Геометрия7_9
.pdf8 KJiac |
140 |
MaJI. 209 |
MaJI. 210 |
2) Ilo6yp;yiiTe <l>irypy, B aKy nepexop;HTL sip;piaoK AB npn nosopoTi H8BKOJIO TO'IKH0 Ha KYT 60° aa rop;HHHHKOBOIO CTpiJIKOIO.
26.II06y,11;yiiTe <l>irypy, B aKy nepexop;HTL TPHKYTHHK ABC npn nosopoTi iioro Has:KoJio sepmHHH C Ha KYT 60°.
27.,ZJ;aHo TO'IKHA, B, C. II06y,11;yiiTe TO'IKYC', B aKy nepexop;HTL TO'IKac npH napaJieJILHOMY nepeHeceHHi, ~o nepeBO,ll;HTL TO'IKY
A y B. |
+ 1, |
28; IlapaJieJILHe nepeHeCeHHS: 3a,11;86TLCa <l>oPMYJI8MH x' = x |
y' = y - 1. B a:Ki TO'IKHnpn ~LOMY napaJieJILHOMY nepeHeceHHi nepexop;aTL TO'IKH(0; 0), (1; 0), (0; 2)?
29.3HaH:,a;iTL seJIH'IHHHa i b y <l>opMyJiax napaJ1eJ11>Horo nepeHe-
+a, y' = y + b, aK~o: 1) TO'IKa(1; 2) nepe-
xo,a;HTb y TO'IKY(3; 4); 2) TO'IK8(2; -3) - y TO'IKY(-1; 5);
3)TO'IKa(-1; - 3) - y TO'IKY(0; -2).
30.Ilpn napaJieJihHOMY nepeHeceHHi TO'IKa(1; 1) nepexo,a;HTb y TO'IKY(-1; 0). B aKy TO'IKYnepeXO,ll;HTb Il0'18TOKKOOp,ll;HHaT?
31."llH icHye napaJieJibHe nepeHeceHHa, npH aKoMy: 1) TO'IKa(1; 2)
nepexo,a;HTL y TO'IKY(3; 4), a TO'IKa(0; 1) - y TO'IKY(-1; O); 2) TO'IKa(2;-l)nepexo'AHTbYTO'IKy(l;0), 8TO'IK8(-1;3)- y TO'IKY(0; 4)?
32.IlpaMi AB i CD - napaJieJibHi. To11KH Ai D JiemaTb a o,11;Horo 6oKy sip; ci11Hoi BC. ,ZJ;ose,a;iTh, ~o npoMeHi BA i CD o,a;HaKoeo HanpaMJieHi.
33.,ZJ;osep;iTb, ~o B aa,11;a11i 32 npoMeHi BA i CD npoTHJiemHo HanpaMJieHi, a~o TO'IKHAi D JiemaTb no piaHi 6oKH sip; ci11Hoi
BC.
34."l!oTHPHKYTHHK ABCD - napaJieJiorpaM. Cepe,11; npoMeHiB AB, BA, BC, CB, CD, DC, AD, DA HaasiTb napn op;HaKOBO HanpaM-
JieHHX i npOTHJI0)KH0 Hanpa:MJieHHX npoMeHiB.
35.,ZJ;ose,a;iTb, ~o sip;piaKH O,ll;H8KOBOl ,ll;OB)KHHH i KYTH a O,ll;H8KOBOIO rpap;yCHOIO MipoIO cyM~8lOTbCa pyxoM.
§ 9. Pyx |
141 |
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36!'Y |
rrapaJieJiorpaMiB ABCD i A1B1C1D1: AB= A1B1, AD= |
= |
A,D, i LA= LA,. ,IJ;oBep;iTL, m;o rrapaJieJiorpaMH piBHi, |
To6To cyM~aIOTLCa pyxoM.
37'!',IJ;oBep;iTb, m;o poM6H piBHi, RKID;O BOHH MaIOTb piBHi p;iaroHaJii.
38.,IJ;oBep;iTL, m;o p;Ba KOJia op;HaKoBoro pap;iyca piBHi, To6To cyMim;aIOTbCR pyxoM.
§10. BEKTOPH
91. ABCOJIIOTHA BEJIH'IHHA I HAIIP.HM BEKTOPA
BeKropoM HaauBaTHMeMo HarrpaMJieHHii Bip;piaoK (MaJI. 211). HarrpRM BeKTopa BH3Ha11aeTLca iioro rro11aTKOM i KiHa;eM. Ha MaJIIOHKY HarrpaM BeKTOpa II03Ha'l&IOTbCTpiJIKOIO. ,IJ;Jia II03Ha'leHHR BeKTopiB BHKOPHCTOByBaTHMeMO M&Jli 6yKBH JiaTHHCbKOro ampaBiTY a, b, c, .... MoxcHa TaKO)K rroaHa'IHTHBeKTop, BKaaaBmH iioro IIO''laTOKi KiHea;b. IlpH TaKOMY crroco6i II03Ha'leHHRBeKTopa Ha rrepme Mica;e CT&BJIRTL iforo rro11aTOK. IHop;i aaMiCTb CJIOBa •BeKTop• Hap; 6yKBeHHM rroaHa11eHHRM BeKTopa cTaBJIRTb CTPiJIKY ~60 pucA. BeKTop Ha MaJIIoHKY 211 MO)KHa rroaHa'IHTHTaK: a, a a6o A .
BeKTOPH AB i CD Ha3HBalOTbCR oana1C060 nanpSLM.AeHUM_Jb RKID;O op;HaKOBO HarrpaMJieHi ii rriBrrpaMi AB i CD. BeKTOPH AB i CD HaauBaIOTLCR nporu.n.e:JH:HO nanp.RM.Jt.eHUM.U, RKID;O rrpoTHJl~)K!!_O HarrpaMJieHi ii rriBrrpaMi AB i CD. Ha MaJIIOHKY 212 BeKTopu a i b op;HaKoBo HarrpaMJieHi, a BeKTOPH a i c rrpoTHJie)KHO HarrpaMJieHi.
A6co.n.10rno10 ee.n.u'tuno10 (a6o Mop;yJieM) BeKTopa HaauBaeTLCR AOBX<HHa _Bip;piaKa, m;o ao§pa)Kae BeKTOp. A6coJIIOTHa BeJIH'IHHa BeKTopa a rroaHa11aeTbCR !al.
Ilo'laToKBeKTopa MO)Ke a6iraTHca a iioro KiHa;eM. TaKHii BeKTop Ha3HBaTHMeMo HYAbOB'!_M BeKTopoM. HyJILOBHii BeKTop rroaHa'laeTLca HyJieM a pucKoIO (0). Ilpo HarrpaM HYJILOBoro BeKTopa He roBopaTL. BBa)KaGTLca, m;o a6coJIIOTHa BeJIH'IHHaHYJILOBoro BeKTopa p;opiBHIOe HYJIIO.
8
A
Man. 211 |
Man. 212 |
8 KJiac |
142 |
92. PIBHICTb BEKTOPIB
,IJ;ua BeKTOpH HR3HBRIOTbC.H pi6HUMU, HKlll;O BOHH cyMiw;aIOTbC.11 rrapaJieJibHHM rrepeHeceHHHM. D;e oaHa'lae,w;o icHye rrapaneJibHe rrepeHeceHH.11, .HKe rrepeBO,IJ;HTb IIO'IRTOKi KiHe~ O,ll;HOro BeKTOpa Bi,ll;IIOBi.AHO y IIO'IRTOKi KiHe~ ,zi;pyroro BeKTOpa.
3 03Ha'leHH.H piBHOCTi BeKTOpiB BHIIJIHBRG, w;o oOHaH:080 HanplUIClU!Hi u pi8Hi 3a a6COJUOTHOK> 8eJr.U'tUHOK>.
I HRBIIRKH, JlH:UfO 8eH:TOpu oih'4H:080 HanpRMJU!Hi u pi8Hi aa a6cOJr.K>THOK> 8eJr.U'tUHOK>, TO 80HU pi8Hi.
Crrpau,zi;i, Hexaii AB i CD - o,zi;HaKOBO HarrpHMJieHi BeKTOpH, piBHi aa a6coJIIOTHOIO ueJIH'IHHOIO(MaJI. 213). IIapaneJibHe rrepeHeceHHH, .HKe rrepeuo,zi;HTb TO'IKYC y TO'IKYA, cyMiw;ae rriBnpHMy CD a rriurrp.11Mo10 AB, ocKiJibKH BOHH o,zi;HaKoBo HanpSIMJieHi. A ttepea Te w;o ui,zi;piaoK AB ,zi;opiBHIOG ui.ApiaKy CD, To TO'IKaD a6iraeTbCH a TO'IKOIO B, T06To rrapaneJihHe rrepeHeceHHSI rrepeuo,zi;uTi. BeKTop CD y BeKTop AB. 0Tme, BeKTopH AB i CD piBHi, w;o ii Tpe6a 6yno ,ll;OBecTH.
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MaJI. |
214 |
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3 a A a 'Ia (2). "tiOTHpHKYTHHK_ABCD - |
napaJieJiorpaM. |
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,IJ;oue,zi;iTb piBHiCTh BeKTopiB AB i DC. |
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P o a u'SI a a H H SI. 3acTocyeMo ,zi;o BeKropa AB napaneJib- |
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He rrepeHeceHHSI, rrpu SIKOMY TO'IKaA nepexo,zi;HTb y TO'IKYD |
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(Man. 214). B peayJibTaTi ~oro nepeHeceHHSI TO'IKaA rrepeMi- |
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w;yeTbCH ua,zi;oum np.11Mo'iAD, a oTme, TO'IKaB rrepeMiw;yeTbCSI |
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ua,zi;oum |
rrapaJieJibHO'i rrpSIMo'i BC. Ilp.11Ma |
AB nepexo,zi;HTb |
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y rrapaneJibHY rrpaMy, T06To y rrpaMy DC. 0Tme, TOttKa B |
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nepeii,zi;e |
y TO'IKYC. TaKHM 'IHHOM,HaIIIe !!!!}>aJieJibHe rrepe- |
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HeceHH.11 rrepeuo,zi;HTb BeKTOP AB y BeKTOp DC, a n;e oaHa'laG, |
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w;o n;i_BeKTOPH piBHi. |
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Hexaii a - |
BeKTop i A - ,zi;ouiJibHa TO'IKa. To,zi;i Bi.A TO'IKHA |
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MOJKH~ Bi.AKJI8CTH O,ll;HH i TiJibKH O,ll;HH BeKTOp a', w;o ,zi;opiBHIOG BeKTOpy a.
Cnpau,zi;i, icHye e,zi;HHe rrapaJieJibHe rrepeHeceHHSI, rrpH SIKOMY rroqaTOK BeKTopa arrepexo,zi;~Th y TO'IKY4_. BeKTop, B SIKHii rrepe-
xo,n;HTb rrpH D;bOMY BeKTOp a, i G BeKTOP a'.
§ 10. BeKTOpH |
143 |
,ll;JUI rrpaKTH"'IHOrOBi,ll.KJiap;aHHH Bi,u; A8HOl TO"'IKH(D) BeKTopa, ~o p;opiBHIOG A8HOMY (AB), MOmHa CKOpHCTaTHCH TBep,u;meHHHM aap;aqi 2.
93. KOOP~HATH BEKTOPA
Hexaii BeKTop a Mae rroqaTKOM TO"<IKf A1(x1; Y1), a KiH~eM
TO'IKYA2(x2; y2). Koop,u.HHaTaMH seKTopa a HaaHBaTHMeMo 11ucJia |
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ai = X2 - |
Xi, a 2 = y2 - Yt · KoopAlfH&TH BeKropa cTaBHTHMeMo |
IIOPHA 3 |
6yKBeHHM II03H8"'1eHHBM BeKTOpa, y ,ll.8HOMY BHII8,ll.KY |
a(a1; a2) a6o npOCTO (ai; a 2). Koopp;HH8TH HyJibOBOro BeKTOpa ,u.opiBHIOIOTb HYJIIO.
3 <l>opMyJIH, HK8 BHpamae BiACT8Hb Mim ABOM8 TO'IKaMH11epea ix KOOPAHHaTH, BHIIJIHB86, ~o a6coJIIOTHa BeJIH'IHH8 BeKTOpa 3 KOOp,ll.HH8T8MH ai, a2 p;opiBHIOG ./at+ at
PiBHi Bexropu MalOTb piBHi 8iiln08i0Hi xoopiluHaTU. I HasrraKH,
"'"UfO y Bexropi8 BiilnOBiiJHi xoopiluHaru pi8Hi, ro Bexropu piBHi.
Crrpa~i, Hexaii A1(x1; Yi) i A2(x2; y2) - rro11aTOK i KiHe~b seKTopa a. OcKiJI&KH seKTOP a', ~o ,u;opiBHIOG iioMy, ,u;icTaIOTb a BeKTopa aIIl;tpaJieJibHHM rrepeHeceHHHM, TO HOrO IIO'laTKOMi KiH~eM 6y,u.yT& ei,u;nosi,u.Ho: A f(x1 +_ c;_y1 + d), A2(x2 + c; y2 + d). 3si,u;cH BHAHO, ~o o6H,U.Ba BeKTOPH a i a' M8IOTb Op;Hi i Ti caMi KOOPAHH8TH:
X2 - Xi, |
Y2 - |
y,. |
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,ll;osep;eMo Terrep o6epHeHe TBep,u;meHHH. Hexaii |
si,u.rrosi,u;Hi |
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Koop,u.uHaTH seKTopiB A 1A 2 i |
A IA2 |
pisHi. ,ll;osep;eMo, ~o BeKTopu |
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piBHi. |
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Koopp;HHaTH TO'IKHA f. a x2, y2 - |
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Hexaii xf i yf - |
Koopp;HHaTH |
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TO'IKHA2. 3a YMOBOIO TeopeMH X2 - |
X1 |
= x2 - |
xf, Y2 - |
Yt = y 2 - |
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- yf. 3si,u;cH x2 = X2 + xf - |
xi, y2 = Y2 |
+ yf - |
yi. IlapaJieJI&He |
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rrepeHeCeHHH, ~o aa,u.aGTbCH <l>oPMYJiaMH |
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x' = x + xi - |
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xi, y' = y |
+ yf - |
Yi. |
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rrepeBO,ll.HTb TO'IK~ TO'IKYAf, a TO'IKYA2 y |
TO'IKYA 2, To6To |
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seKTopH AiA 2 i AfA2 piBHi, ~o ii Tpe6a 6yJio ,u.osecTH. |
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0 |
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3 a ,u. a 11 a |
(7). ,ll;aHo TPH TO'IKH:A(l; 1), B( |
1; Q),_C(O; 1). |
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3Haii,u.iT& TaKy TO'IKYD(x; y), ~06 seKTOPH AB i CD 6yJIH |
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~~ |
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-1. |
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PoaB'HaaHHH. Koop,ll.HHaTH |
BeKTopa |
AB: -2; |
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Koop,u.HHaTH BeKTopa CD: x - |
0, y |
- |
1. OcKiJI&KH AB = |
CD, |
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TO |
X - |
Q = -2, y - |
1 = -1. 3Bi,ll.CH . 3H8XO,ll.HMO KOOp,ll.H- |
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HaTH TO'IKHD: x = -2, |
y = 0. |
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94. lJ,OlJ,ABAHH11 BEKTOPIB |
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CyM<?_106elCTOpi6 ai b 3 KOOPAHH8T8MH ai, a 2i bi, b2Ha3HBa6TbC$1 |
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BeKTop c a Koopp;HHaTaMH ai |
+ bi, |
a 2 |
+ b2, TOOTO |
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a(ai; a2) + b(bi; |
b2) = c(ai |
+ bi; |
a2 + b2). |
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8 KJiac |
144 |
,lJ;Jia 6yAI>-$1'.KHXBeKTOpiB a(a1; a2), b(b1; |
b2), c(c1; C2) |
a+ ii= 5 +a, a+ <ii+ c) =<ti+ 5) + c.
,lJ;Jia AOBeAeBHJI AOCHTI> nopiBHJITH siAnoBiAHi KOOPAHHaTH BeKTOpis, ~o cToaTL y .npasiii i JiiBiii 11acTHHax piBHOCTeii. Mu 6a11HMo, ~o BOHH pisHi. A seKTOPH a siAnoBiAHO piBHHMH KOOPAHHa-
TaMH piBHi. |
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Teo p e Ma 10.1. HKi 6 He 6yau To""" A, B, C, cnptU1ibcyencsi |
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AB+ BC= AC . |
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.n; o Be A e H Ha. Hexaii A(x1; y1), B(xlj__y_2), |
C(xa; |
Ya) - |
AaHi |
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TO'IKH (MaJI. 215). KOQIU\HH8TH |
seKTopa AB: |
x2 - |
Xi, Y2 - |
y,, |
a KOOPAHH8TH BeKTopa BC: xa - |
x2, Ya - y2. OTme, KOOPJJ;HHaTaMH |
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BeKTopa AB + BC~ - ~3 - |
y1. A o;e i e KOOM!fH8TH eeKTopa |
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AC. 0Tme, seKTop AB+ BC AOPiBHIOe seKTopy AC. TeopeMy AOBeAeHo.
A(x1 ; y 1 )
Ma.n. 215 |
Ma.n. 216 |
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A
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Ma.n. 217 |
Ma.n. 218 |
3 TeopeMH 10.1 M&E:MO TaKHA cnoci6 no6yAoBH cyMH AOBiJJLHHX BeKTOpiB ai 1i. Tpe6a BiA KiHD;JI BeKTOpa aBiAKJJ8CTH BeKTOp b'' ID;O
AOpiBHJOE: |
eeKTopy b. |
ToAi BeKTop, no'laToK .flKoro 36irau1>ca 3 |
Il0'18TKOMBeKTOpa a, |
a KiHeD;b - 3 KiHD;eM BeKTOpa b', 6yAe CyYOJO |
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BeKTOpiB |
ai b (MaJI. 216). TaKHA cnoci6 3H8XOJpKeHHJI cyMH ,ABOX |
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§ 10. BeKTopu |
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BeKTOpiB Ha3HBaGThC$1 •npa6UJl.OM rpu- |
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KYTHUKa• |
A0,11,aBaHHSI BeKTopiB. |
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Ix |
,ll;JISI BeKTOpiB ia cniJibHHM Il0'18TKOM |
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cyMa |
ao6pamaGTbCSI |
,11,iaroHaJIJIIO |
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napa;re;rorpaMa, no6y,11,0BaHOro Ha |
~x |
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BeKTopax |
( •npaeu.1w |
llfH!flJLeJLotpaMa•) |
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(MaJI. 217),__QnpaB,11,i, AB ±_BC |
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AC, |
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a |
BC = AD. |
0Tme, |
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AB + AD = |
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=AC. |
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Pi3HUijeJO BeKTOpiB a(a1; a2),_b(b1; b2) |
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MaJI. 219 |
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Ha3HBaGTbCSI TaKHH BeKTOP c(c1; C2), |
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SIKHH |
y _ |
c~•Mi |
_a |
l!_eKTopoM |
b |
,11,as |
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J!eKTO_p |
a:.. b + c = |
a. |
3Bi,ll,CH |
3HaX0,11,HMO |
KOOp,11,HHRTH |
BeKTOpa |
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c =a - |
b: |
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Ct |
= |
a1 - |
bi, c2 = |
a2 - |
b2. |
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_ 3 R_A.!l q |
a (11}. ,l(aHO |
BeKTOPH ia |
cniJibHHM |
no11aTKOM: |
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AB i |
AC (MaJI. |
218). ,l(osecTH, ..illQ AC - |
Al} - |
BC. |
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Po as'sraaHHSI. MasMo: AB+ BC= AC. A ~e oaHa- |
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11as, |
~o AC - |
AB = |
BC. |
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3Bi,ll,CH ,11,icTaGMO TaKe npaBHJIO ,11,JISI. no6y,11,0BH piaHHu,i ,11,BOX |
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BeKTopis. _ ~6 no6y,11,ysaTH |
BeKTop, |
SIKHH |
,11,opiBHIOG |
p!aHH_gi |
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BeKTOpiB a i b, Tpe6a Bi,11, 0,11,HiGi'.TO'IKHBi,11,KJIRCTH BeKTOPH a' i b',
~o AOPiBHIOIOTD |
i'.M. To,11,i BeKTop, no11aTOK S1Koro |
a6iraGThCSI |
3 KiH~eM BeKTOpa ~ a KiHe~ - 3 KiH~eM BeKTOpa a', |
6y,11,e pi3HH- |
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~eIO BeKTOpiB ii i |
b (MaJI. 219). |
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95. ,IJ;0,11.ABAHHH CHJI
CHJiy, npHKJia,11,eHy ,11,0 TiJia, apy11Ho ao6pamaTH BeKTopoM, HanpSIM SKOro a6iraGTbCS 3 Hailp$1MOM Ail CHJIH, a a6cOJIIOTH8 BeJIH'IHHanponopu,iH:Ha AO BeJIH'IHHHCHJIH. HK cBi,11,'IHTb,11,ocBi,ll,, npH TRKOMY cnoco6i ao6pa:>KeHHSI CHJI piBHOAiHHa ,11,BOX a6o p;eKiJibKOX CHJI, npHKJia,11,eHHX AO TiJia B O,ll;HiH TO'lu,i, ao6pa:>KaGTbCSI cyMoIO Bi,D;noBi,n;HHX i'.M seKTopiB. Ha MaJIIOHKY 220, _a Jf,O TiJia B TO'lu,iA npHKJiaAeHi ,D;Bi CHJIH, ao6pameHi seKTopaMH a i b. PiBHO- p;iAHa ~HX CHJI ao6pa:>KaGTbCSI BeKTOpOM
c =a+ 5.
3o6pa:>KeHHSI CHJIH y BHrJISl,ll;i cyMH CHJI, SIKi ,D;iIOTb y ,ll;BOX 38,D;a- HHX HanpSIMax, H83HBaGThCSI p03KJI8,ll;81_!HSIM CHJIH aa ~HMH Ha- 1!PSl1'!!8MH.TaK, Ha MaJIIOl!KY 220, a CHJiy c poaKJiap;eHo Ha CYMY CHJI a i b - CKJI8,11;0Bi CHJIH C.
PoaKJiaARHHSI seKTopa apy11Ho po6HTH aa ,ll;BOMa nepneH,D;HKyJISPHHMH OCSIMH. y ~OMY BHil8,ll;KY CKJia,D;OBi BeKTOpa H83HB8IOTbCSI npoe1cu,istMu BeKTopa Ha oci (MaJI. 220, 6).
8 KJiac |
146 |
y
A |
b |
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a) |
0 |
ti |
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6) |
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MaJI. 220 |
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MaJI. 221
3 a .u; a 11 a (16). 3 mc010 CHJIOIO F Tpe6a YTPHMyBaTH BaHTam BaroIO p Ha IlOXHJiiH IlJIO~Hi, m;o6 BiH He CKO'lyBaB- c.s BHH3 (MaJI. 221)?
Po a B'SI a a H HSI. Hexaii 0 - ~eHTP MacH BaHTamy, ,11;0 mcoro npHKJia,u;eHo CHJIY P. PoaKJia,D;eMo BeKTop P aa ,D;BOMa
B3a6MHO nepneH,D;HKYJISIPHHMH HanpSIMaMH, SIK IlOKa3aHO Ha MaJIIOHKY 221. CHJia OA nepneH,D;HKYJISlpHa ,11;0 nO_!HJioi nJIOID;HHH i He BHKJIHKa6 nepeMim;eHHSI BaHTamy. C.HJia F, SIKa YTPHMYEl BaHTam, Mas .u;opiBHIOBaTli aa BeJIH'IHHOIO i 6yTH npOTHJie:>KHOIO aa HanpSIMOM CHJii OB. ToMy F = P sin a.
96. MHOmEHH.H BEKTOPA HA 'IHCJIO
llo6YTKOM 6€K~_(a,; az) Ha '1.UC.1£0 ').., Ha3HBa6TLCSI
(w1; A.a2), To6To (a1; |
a 2)'A, = |
(A,a,; |
w2). |
3a 03Ha11eHHSIM |
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(a1; |
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'A,(a1; a2)· . |
3 03Ha11eHHSI onepa~~ MHO!KeHHSI BeKTOpa Ha 11.HCJIO BHilJIHBae, |
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m;o iJ1ui 6yiJb-R.KOto |
6e-,cropa a~ i 'tUc_ea ')..., p. |
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(A,+ µ)a=~-+=_ µa. |
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,l{aa 6yiJb-1'KUX iJBox_ 6eKT(ljJi6 |
a ! b i 1!UCl£a A. |
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A,(a |
+ b) = |
'A,a + 'A,b. |
§ 10. BeKTopH |
147 |
y |
y |
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a 1-----....,,A |
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0 |
"Aa 1 a, |
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(;>, >0) |
B |
A02 (ii< 0) |
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MaJI. 222
T ~ope Ma 10.2. A6co!J10THa 6~J£U1l.UHa 6e1'To-pa /..4 iJopiBn10e IA.I • lal. HanpSiM 6eKTO-pa A.a, RKJ40 a =I=- 0, a6izaerbca a nanpaMOJ!!. 6eKTopa a, R'ICJ40 A> 0, i nporu.n.e:HCHUU HanpR.My 6e1'TOpa a,
RKJ40 A< 0. |
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,IJ; o B e /1. e H H H. |
IT06y11.yeMo BeKTOpH OA i OB, HKi Bi11.n0Bi1I.HO |
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11.opiBHIOIOTb |
a i |
A.a |
(0 ::- no11aTOK KOOP/I,HHaT). Hexaii a1 i az - |
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KOOPII.HHaTH |
BeKTopa a. To11.i KOOPII.HHaTaMH TO'IKH A |
6YII.YTb |
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'IHCJia a1 i az, a |
KOOPII.HHaTaMH TO'IKH B - |
A.ai. Aaz (MaJI. 222). |
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PiBHHHHH np.HMO'iOA Mae BHrJIH/I,: |
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ax+ ~y = 0. |
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OcKiJibKH |
~e |
piBHHHHH aa11.oBOJibHHIOTb |
KOOPII.HHaTH |
TO'IKH |
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A(a1; az), TO iioro aa11.oBOJibHHIOTb i Koop/I,HHaTH TO'IKHB("Aa1; "Aaz).
3Bi/I,cH BHnJIHBae, ~o TO'IKaB JieJKHTb Ha npHMiH OA. Koop/I,HHaTH c1 i Cz 6y11.1>-HKO'iTO'IKHC, .HKa JiemHTb Ha niBnpHMiH OA, MaIOTb Ti caMi aHaKH, ~o ii KOOPII.HHaTH a1 i az TO'IKH A, a KOOPII.HHaTH 6y11.1>-HKO'iTO'IKH,~o JieJKHTb Ha niBnpHMiH, 11.onoBHHJibHiH 11.0 OA,
MaIOTb npOTHJieJKHi 3HaKH•
.HK~o A,> 0, TO TO'IKaB JieJKHTb Ha niBnpHMiH OA, a oTme, BeKTOPH a i A.a O/I,HaKOBO Hanp.HMJieHi. HK~O A_< 0, -TO TO'IKaB
JieJKHTb Ha /I,OilOBH.HJibHiH niBnpHMiH, i BeKTOpH a Ta A.a npOTHJiemHO Hanp.HMJieHi.
A6coJI10THa BeJIH'IHHaBeKTopa /,a 11.opiBHIOe:
IA.al = ../{fQ.1)2 + ("Aa2)2 = l"Ah/ar + a§ = IA.I lal.
TeopeMy 11.oBe/I,eHo. |
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t1\ |
3 |
a /1. a 11 a (17). ,ll;aH~O'IKH A(x1; y1) i B(xz; |
yz). ,ll;oBe- |
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/I,i•ri., |
~o BeKTOpH AB i |
BA npoTHJiemHo Hanp.HMJieHi. |
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Po a B'Ha a H H H. |
Koop/I,HHaTH BeKTopa AB: |
X2 - |
X1 i |
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Yz - |
Yi· Koop/I,HHaTH BeKTopa BA: X1 - X2 i y1 - |
yz. MH 6a- |
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'IHMo, ~o AB= (-l)BA. A ~e oaHa11ae, ~o BeKTOPH |
AB |
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i BA npoTHJiemHo HanpHMJieHi. |
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8 KJiac |
148 |
97.P03KJIA,z:t;AHH.H BEKTOPA 3A ,z:t;BOMA HEKOJIIHEAPHHMH BEKTOPAMH
,ll;Ba HeHyJILOBi BeKTOpH Ha3HBaIOTbCH ICOJLineapHUMU, HKI.IW BOHH JiemaTb Ha O,ll;HiH nps:Mili a6o Ha napaJieJibHHX npHMHX (MaJI. 223). KoJiiHeapHi BeKTOpH a6o O,ll;HaKOBO HanpHMJieHi, a6o npOTHJiemJ!O l!_anpHMJieHi.
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Hexau a i |
b ~ BiiJMiHHi 6W ny1tsi 1Conineapni seKTopu. ,!I;oBe,ll;e- |
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MO, |
ll.\O |
icnye 'tUCJtO A, |
TaJCe, |
~o |
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b = |
'Aa. |
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IlpHilYCTHMO, BeKTOpH a i ii |
O,ll;HaKOBO HanpHMJieHi. BeKTOPH |
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-. lbl |
- |
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6 |
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b l |
liil . |
a O)J;HaKOBO HanpHMJieHl l |
MaIOTb O,D;HY l |
TY m a COJIIOTHY |
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BeJIH'IHHY liil. |
0Tme, BOHH piBHi: |
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- |
lbl |
- |
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lbl |
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b=-·a=A.a |
A.=- |
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lal |
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' la!" |
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MaJI. |
223 |
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MaJI. |
224 |
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,Ll;JIH Blilla,lJ;KY npOTHJIC)KHO HanpHMJieHHX BeKTOpiB ai |
ii aHaJIO- |
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ri'IHOp06HMO BHCHOBOK, JJ..\O |
- |
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liil |
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b= |
-lal·a=A.a, |
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ll.\O H: Tpe6a 6yJio ,ll;OBecTH. |
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HexaH: |
a i |
ii |
- Bi,lJ;MiHHi Bi,ll; |
_HYJIH HeKOJiiHeapHi |
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BeKTOpH. |
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,Ll;oBe)J;eMO, |
ll.\O |
6yiJr.-1'1CUU 6e1CTOp |
c MO:Ht:1'a aanucaTU |
y |
8UZJt1'8i |
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HexaH: A i B - |
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c = 1vii |
+µii. |
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no'laTOKi |
KiHe~ BeKTopa c (MaJI. 2_24)!. qepea |
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TO'IKHA i |
B npoBep;eMo nps:Mi, napaJieJILHi BeKTopaM a i |
b. BoHH |
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nepeTHYTLCH y |
,D;es:KiH TO'IIVC. MaeMo: |
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AB= AC+ CB. |
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OcKiJibKH B~KTOPH ai AC KOJiiHeapH_!, TO AC = |
All. |
OcKiJibKH |
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BeKTOPH CB i |
b KO~iHeaP-_Hi, |
To_ CB = |
µb. |
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TaKHM |
'IHHOM,c = |
Iva+ µb, ll.\O ii |
Tpe6a 6yJio ,ll;OBeCTH. |
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§ 10. BeKTOPH |
149 . |
98. CKAJUIPBH:O )J;OBYTOK BEKTOPIB
C1CaJLapnuM ao6yT1COM BeKTOpiB a (a1; a2) i b( b1; b2) Ha3HBa6TbCH 'lHCJIOa1b1 + a2b2.
CKaJtHPHHH ,11;06yToK BeKTopiB aan11cy10Tb TaK caMo, HK ,11;06yToK |
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tIHCeJI. CKaJIHPHHH ,11;06yTOK a. a noaHa11aIOTb a2 i Ha3HBaIOTb |
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CKaJIHPHHM KBa,D;paTOM. OtieBH,ll;HO, a2 = |
lal2• |
3 oaHa11eHHH cKaJt~pHoro ,11;o~TKY BeKT~piB BHilJIHBae, ~o iJ;i1t |
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6yiJb -ll1'UX seJCropis a(a1; a2), b(b1; b 2) , c(c1; c2) |
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(a+bJc=ac+bc. |
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CnpaB)J;i, J1iB010 11acTHHOIO piBHOCTi e (a1 + b1 ) c1 + (a 2 + b2) c2, |
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a npaBOIO a1C1 |
+ a2C2 + b 1C1 |
+ b 2C2. 011g_BH,ll;HO, BOHH piBHi. |
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KyroM Mim HeHyJibOBHMH BeKTopaMH AB i AC Haa11BaeThcH KYT |
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/}AC..:. KyTOM |
Mim 6yp;b-HKHMH ,D;BOMa HeHyJibOBHMH BeKTopaMH |
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a i b Ha3HBa6TbCH KYT Mim |
BeKTOpaMH, ~o p;opiBHIOIOTb ,ll;aHHM |
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i MaIOTb cniJtbHHH notiaToK. BBamaIOTb, ~o KYT Mim |
o,11;HaK0Bo |
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HanpHMJleHHMH BeKTopaMH AOPiBHJOe HYJIIO. |
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T e op e M a 10.3. CKa;i1tpHuu |
iJo6yrox sexropis |
iJopi8n10e |
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iJo6yrxy ix a6coJi'IOTHUX |
8eJ!U1'f!H |
Ha 1'0CUHYC xyra Mi:HC HUMU. |
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A o B e A e H H H. Hexaii a i |
b - |
)J;aHi BeKTOpH i <p - |
KYT Mim |
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HHMH. MaeMo: |
= (a + ii) (a + ii) = (a + ii)ti + (a + ii)ii = |
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(a + 6)2 |
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= |
aa + ba + ab+ bb = a2 + 2ab + b2, |
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a6o |
1li + "b1 |
2 |
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1li1 |
2 |
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= |
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+ 1"b1 + 2a6. |
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3Bi,ll;CH BH)l;HO, ~o ~KaJ!HPH~H ,11;06yTOK ab BHpama6TbCH 11epea ,ll;OB)KHHH BeKTOpiB a, b i a + b, a TOMY He aaJle)KHTb Bi)!; BH6opy
CHCTeMH KOOp)J;HHaT, T06TO CKaJIHpHd ,ll;06yTOK He 3MiHHTbCH, HK~O CHCTeMy KOOp,D;HHaT BH6paTH cne~iaJibHO. BiabMeMO CHCTeMy Koop-
MaJI. 2 25