH BeKTOpa Ha "IHCJIO.
8 KJiac |
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152 |
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18. |
,ll;oB~,ll;iTh, w;o a6COJUOTH!J. |
BeJIH'!HH!! BeKTOpa 'Aa ,11;opiBHI06 |
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l'AI ial, Hanp.HM BeKTopa Aa, KOJIH a:;t=O, a6ira6ThC.fl 3 Hanp.HMOM |
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BeKTOpa |
a, .HKill;O A > 0, i |
npOTHJie.>KHHH Hanp.HMY BeKTopa |
19. |
a, .flKw;o |
A< o. |
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HKi BeKTOpH Ha3HBaIOThC.fl Ko~iHea~HHMH? |
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20.,l1;0Be,11;iTh, w;o KOJIH BeKTOpH a Ta b Bi,11;MiHHi_ Bi,11; HYJihOBoro
BeKTOpa i H~ KOJii~eapH~ TO ,ll;OBiJihHHH BeKTOp c MO.>KHa no,11;aTH y BHrJI.H,IJ;i C = Aa + µb.
21.,ll;aiiTe 03Ha'leHH.HCKamlpHoro ,11;06yTKy BeKTopi~.
c:(a+
+b) c = ac + be.
23.HK 03Ha'la6ThC.flKYT Mi.>K BeKTopaMH?
24.1IoMy ,11;opiBHI06 KYT Mi.>K O,IJ;HaKOBO Hanp.HMJieHHMH BeKTO- ·paMH?
25.,ll;oBe,IJ;iTh, w;o CKaJI.HpHHH ,11;06yTOK BeKTOpiB ,11;opiBHI06 ,11;06yTKY
'ix a6COJIIOTHHX BeJIH'IHHHa KOCHHYC KyTa Mi.>K HHMH.
26.,ll;oBe,ZJ;iTh, w;o KOJIH BeKTOPH nepneH,ZJ;HKyJIBpHi, TO ix CKaJI.HpHHH ,11;06yTOK ,11;opiBHI06 HYJIIO. I HaBnaKH, .HKW:O cKaJI.HpHHH
,11;06yTOK Bi,ZJ;MiHHHX Bi,ZJ; HYJl.H BeKTOpiB ,11;opiBHI06 HYJIIO, TO BeKTOpH nepneH,ZJ;HKYJI.HpHi•
3A,ll;A'U
1. Ha npHMiii ,ZJ;aHo TPH TO'IKH:A, B, C, npH'lOMYTO'IKaB Jie.>KHTh |
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•Mim TO'IKaMHA i C. Cepe,ZJ; BeKTopiB AB, AC, BA Ta BC HaaBiTh |
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o,u;HaKOBO HanpBMJieHi Ta npOTHJielKHO Hanp.HMJieHi. |
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2. |
1IoTHPHKYTHHK ~JJCD - |
napaJieJiorpaM. |
,ll;oBe,ZJ;iTh piBHiCTh |
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BeKTopiB AB i DC. |
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3. |
,ll;aHo BeKTop AB i TO'IKYC. Bi,11;KJia,ZJ;iTh |
Bi,ZJ; TO'IKHC BeKTop, |
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w;o ,11;opiBHI06 aeKTopy AB, BKw;o: 1) |
TO'!KaC JJe.>KHTh Ha np.H- |
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Mi:H AB; ~) TO'IKI!_C He Jie.>K!ITh Ha |
np.HMiii AB. |
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4. |
BeKTOpH a (2; 4), b(-1; |
2) i c(c,; c2) |
Bi,ll;KJiaµ;eHO |
Bi,11; llO'laTKY |
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KOOPAHHaT. 1IoMy ,11;opiBHIOIOTh Koop,IJ;HHaTH ix KiHu;iB?
5.~6COJIIOTHa BeJIH'IHHaBeKTOpa a (5; m) ,ll;OpiBHI06 13, a BeKTOpa b(n ; 24) ,ZJ;OpiBHI06 25 . 3HaH,ZJ;iTh m Ta n.
6.,ll;aHo TO'IKHA(0;_!2! B(l; 0), C(l; 2), D(2; 1). ,ll;oBe,IJ;iTh piBHiCTh
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BeKTopiB AB Ta CD. |
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7. |
,ll;aHO TPH TO'IKHA (l; 1), B(-1; 0 ), C(O; 1). 3Haii,11;iTh TaKy TO'IKY |
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D(x; y), w;o6 BeKTop AB ,11;opiBH10BaB CD. |
b Ta a6co- |
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8. |
3HaH,IJ;iTh BeKTOp C, Ill;O ,ZJ;OpiBHI06 CYMi BeKTOpiB ai |
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.JIIO~HY |
BeJil_!'IHHY BeKTOpa C, .HKill;O: 1) a (l; -4), |
b(-4; 8); |
9. |
2) a (2; |
5), b(4; 3). |
1) AC Ta |
,ll;aHo TPHKYTHHK ABC. 3Haii,ZJ;iTh cyMy BeKTopiB: |
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CB; 2) |
AB i CB; 3) AC i AB; 4) CA i CB. |
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§ 10. BeKTOpH |
153 |
MaJI. |
228 |
MaJI. 229 |
MaJI. |
230 |
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10. 3HaH,ZJ;iTb BeKTOp C= |
a- |
b Ta |
HOrO a6COJIIOTHY |
BeJil:l'IHHy, |
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HKIIl;O: |
1) a((l; -4), |
b(-4; |
8); 2) a(-_?i_ 7), b(4; |
-1). |
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11.,ll;aHo BeKTOPH i3 cniJibHHM no11aTKOM: AB i AC. ,ll;oBe,ZJ;iTb, w;o
AC-AB= BC.
12.Y napaJieJiorpaMi ABCD µ;iaroHaJii nepeTHHaIOTbCH B To11n;i M.
BHpa3iTb BeKTOpH AB CD 11epe3 BeKTOpH a= AM b =i i
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=BM (MaJI. 228). |
_ _ _ |
13. |
HaKpecJiiTb TPH 6y,ZJ;b-HKi BeKTOPH a, b, c TaK, HK HI! Ma.1!_IOHK_y |
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229_. II~6yµ;~H:Te Be~Top~, II\.O |
µ;opiBHIOIOTb: 1) a + b + c; |
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2) a - b + c; 3) -a + b + c. |
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14. |
1) ,ll;oBe,ZJ;iTb, w;o ,ZJ;JIH BeKTopiB |
AB, BC i AC cnpaBµ;myeTbCH |
HepiBHiCTb IACI ~ IABI + IBCI.
2) ,lJ;oBe,ZJ;iTb, IIl;O ,ZJ;JIH 6y,ZJ;b-HKHX BeKTOpiB ai b cnpaB,ZJ;myGTbCH
HepiBHicTb la + bl ~ 1a.1 + 1bl.
15.,n;o ropH30HTaJibHOi 6aJIKH Ha ,Zl;BOX 0,ZJ;HaKOBHX HHTKaX ni,ZJ;- BirueHO BaHTam MaCOIO P. BH3Ha'ITe CHJIY HaTnry HHTOK (MaJI. 230).
16.3 HKOIO CHJIOIO F Tpe6a yTpHMyBaTH BaHTam Maco10 P Ha noxHJiiH IlJIOIIl;HHi, w;o6 BiH He CKO'lyBaBCHBHH3?
17.ll!!_Ho TO'IKHA(a1; Y1) i B(x2; Y2). ,ll;oBe,ZJ;iTb, w;o BeKTOpH AB i BA npoTHJiemHo HanpnMJieHi.
18. |
,ll;oBe,ZJ;iTb, II\.O BeKTOpH a(].; 2) i |
b(0,5; 1) O,ZJ;HaKOBO HanpHMJieHi, |
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a BeKTOPH c(-_1; 2) i ~(0,5; |
-1) npOTHJieJKHO J!ailpHMJieHi. |
19. |
,ll;aH~ BeKTOPH a(3; 2) i b(O; -1). 3HaH,ZJ;iTb BeKTOP c = -2a + |
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+4b Ta H:oro a6COJIIOTHY BeJIH'IHHy.
20.A6coJIIOTHa BeJIH'IHHaBeKropa /..a µ;opiBHIOG 5. 3HaH,ZJ;iTb /.., HKIIl;O: 1) a(-6; 8); 2) a(3; -4); 3) a(5; 12).
21.Y TPHKYTHHKY ABC npoBe,ZJ;eHo Meµ;iaHy AM. ,ll;oBeµ;iTb, w;o
AM = T (AB + AC).
22.To11KH M i N e cepe,ZJ;HHaMH Biµ;pi3KiB Biµ;noBi,ZJ;HO AB i CD.
-- |
1 -- |
- |
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,ll;oBe,ZJ;iTb BeKTopHy piBHiCTb MN= 2 (AC |
+ BD) (MaJI. 231). |
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23. ,ll;aHO napaJieJiorpa_l\!_ABCD, AC _a, DB - |
b_(MaJI. 232). |
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BHpa3iTb BeKTopH AB, CB, CD i |
AD 11epe3 a i |
b. |
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8 KJiac |
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154 |
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[ |
8 |
[ |
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A_~-------i |
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0 |
A |
0 |
MaJI. 231 |
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MaJI. 232 |
24!" ,IJ;oBeAiTb, m;o BiAilOBiAHi KOOPAHHaTH KOJiiHeapHHX BeKTOpiB nponopn;i.HHi. I HaBnaKH, aKm;o BiAnOBi,11;Hi Koop,11;HHaTH ABOX BeKTOpiB nponopn;i.HHi, TO_n;i BeKT!)pH KOJiiH!l8PHi.
25.,IJ;aHO BeKTOPH a(2; -4), b(l; 1), c(l; -2), d(-2; -4).
BKamiTh napH KOJiiHeapHHX BeKTopiB. HKi a ,11;aHHX BeKTopiB O,ll;H8KOBO HanpaMJieH~ a HKi npOT!;(JleJKHO HanpHMJieHi?
26.Bi,ll;OMO, m;o BeKTOPH a (l; -1 ) Ta b(-2; m ) KOJiiHeapHi. 3Ha-
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H,ll;iTb, 'IOMYAOPJBHI06 |
m.:.. |
_ |
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27. |
,IJ;aHO BeKTOPH |
a (l; |
0), |
b(l; 1) Ta |
c(-1; 0). 3HaH,ll;i'!'bTa~i |
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'IHCJia A. i µ, |
m;o6 |
cnpaB,11;myBaJ1aca |
piBJ!iCTb_ c |
_/..a +-f,b. |
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28. |
,IJ;oBe)l;iTb, m:o AJIH 6y,11;h-HKHX BeKTOpiB a i |
b: |
(ab)2 ~a b2• |
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29. |
3H8HAiTb K~T MiJK BeKTOpaMH a (l; |
2) |
i b(1; |
-+). |
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30!< ,lJ;aHO BeKTOpH ai b. 3H8HAiTb a6COJIIOTHY BeJIH'IHHYBeKTOpa a+ b, HKlIJ;O a6COJIIOTHi BeJIH'IHHHBeKTOpiB ai b ,ll;OpiBHIOIOTb 1, 8 KYT MiJK HHMH 60°.
31.3HaAAiTb KYT MiJK BeKTOpaMH ai a+ b nonepe,ll;HbOl 38A8'1i.
32.,IJ;aHo BepWHHH TpHKyTHHKa A (l; 1), .8(4; 1); C(4; 5). 3HaiiAiTb KOCHHYCH KYTiB TpHKYTHHK8.
33.3H8HAiTb KYTH TpHKYTHHKa 3 BepmHHaMH A (O; -J3), .8(2; -J3),
c(: ; ~).
34.,IJ;oBe)J;iTb, lIJ;O BeKTOpH a(m; n) i b(-n; m) nepneH)J;HKYJIHpHi a6o AOPiBHIOIOTb HYJIIO.
35.)J;aHO BeKTOpH a(3; 4) i b(m; 2). IIpH HKOMY 3H8'1eHHim n;i BeKTOpH nepneHAHKYJIHpHi?
36.,IJ;aHO Be_!(TOpH_ a(l; 0) i b(l; 1 ). 3H8H,ll;iTb T8Ke 'IHCJiqA, m;o6
BeKTOp a+ A.b 6yB nepneHAHKYJIHPHHM AO BeKTOpa a. |
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37. )J;oBe)l;iTb, lIJ;O KOJIH |
ai b - OAHHH'IHiHeKOJiiHeapHi BeKTOpH, |
TO BeKTOpH a+ b i |
a- b Bi,ll;MiHHi Bi,11; HYJIH i nepneHAHKy- |
JIHpHi.
38!" ,IJ;oBe,11;iTb, m;o cyMa KBa,11;paTiB AiaroHaJieii napaJ1eJ1orpaMa ,11;opiBHI06 cyMi KBa,11;paTiB :Horo CTopiH.
39!< ,IJ;aHO CTOPOHH TPHKYTHHKa a, b, c. 3H8HAiTb :Horo MeAi8HH m a, m b, me.
§ 10. |
BeKTOpH |
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155 |
40. |
,lJ;oBe,D;iTb, w;o reOl\leTPH'IHeMicu;e TO'IOK,cyMa KBaµ;paTiB Bi,D;- |
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CTaHeii Bi,D; HKHX ,D;O ,D;BOX ,D;aHHX TO'IOKCTaJia, 6 KOJIOM 3 a;eHT- |
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poM y cep~,D;HHl Biµ;~i3Ka.i._ HKHH_ cnoJiy11as µ;aHi |
TO'IKH. |
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41. |
BeKTOPH a+ b i a - b |
nepneH,D;HKYJIHpHi. |
,D;oBe,D;iTb, |
w;o |
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42. |
1ti1=161. |
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,lJ;oBe,D;iT:b 3a µ;on0Moro10 BeKTopiB, w;o µ;iaroHaJii poM6a nepneH- |
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AHKYJIHpHi. |
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43. |
,lJ;aHo 'IOTHPHTO'IKH:A(l; 1), B(2; 3), C(O; 4), D(-1; 2). ,lJ;oBe- |
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44. |
µ;iTb, w;o 'IOTHPHKYTHHKABCD - npHMoKyTHHK. |
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,lJ;aHo 'IOTHPHTO'IKHA(O; 0), B(l; 1), C(O; 2), D(-1; 1). ,D;oBe,D;iTb, |
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w;o 'IOTHPHKYTHHKABCD - |
KBa,11;paT. |
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45. |
Cepe,D; BeKTOpiB a(- |
~ ; : ), b ( : ; : ), C(0; -1 ), d (: ; - |
: ) |
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3Haii,D;iTb 0,IJ;HHH'IHii |
3a3Ha'ITe_,HKi 3 HHX KOJiiHeapHi. - |
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46. |
3Haiiµ;iTb O,D;HHH'IHHHBeKTOp e, KOJiiHeapHHH BeKTopy a(6; |
8) |
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i 0,D;HaKOBO 3 HHM HanpHMJieHHH.
47.,lJ;aHO KOOp,D;HHaTHi BeK~PH e1(J; O)i e2(0; 1). "lloMy µ;opiBHIOIOTb KOOp,D;HHaTH BeKTopa 2e1 - 3e2?
48!'1) ,D;aHo TPH TO'IKH 0, A, B. To11Ka X µ;iJIHTb Biµ;pfaoK AB
YJ!jµ;aomeHHi /... : µ, no'IHHa_IO'IHBiµ; TO_'IKHA. BHpa3iTb BeKTop OX qepe3 BeKTopH OA = a i OB = b.
2) ,D;oBe,D;iTb, w;o Me,D;iaHH TPHKYTHHKa nepeTHHaIOTbC$1 B O,D;Hiii TO'la;i,HKa µ;iJIHTb i'xy Bi,D;HOmeHHi 2: 1, IlO'IHHaIO'IHBi,D; Bi,D;no-
Bi,D;HHX BepmHH.
49. ,lJ;oBe,D;iTb, IIJ;O npoe~a;iH aBeKTOpa CHa BiCb a6ca;m~ 3 KO<!PAH-
HaTHHM BeKTOPOM e1(l; 0) aaµ;a6TbC$1 cPOPMYJIOIO a= kei, µ;e
k = ce,.
50. ,lJ;oBe,D;iTb, w;o npoeKa;iH CYMH BeKTopiB Ha BiCb µ;opiBHI06 cyMi npoeKu;i:H µ;oµ;aHKiB Ha TY caMy Bicb.
9 1tJiac
§ 11. ITOAIBHICTb «l>IrYP
100. IIEPETBOPEHHH 110,IJ;IBHOCTI
IIepeTBopeHHH <l>irypH F y <l>irypy F' HaaHBaeT1>CH neper6open- H.HM noiJi6HOCTi, HK~O npH D;bOMY nepeTBopeHHi Bi,Zl;CTaHi Mi)K TO"tJKaMH aMiHIOIOTl>CH B o,n;Hy :H Ty caMy KiJI1>KiCT1> paaiB (MaJI.
II;e oaHa"tJae, ~o KOJIH ,n;oBiJI1>Hi TO"'IKHXi Y <l>irypH F npH nepeTBopeuHi no,n;i6HocTi nepexo,n;HTI> y TO"'IKHX', Y' cl>irypH F', TO X'Y' =
= k · XY, npH"tJOMY "'IHCJIO k - o,n;He i Te caMe ,Zl;JIH Bcix TO"'IOK Xi Y. 't{HcJiok Ha3HBaeT1>CH 1coe<jJiqienroM noiJi6nocri. SIK~o k =
= l, nepeTBopeHHH no,n;i6HocTi, O"'leBH,n;Ho,e pyxoM. |
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Hexa:H F - ,n;aHa <Pirypa i 0 - |
<l>iKcoBaHa |
TO"tJKa (MaJI. 234). |
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't{epea,n;oBiJI1>Hy TO"'IKYX <l>irypH F npoBe,n;eMo npoMiH1> OX i |
Bi,n;- |
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KJia,n;eMo |
Ha Hl>OMY Bi,n;piaoK OX', ~o ,n;opiBHIOe k · OX, ,n;e |
k - |
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,n;o,n;aTHe |
"'IHCJIO. IIepeTBopeHHH <l>irypH F, npH |
HKOMy |
KO)KHa n: |
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TO"'IKa X |
nepexo,n;HTb y TO"'IKY X', |
no6y,n;oBaHy |
TaKHM |
cnoco6oM, |
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Ha3HBa6TbCH ZOMOTeTie10 6iiJHOCH.O qenrpa 0. qHCJIO k Ha3HBa6TbCH
JCoe<jJiqienroM zoMorerii, <l>irypH F i F' TU'LH.UMU.
F
xG;
MaJI. |
233 |
0 |
MaJI. 234 |
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Teo p e Ma |
11.1. I'oMoTeTisi |
e nepeTBopennRM |
noiJi6nocTi. |
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.n; o B e ,n; e H H s. Hexa:H 0 - u;eHTP roMoTeTi'i,k - |
Koe<Piu;ieHT |
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roMOTeTii, xi y - |
,Zl;Bi ,Zl;OBiJil>Hi TO"'IKH<PirypH (MaJI. 235). y peayJib- |
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TaTi roMoTeTii TO"'IKHXi Y nepexo,n;HTb y TO"'IKHX' i Y' Bi,n;noBi,n;Ho
§ 11. Ilo,11,i6HiCTb <Piryp |
157 |
Ha rrpoMeHJIX OX i OY, rrpH'IOMYOX' = k · OX, OY' = k |
OY. |
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3BiACH MaGMO BeKTOpHi piBHOCTi: |
-- |
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- - |
- - |
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OX'= kOX, OY' = kOY. |
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BiAHHBlliH n;i piaHocTi rro'!JieHHO,AicTaHeMo: |
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OY ' - |
OX' = |
k(OY - OX). |
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Ocaj~!>KH OY' = -OX'= 4°! Y', OY - OX= XY, TO X'Y' = |
k X |
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X XY. 0Tme, IX'Y'l - klXYI, T06To X'Y' = kXY. TaKRM 'IHHOM, roMoTeTiH G rrepeTaopeHHHM rroAi6HocTi. TeopeMy AOBeAeHo.
IlepeTBopeHHH rro,n;i6HOCTi lliHpOKO BHKOp~CTOBYGTbCH Ha npaKTHD;i npn BHKOHaHHi KpecJieHb AeTaJieii MaWHH, cnopyA, miaHiB Micn;eaocTi Toxn;o. TaKi ao6pameHHH - IIOAi6Hi nepeTBopeHHH yHBHHX ao6pameH1> B HaTypaJibHY aeJIH'IHHy.Koe4>in;iGHT IIOAi6HOCTi IIPH ll;bOMY Ha3HBaeTbCH MacmTa6oM. HarrpHKJiaA, HKID;O AiJIHHKa Micn;eaocTi ao6pamyeTbCH B MaCWTa6i TO n;e 03HaqaG, In;O OAHOMY caHTHMeTpy Ha IlJiaHi BiAilOBiAaG 1 M Ha Micn;e- BOCTi.
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Man. 235 |
MaJI. 236 |
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0 |
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3 |
a A a q a |
(4). Ha MamoHKy 236 ao6pameHo rrJiaH caAR6H |
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y |
MacmTa6i |
1: 1000. |
BHaHa'ITe poaMipR caAR6H (,n;oB:>KHHY |
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i |
lliHpHHy). |
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p |
0 a B, H a a H H JI. |
,D;oB:>KHHa i lliHPHHa ca,n;H6H Ha IIJiaHi |
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4 CM i 2, 7 CM. OcKiJibKH rrJiaH BHKOHaHO y MaCWTa6i 1 : 1000, TO poaMipH caAH6H AOPiBHIOIOTb aiArroBiAHO 4 · 1000 CM=
= 40 (M), 2, 7 · 1000 CM = 27 (M).
101. BJIACTHBOCTI IIEPETBOPEHH.H IIO~IBHOCTI
TaK caMo HK i AJIH pyxy, AOBOAHMO, xn;o npH nepeTaopeHHi IIOAi6HocTi TPH TO'IKHA, B, C, HKi JiemaTh Ha o,n;Hiii npHMiii, nepeXOAHTb y TPH TOl{KH A1, B1, C1, HKi Tem Jie:>KaTh Ha OAHiii rrpnMiii. Ilp111I0My, nKxn;o TO'!KaB Jie:>KHTb Mim TO'!KaM11Ai C, TO B 1nemHTb Mim TO'IKaMHA 1 i CI· 3aiACH a11rrJ111aae, xn;o neperBopennsi nooi6nocri nepeBoouT"b npsiMi y npsiMi, niBnpsiMi - y ni6npsiMi, Biopiaxu - y Biopiaxu.
9 KJIRC |
158 |
B,
MaJI. 237
,ll;oBe,u;eMo, ~o nepersopenHsi noili6HocTi a6epizae Kyru Mi31C nisnpsiMUMU.
CrrpaB,u;i, Hexaii KYT ABC rrepeTBopeHH.HM no,u;i6HocTi a Koe<l>in;i- eHTOM k nepeBO,Zl;HTbC.H B KYT A1B1C 1 (MaJI. 237). 3acTocyeMo ,u;o KyTa ABC nepeTBopeHH.H roMoTeTi'iBi,u;HocHo iioro BepmHHH Ba Koe- <l>in;ieHTOM roMoTeTi'ik. IlpH IJ;bOMY TO'IKHA i C nepeii,u;yTb y TO'IKH A z i C2. TpHKYTHHK A zBC2 ,u;opiBHIOe TPHKYTHHKY A1B1C1 aa TpeTbOIO OOHaKOIO. 3 piBHOCTi TpHKYTHHKiB BHIIJIHBae piBHiCTb
KyTiB AzBC2 i |
A1B1C1. 0Tme, KYT ABC ,u;opiBHIOe KYTY A1B1Ci, ~o |
ii Tpe6a 6yJio |
,u;oBeCTH. |
102. IlO,lI;IBHICTb Cl>IrYP
,ll;Bi <l>irypH Ha3HBaIOTbC.H noiJi6HUMU, .HK~O BOHH nepeBO,Zl;.HTbC.H o,u;Ha B O,Zl;HY nepeTBOpeHH.HM no,u;i6HOCTi. Ilo,u;i6HiCTb <l>iryp II03Ha- 'laIOTbcnen;iaJibHHM 3HaKoM: =·3arrHc F = F' 'IHTaeTLcH:•<l>irypa F no,u;i6Ha <Pirypi
,ll;oBe,u;eMo, ~o Ko.n.u tjJizypa F1 noiH.6Ha tjJizypi F2, a tjJizypa F2 noiJi6Ha tjJizypi F 3, To tjJizypu Fi i F 3 noili6Hi.
Hexaii X 1i Y 1 - ,u;Bi ,u;oBiJibHi TO'IKH<l>irypH F,. IlepeTBopeHH.H no,u;i6HocTeii, .HKe nepeBOAHTb <Pirypy F 1B F 2, nepeBOAHTb n;i TO'IKH y TO'IKHX 2, Y2, ,Zl;Jl.H .HKHX X 2Y 2 = k1X1Y1.
IlepeTBopeHiin: no,u;i6HOCTi, n:Ke nepeBOAHTL <l>irypy F 2B F 3, nepeBOAHTL TO'IKHX 2, Y2 y TO'IKHX 3, Y3, ,Zl;JIH HKHX X 3Y3 = k2 . X 2Y2.
3 piBHOCTeH |
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X 2Y 2 = |
k1X1Y1, X3Y3 = k2X2Y 2 |
BHIIJIHBas, ~o X 3Y3 = k1 |
· k2X1Y1. A n;e oaHa'lae,~o nepeTBopeH- |
HH <PirypH F 1B F 3, n:Ke p;icTaeMo npH nocJii,u;OBHOMY BHKOHaHHi ,u;Box nepeTBopeHL no,u;i6HocTi, e no,u;i6HiCTb. Orme, <PirypH F 1i F3 no,u;i6Hi, ~o ii Tpe6a 6yJIO ,u;oBeCTH.
Y aarrHcy no,u;i6ttocTi TpHKYTHHKiB: !:::,. rrepe,u;- 6aqaeTLCH, ~o BepIIIHHH, .HKi cyMi~aIOTbC.H nepeTBOpeHH.HM no,u;i6-
HOCTi, CTOHTb Ha Bi,u;rroBi,u;HHX Micn;n:x, To6To A nepexo,u;HTb B A 1,
B - B B1, c - B C1.
§ 11. IIoAi6HicTb <l>iryp |
159 |
3 BJiaCTHBOCTeH rrepeTBOpeHHH IIOAi6HOCTi BHIIJIHBae, ~o y noiJi6nux t}Jizyp BiiJno8iiJHi ICYTU piBHi, a BiiJnoBiiJni BiiJpi31CU nponOplfiUHi. 3oKpeMa, y nooi6nux TpU1CYTHU1CaX ABC i A1B1C1
103. 03HAKA 110,l:t;IBHOCTI TPHKYTHHKIB
3A ,l:t;BOMA KYTAMH
T e o p e M a 11.2. HKUfO iJBa JCyru oiJnozo TpuJCyrnuJCa BiiJnoBiiJno iJopi8H1010Tb iJBoM JCyTaM iJpyzozo rpuJCyTHUJCa, TO TaJCi TpU1CYTHU1CU no0i6ni.
,II; o e e A e H H a. Hexaii y TPHKYTHHKiB ABC i A 1B1C 1 MaeMo
LA= L Ai. LB= L B1. ,ZI;oBeAeMo, ~o 6. ABC= 6. A1B1C1.
Hexaii k = :~,. 3acTocyeMo AO TPHKYTHHKa A1B1C1 rrepeTBO-
peHHH IIOAi6HOCTi 3 Koe<l>i~ieHTOM IIOAi6HOCTi k, HarrpHKJiaA rOMOTeTiIO (MaJI. 238). IlpH ~1>0My AicTaHeMo AeHKHH TPHKYTHHK A 2B 2C2, ~o AOPiBHIOe TPHKYTHHKY ABC. CrrpaBAi, OCKiJihKH rrepeTBopeHHH
rroAi6HocTi a6epirae KYTH, To L |
A 2 = |
L Ai. L B 2 = L Bi. 0Tme, |
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y TPHKYTHHKiB |
ABC i |
A 2B2C2: |
LA= L A 2, LB= L B 2. ,II;aJii |
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A 2B2 = kA 1B1 = AB. 0Tme, TPHKYTHHK ABC AOPiBHIOe TPHKYTHH- |
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KY A 2B 2C2 aa |
Apyro10 |
oaHaKoIO |
(aa |
cTopoHoIO i rrpHJierJIHMH AO |
Hei KyTaMH). |
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[
0
MaJI. 238
OcKiJihKH TPHKYTHHKH A1B1C1 i A 2B2C2 roMoTeTH'IHi,i oTme, rroAi6Hi, a TPHKYTHHKH A 2B 2C2 i ABC piBHi i TOMY rem rroAi6Hi, TO TPHKYTHHKH A1B1C 1 i ABC rroAi6Hi. TeopeMy AOBeAeHo.
3 a A a q a (15). IlpaMa, napaJieJI1>Ha cTopoHi AB TpHKYTHHKa ABC, rrepeTHHae iioro cTopoHy AC y TO'l~i Ai. a CTOpoHy BC y TO'I~ B1 . ,II;oeeAiTh, ~o 6. ABC= 6.A1B1C1.