ࡱ>  UG ֚bjbjَ  WK]zzzz~~~vvvvv<TvX: /.]q QWSWSWSWSWSWSW$Yz[DwWY~}l/}}wW3Hzz :3H3H3H}"z~QWrrzzzz}QW3H3HK7U4J~QW4`8vvg@W6!!" " =0;VB8G=0 35><5B@VO : ?@O<0 V ?;>I8=0 1. V2=O==O ?@O<>W =0 ?;>I8=V. 1.1.  45:0@B>28E :>>@48=0B0E :>6=0 ?@O<0 287=0G0TBLAO @V2=O==O< ?5@H>3> AB5?5=O V, =02?0:8, :>6=5 @V2=O==O ?5@H>3> AB5?5=O 287=0G0T ?@O<C. 1.2. V2=O==O 284C  EMBED Equation.3  (1) =07820TBLAO 7030;L=8< 01> :0=>=VG=8< @V2=O==O< ?@O<>W =0 ?;>I8=V. /:I> ?@O<0 7040=0 7030;L=8< @V2=O==O<, B> WW :CB>289 :>5DVFVT=B 287=0G0TBLAO 70 D>@<C;>N  EMBED Equation.3 . 1.3. 03040T<>, I> :CB>289 :>5DVFVT=B T B0=35=A>< :CB0 =0E8;C ?@O<>W 4> >AV E, 0 @V2=O==O  EMBED Equation.3  =07820TBLAO @V2=O==O< ?@O<>W 7 :CB>28< :>5DVFVT=B><. V;L=89 G;5= b 2 FL><C @V2=O==V 4>@V2=NT 4>268=V 2V4@V7:0, I> 2V4B8=0T =0 >AV C 40=0 ?@O<0, @0ECNG8 2V4 ?>G0B:C :>>@48=0B. 1.4. V2=O==O  EMBED Equation.3  T @V2=O==O< ?@O<>W, I> ?@>E>48BL G5@57 B>G:C  EMBED Equation.3  V <0T :CB>289 :>5DVFVT=B k. 1.5. /:I> ?@O<0 ?@>E>48BL G5@57 B>G:8  EMBED Equation.3  B0  EMBED Equation.3 , B> WW :CB>289 :>5DVFVT=B 287=0G0TBLAO 70 D>@<C;>N  EMBED Equation.3 . V2=O==O  EMBED Equation.3   EMBED Equation.3  T @V2=O==O< ?@O<>W, I> ?@>E>48BL G5@57 42V B>G:8  EMBED Equation.3  B0  EMBED Equation.3 . 1.6. 7=0:>N ?0@0;5;L=>ABV 42>E ?@O<8E T @V2=VABL WE :CB>28E :>5DVFVT=BV2  EMBED Equation.3  7=0:>N ?5@?5=48:C;O@=>ABV 42>E ?@O<8E T >15@=5=V 70 01A>;NB=>N 25;8G8=>N B0 ?@>B8;56=V 70 7=0:>< :CB>2V :>5DVFVT=B8:  EMBED Equation.3  01>  EMBED Equation.3 . 0240==O. 825AB8 C<>28 ?0@0;5;L=>ABV B0 ?5@?5=48:C;O@=>ABV ?@O<8E, 7040=8E 7030;L=8< @V2=O==O<N 1.7. >@<0;L=5 @V2=O==O ?@O<>W. 5E09 =0 ?;>I8=V EC 7040=0 ?@O<0. @>2545< G5@57 ?>G0B>: :>>@48=0B ?5@?5=48:C;O@ 4> 40=>W ?@O<>W V =0725<> 9>3> =>@<0;;N . >7=0G8<> G5@57  B>G:C ?5@5B8=C =>@<0;V 7 40=>N ?@O<>N V 22815@5<> 70 4>40B=V9 =0?@O<>: =>@<0;V 2V4 B>G:8  4> B>G:8 . /:I>  EMBED Equation.3  T ?>;O@=89 :CB =>@<0;V, @  4>268=0 2V4@V7:0  EMBED Equation.3 B> @V2=O==O 40=>W ?@O<>W <>65 1CB8 70?8A0=> C 283;O4V  EMBED Equation.3 . V2=O==O B0:>3> 284C =07820TBLAO =>@<0;L=8<. /:I> 40=> 7030;L=5 @V2=O==O ?@O<>W  EMBED Equation.3 , B>, I>1 725AB8 9>3> 4> =>@<0;L=>3> 284C , B@510 2AV G;5=8 FL>3> @V2=O==O 4><=>68B8 =0 =>@<CNG89 <=>6=8:  EMBED Equation.3 . =0: =>@<CNG>3> <=>6=8:0 2818@0T<> ?@>B8;56=8< 4> 7=0:C 2V;L=>3> G;5=0 @V2=O==O, I> =>@<CTBLAO. 1.8. V4AB0=L 2V4 B>G:8 4> ?@O<>W. 5E09 40=> 45O:C ?@O<C V 4>2V;L=C B>G:C ; ?>7=0G8<> G5@57 d 2V4AB0=L 2V4 B>G:8 4> ?@O<>W. V4E8;5==O<  EMBED Equation.3  B>G:8  =07820TBLAO G8A;> +d, O:I> FO B>G:0 B0 ?>G0B>: :>>@48=0B @>7B0H>20=V ?> @V7=V 1>:8 2V4 ?@O<>W, B0  d, O:I> FO B>G:0 B0 ?>G0B>: :>>@48=0B @>7B0H>20=V ?> >48= 1V: 2V4 40=>W ?@O<>W ( 4;O B>G>: , I> ;560BL =0 40=V9 ?@O<V9  EMBED Equation.3 ). /:I> 7040=> :>>@48=0B8  EMBED Equation.3  EMBED Equation.3 B>G:8 , B0 =>@<0;L=5 @V2=O==O ?@O<>W  EMBED Equation.3 , B> 2V4E8;5==O  EMBED Equation.3  B>G:8  >1G8A;NTBLAO 70 D>@<C;>N:  EMBED Equation.3  )>1 7=09B8 2V4AB0=L d 2V4 B>G:8 4> ?@O<>W, 4>AB0B=L> >1G8A;8B8 2V4E8;5==O V 27OB8 9>3> <>4C;L: EMBED Equation.3  2. @O<0 V ?;>I8=0 2 ?@>AB>@V. 2.1. @>5:FVTN 25:B>@0  EMBED Equation.3  =0 2VAL u =07820TBLAO 2V4@V7>:  EMBED Equation.3 , 45 B>G:0  EMBED Equation.3  T ?@>5:FVTN =0 2VAL u B>G:8 , 0 B>G:0  EMBED Equation.3  - ?@>5:FVTN =0 FN 2VAL B>G:8 . @>5:FVO 25:B>@0  EMBED Equation.3  =0 2VAL u ?>7=0G0TBLAO  EMBED Equation.3  @>5:FVO 25:B>@0 a =0 2VAL u ?>7=0G0TBLAO  EMBED Equation.3  ;0AB82>ABV ?@>5:FVW: 2.1.1.  EMBED Equation.3 , 45  EMBED Equation.3  - :CB =0E8;C 25:B>@0 4> >AV u. 2.1.2. @>5:FVO AC<8 25:B>@V2 =0 45O:C 2VAL 4>@V2=NT AC<V WE ?@>5:FV9 =0 FN 2VAL  EMBED Equation.3  2.1.2. @8 <=>65==V 25:B>@0 =0 G8A;> 9>3> ?@>5:FVO <=>68BLAO =0 B56 A0<5 G8A;>.  EMBED Equation.3  2.1.3. @>5:FVO 4>2V;L=>3> 25:B>@0 S={X;Y;Z} =0 O:C-=51C4L 2VAL u 287=0G0TBLAO D>@<C;>N EMBED Equation.3, 45 e  >48=8G=89 25:B>@, =0?@02;5=89 70 2VAAN u. /:I> 40=V :CB8 , , , O:V 2VAL u CB2>@NT 7 :>>@48=0B=8<8 >AO<8, B> 5={cos;cos;cos} V 4;O >1G8A;5==O ?@>5:FVW 25:B>@0 S <>65 28:>@8AB>2C20B8AL D>@<C;0 EMBED Equation.3. 2.1.4. >268=8 ?@>5:FV9 25:B>@0 =0 :>>@48=0B=V >AV =07820NBLAO 9>3> (45:0@B>28<8) :>>@48=0B0<8. 2.2. /:I>  EMBED Equation.3  :CB8, O:V CB2>@NT 25:B>@ 0 7 :>>@48=0B=8<8 >AO<8, B>  EMBED Equation.3  =07820NBLAO =0?@O<;ONG8<8 :>A8=CA0<8 25:B>@0 0. 0?@O<;ONGV :>A8=CA8 T :>>@48=0B0<8 >48=8G=>3> 25:B>@0, O:89 <0T B>9 A0<89 =0?@O<>:, I> V 25:B>@ 0. 48=8G=89 25:B>@, O:89 <0T B>9 A0<89 =0?@O<>:, I> V 25:B>@ 0, =07820TBLAO >@B>< 25:B>@0 0 V ?>7=0G0TBLAO  EMBED Equation.3 . /:I>  EMBED Equation.3 ,  EMBED Equation.3 , B>  EMBED Equation.3 , 0 B0:>6  EMBED Equation.3  3. 5:B>@=89 4>1CB>: 25:B>@V2. 5:B>@=8< 4>1CB:>< 25:B>@0 a =0 25:B>@ b =07820TBLAO 25:B>@, I> ?>7=0G0TBLAO A8<2>;0<8 [ab] V 287=0G0TBLAO =0ABC?=8<8 B@L><C C<>20<8: <>4C;L 25:B>@0 [ab] 4>@V2=NT [a][b]sin, 45   :CB <V6 25:B>@0<8 aVb; 25:B>@ [ab] ?5@?5=48:C;O@=89 4> :>6=>3> 7 25:B>@V2 a i b; =0?@O<>: 25:B>@0 [ab] 2V4?>2V40T ?@028;C  ?@02>W @C:8 . &5 >7=0G0T, I> O:I> 25:B>@8 a, b V [ab] 72545=V 4> 7030;L=>3> ?>G0B:C, B> 25:B>@ [ab] <0T 1CB8 A?@O<>20=8< B0:, O: A?@O<>20=89 A5@54=V9 ?0;5FL ?@02>W @C:8, 25;8:89 ?0;5FL O:>W A?@O<>20=89 70 ?5@H8< A?V2<=>6=8:>< (B>1B> 70 25:B>@>< a), 0 2:07V2=89  70 4@C38< (B>1B> 70 25:B>@>< b). 5:B>@=89 4>1CB>: 70;568BL 2V4 ?>@O4:C A?V2<=>6=8:V2, 0 A0<5:EMBED Equation.3. >4C;L 25:B>@=>3> 4>1CB:C [ab] 4>@V2=NT ?;>IV ?0@0;5;>3@0<0, ?>1C4>20=>3> =0 25:B>@0E a V b:EMBED Equation.3. !0< 25:B>@=89 4>1CB>: <>65 28@060B8AO D>@<C;>N EMBED Equation.3, 45 5  >@B 25:B>@=>3> 4>1CB:C. 5:B>@=89 4>1CB>: [ab] 4>@V2=NT =C;N B>4V V BV;L:8 B>4V, :>;8 25:B>@8 a V b :>;V=50@=V. >:@5<0 [00]=0. /:I> A8AB5<0 :>>@48=0B=8E >A59 ?@020 V 25:B>@8 a i b 7040=V 2 FV9 A8AB5<V A2>W<8 :>>@48=0B0<8: a={X1;Y1;Z1}, b={X2;Y2;Z2}, B> 25:B>@=89 4>1CB>: 25:B>@0 a =0 25:B>@ b 287=0G0TBLAO 70 D>@<C;>N EMBED Equation.3 01> EMBED Equation.3. <VH0=89 4>1CB>: B@L>E 25:B>@V2. "@V9:>N 25:B>@V2 =07820NBLAO B@8 25:B>@8, O:I> 2:070=>, O:89 7 =8E 22060TBLAO ?5@H8<, O:89 4@C38< V O:89 B@5BV<. "@V9:C 25:B>@V2 70?8ACNBL 70 ?>@O4:>< =C<5@0FVW; =0?@8:;04, 70?8A a, b, c >7=0G0T, I> 25:B>@ a 22060TBLAO ?5@H8<, 25:B>@ b  4@C38<, A  B@5BV<. "@V9:0 =5:><?;0=0@=8E 25:B>@V2 a, b, c =07820TBLAO ?@02>N, O:I> 25:B>@8, I> WW CB2>@NNBL, ?VA;O ?@82545==O 4> 7030;L=>3> ?>G0B:C, @>7B0H>2CNBLAO C ?>@O4:C =C<5@0FVW 0=0;>3VG=> 4> B>3>, O: @>7B0H>2CNBLAO 25;8:89, 2:07V2=89 B0 A5@54=V9 ?0;LFV ?@02>W @C:8. /:I> 25:B>@8 a, b, c @>7B0H>20=V 0=0;>3VG=> 4> B>3>, O: @>7B0H>20=V 25;8:89, 2:07V2=89 B0 A5@54=V9 ?0;LFV ;V2>W @C:8, B> B@V9:0 F8E 25:B>@V2 =07820TBLAO ;V2>N. <VH0=8< 4>1CB:>< B@L>E 25:B>@V2 =07820TBLAO G8A;>, O:5 4>@V2=NT 25:B>@=><C 4>1CB:C [ab], ?><=>65=><C A:0;O@=> =0 25:B>@ A, B>1B> [ab]c. !?@0246CTBLAO B>B>6=VABL: [ab]c=a[bc]; 72060NG8 =0 F5 4;O ?>7=0G5==O 7<VH0=>3> 4>1CB:C [ab]c 28:>@8AB>2CTBLAO ?@>ABVH89 A8<2>;: abc. "0:8< G8=><, abc=[ab]c,abc=a[bc]. <VH0=89 4>1CB>: abc 4>@V2=NT >1 T<C ?0@0;5;5?V?540, ?>1C4>20=>3> =0 25:B>@0E a, b, c, 27OB><C 7V 7=0:>< ?;NA, O:I> B@V9:0 abc ?@020, 7V 7=0:>< <V=CA, O:I> FO B@V9:0 ;V20. /:I> 25:B>@8 a, b, c :><?;0=0@=V (V BV;L:8 2 FL><C 28?04:C), 7<VH0=89 4>1CB>: abc 4>@V2=NT =C;N; V=H8<8 A;>20<8, @V2=VABL abc=0 T =5>1EV4=>N V 4>AB0B=L>N C<>2>N :><?;0=0@=>ABV 25:B>@V2 a, b, c. /:I> 25:B>@8 a, b, c 7040=V A2>W<8 :>>@48=0B0<8 a={X1;Y1;Z1}, b={X2;Y2;Z2}, A={X3;Y3;Z3}, B> <VH0=89 4>1CB>: abc 287=0G0TBLAO D>@<C;>N EMBED Equation.3. 03040T<>, I> A8AB5<0 :>>@48=0B=8E >A59 22060TBLAO ?@02>N (@07>< 7 B8< T ?@02>N V B@V9:0 25:B>@V2 i, j, k). 030;L=5 @V2=O==O ?;>I8=8. V2=O==O ?;>I8=8, I> ?@>E>48BL G5@57 40=C B>G:C V <0T 40=89 =>@<0;L=89 25:B>@.  45:0@B>28E :>>@48=0B0E :>6=0 ?;>I8=0 287=0G0TBLAO @V2=O==O< ?5@H>3> AB5?5=O V :>6=5 @V2=O==O ?5@H>3> AB5?5=O 287=0G0T ?;>I8=C. C4L-O:89 (=5 @V2=89 =C;N) 25:B>@, ?5@?5=48:C;O@=89 4> 40=>W ?;>I8=8, =07820TBLAO WW =>@<0;L=8< 25:B>@><. V2=O==O EMBED Equation.3 EMBED Equation.3 287=0G0T ?;>I8=C, O:0 ?@>E>48BL G5@57 B>G:C EMBED Equation.3 V <0T =>@<0;L=89 25:B>@ EMBED Equation.3. >7:@820NG8 2 @V2=O==V (1) 4C6:8 V ?>7=0G0NG8 G8A;>  Ax0 By0 Cz0 ;VB5@>N D, ?@54AB028<> 9>3> C 283;O4V EMBED Equation.3. &5 @V2=O==O =07820TBLAO 7030;L=8< @V2=O==O< ?;>I8=8. >@<0;L=5 @V2=O==O ?;>I8=8. V4AB0=L 2V4 B>G:8 4> ?;>I8=8. >@<0;L=8< @V2=O==O< ?;>I8=8 =07820TBLAO WW @V2=O==O, 70?8A0=5 C 283;O4V EMBED Equation.3, 45 cos, cos, cos T =0?@02;ONG8<8 :>A8=CA0<8 =>@<0;V ?;>I8=8, @  2V4AB0=L 4> ?;>I8=8 2V4 ?>G0B:C :>>@48=0B. @8 >1G8A;5==V =0?@02;ONG8E :>A8=CAV2 =>@<0;V A;V4 22060B8, I> 2>=0 =0?@02;5=0 2V4 ?>G0B:C :>>@48=0B 4> ?;>I8=8 (O:I> 6 ?;>I8=0 ?@>E>48BL G5@57 ?>G0B>: :>>@48=0B, B> 281V@ 4>40B=L>3> =0?@O<:C =>@<0;V =5 <0T 7=0G5==O). 5E09 M* - 1C4L-O:0 B>G:0 ?@>AB>@C, d  2V4AB0=L 2V4 =5W 4> 40=>W ?;>I8=8. V4E8;5==O<  B>G:8 * 2V4 40=>W ?;>I8=8 =07820TBLAO G8A;> +d, O:I> B>G:0 * V ?>G0B>: :>>@48=0B ;560BL ?> @V7=V 1>:8 2V4 40=>W ?;>I8=8, V G8A;>  d, O:I> 2>=8 ;560BL ?> >48= 1V: 2V4 40=>W ?;>I8=8 (O:I> * ;568BL =0 A0<V9 ?;>I8=V, B> 2V4E8;5==O 4>@V2=NT =C;N). /:I> B>G:0 * <0T :>>@48=0B8 x*, y*, z*, 0 ?;>I8=0 7040=0 =>@<0;L=8< @V2=O==O< EMBED Equation.3 B> 2V4E8;5==O B>G:8 * 2V4 FVTW ?;>I8=8 ?>40TBLAO D>@<C;>N EMBED Equation.3. G5284=> d=||. 030;L=5 @V2=O==O ?;>I8=8 EMBED Equation.3 72>48BLAO 4> =>@<0;L=>3> 283;O4C (1) <=>65==O< =0 =>@<CNG89 <=>6=8:, I> 287=0G0TBLAO 70 D>@<C;>N  EMBED Equation.3 ; 7=0: =>@<CNG>3> <=>6=8:0 15@5BLAO ?@>B8;56=8< 7=0:C 2V;L=>3> G;5=0 =>@<>20=>3> @V2=O==O. 0?@02;ONG89 25:B>@ ?@O<>W. 0=>=VG=V @V2=O==O ?@O<>W. 0@0<5B@8G=V @V2=O==O ?@O<>W. >65= 25:B>@, I> =5 4>@V2=NT =C;N V ;568BL =0 40=V9 ?@O<V9 01> =0 ?0@0;5;L=V9 W9, =07820TBLAO =0?@02;ONG8< 25:B>@>< FVTW ?@O<>W. 0?@02;ONG89 25:B>@ 4>2V;L=>W ?@O<>W 2 ?>40;LH><C ?>7=0G0TBLAO ;VB5@>N a, 9>3> :>>@48=0B8  ;VB5@0<8 l, m, n: EMBED Equation.3 /:I> 2V4><0 >4=0 B>G:0 EMBED Equation.3 ?@O<>W V =0?@02;ONG89 25:B>@ EMBED Equation.3, B> ?@O<0 <>65 1CB8 287=0G5=0 (42><0) @V2=O==O<8 284C:  EMBED Equation.3  (1) # B0:><C 283;O4V @V2=O==O ?@O<>W =07820NBLAO :0=>=VG=8<8. 0=>=VG=V @V2=O==O ?@O<>W, I> ?@>E>48BL G5@57 42V 40=V B>G:8 EMBED Equation.3 V EMBED Equation.3, <0NBL 283;O4  EMBED Equation.3  (2) >7=0G8<> ;VB5@>N t :>6=5 7 @V2=8E 2V4=>H5=L C :0=>=VG=8E @V2=O==OE (1); <8 >B@8<0T<>:  EMBED Equation.3  2V4A8 EMBED Equation.3 (3) &5  ?0@0<5B@8G=5 @V2=O==O ?@O<>W, I> ?@>E>48BL G5@57 B>G:C EMBED Equation.3 C =0?@O<:C 25:B>@0 EMBED Equation.3.  @V2=O==OE (3) t @>73;O40TBLAO O: ?0@0<5B@, I> 4>2V;L=> 7<V=NTBLAO, x, y, z  O: DC=:FVW 2V4 t; ?@8 7<V=V t 25;8G8=8 x, y, z 7<V=NNBLAO B0:, I> B>G:0 EMBED Equation.3 @CE0TBLAO ?> 7040=V9 ?@O<V9. /:I> ?0@0<5B@ t @>73;O40B8 O: 7<V==89 G0A, 0 @V2=O==O (3) O: @V2=O==O @CEC B>G:8 , B> FV @V2=O==O 1C4CBL 287=0G0B8 ?@O<>;V=V9=89 V @V2=><V@=89 @CE B>G:8 . @8 t=0 B>G:0  A?V2?040T 7 B>G:>N EMBED Equation.3. (284:VABL v B>G:8  ?>ABV9=0 V 287=0G0TBLAO D>@<C;>N  EMBED Equation.3 . 0240==O 7 0=0;VB8G=>W 35><5B@VW B0 0;351@8 =0 B8645=L A0<>ABV9=>W @>1>B8. #<>28 @59BV=30: >1>2 O7:>2V 7040GV  0.2 1.; 4>40B:>2V 7040GV  0,5 1. 0:A8<0;L=89 10; 70 @>1>BC-10 1>2 O7:>2V 7040GV. !:;0AB8 @V2=O==O ?@O<8E, I> ?@>E>4OBL G5@57 25@H8=8 B@8:CB=8:0 A(5;4), B(-1;3), C(-3;-2) ?0@0;5;L=> ?@>B8;56=8< AB>@>=0<. 87=0G8B8 :CB ( <V6 ?@O<8<8 5x-y+7=0, 3x+2y=0. 1G8A;8B8 2V4E8;5==O ( V 2V4AB0=L d 2V4 B>G:8 A(2;-1) 4> ?@O<>W 4x+3y+10=0; . 5:B>@8 0 V 2 CB2>@NNBL :CB = EMBED Equation.3 . =0NG8, I>  EMBED Equation.3  EMBED Equation.3 , EMBED Equation.3 , >1G8A;8B8 :CB  EMBED Equation.3  <V6 25:B>@0<8  EMBED Equation.3 ,  EMBED Equation.3 . 5:B>@8 0 V 2 CB2>@NNBL :CB = EMBED Equation.3  EMBED Equation.3 . =0NG8, I>  EMBED Equation.3  EMBED Equation.3 , EMBED Equation.3 , >1G8A;8B8  EMBED Equation.3 . =09B8 >@B V =0?@O<;ONGV :>A8=CA8 25:B>@0  EMBED Equation.3 . 5:B>@ CB2>@NT 7 >AO<8  EMBED Equation.3  V  EMBED Equation.3  :CB8  EMBED Equation.3  V  EMBED Equation.3 . /:89 :CB 2V= CB2>@NT 7 2VAAN  EMBED Equation.3 ? 0=> <>4C;L 25:B>@0  EMBED Equation.3  V :CB8  EMBED Equation.3 ,  EMBED Equation.3 , EMBED Equation.3 . =09B8 ?@>5:FVW 25:B>@0  EMBED Equation.3 =0 >AV :>>@48=0B. 1G8A;8B8 2V4AB0=L ?;>I8=8 15x  10C + 6z  190 = 0 2V4 ?>G0B:C :>>@48=0B. !:;0AB8 @V2=O==O ?;>I8=8, O:0 ?@>E>48BL G5@57 B>G:C  EMBED Equation.3  V <0T =>@<0;L=89 25:B>@  EMBED Equation.3 . !:;0AB8 @V2=O==O ?;>I8=8, I> ?@>E>48BL G5@57 B>G:C ( 2; +7; +3) ?0@0;5;L=> ?;>I8=V x4C+5z 1=0; 0?8A0B8 @V2=O==O ?;>I8=8, I> ?@>E>48BL G5@57 B>G:C (+3; +1;  2) V G5@57 ?@O<C  EMBED Equation.DSMT4  25AB8 4> :0=>=VG=>3> 284C @V2=O==O ?@O<>W { 2x  3C  3z  9 = 0, x  2C + z + 3 = 0. 0=> B@8 25:B>@8 :  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 . 1G8A;8B8 02A. 87=0G8B8, G8 :><?;0=0@=V 25:B>@8 0, 2, A. 1G8A;8B8 >1 T< B5B@054@0, 25@H8=8 O:>3> @>7<VI5=V 2 B>G:0E  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3  V  EMBED Equation.3 . >40B:>2V 7040GV.(0.5 10;V2) 0=> @V2=O==O 42>E AB>@V= ?0@0;5;>3@0<0 8x+3y+1=0, 2x+y-1=0 V @V2=O==O >4=VTW 7 9>3> 4V03>=0;59 3x+2y+3=0. 87=0G8B8 :>>@48=0B8 25@H8= FL>3> ?0@0;5;>3@0<0. 0=> ?0@0;5;L=V ?@O<V 10x+15y-3=0, 2x+3y+5=0, 2x+3y-9=0. 5@5:>=0B8AL, I> ?5@H0 7 =8E ;568BL <V6 42><0 V=H8<8, V >1G8A;8B8 A?V22V4=>H5==O, 2 O:><C 2>=0 4V;8BL 2V4AB0=L <V6 =8<8. 0=> 25@H8=8 B@8:CB=8:0  EMBED Equation.3 ,  EMBED Equation.3  V  EMBED Equation.3 . 1G8A;8B8 4>268=C 9>3> 28A>B8, >?CI5=>W V7 25@H8=8  =0 AB>@>=C !. 5:B>@8 0 V 2 270T<=> ?5@?5=48:C;O@=V; 25:B>@ A CB2>@NT 7 =8<8 :CB8, @V2=V  EMBED Equation.3 ; 7=0NG8, I>  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , >1G8A;8B8: 0). EMBED Equation.3 ; 1). EMBED Equation.3 . (337). !:;0AB8 @V2=O==O ?@O<>W, I> ?@>E>48BL G5@57 B>G:C P(-2;3) =0 >4=0:>28E 2V4AB0=OE 2V4 B>G>: A(5,-1) V B(3,7). (339.1) !:;0AB8 @V2=O==O 1VA5:B@8A :CBV2, CB2>@5=8E 42><0 ?@O<8<8 x-y+5=0, 3x-y-2=0, I> ?5@5B8=0NBLAO. (236). 0=> 25@H8=8 B@8:CB=8:0 A(1;-1), B(-2;1), C(3;5). !:;0AB8 @V2=O==O ?5@?5=48:C;O@0, >?CI5=>3> 7 25@H8=8 A =0 <54V0=C, ?@>2545=C 7 25@H8=8 B. (296:). >25AB8, I> O:I> B@8 ?@O<V EMBED Equation.3, EMBED Equation.3, EMBED Equation.3 ?5@5B8=0NBLAO 2 >4=V9 B>GFV, B> EMBED Equation.3.  =02?0:8, O:I> EMBED Equation.3=0, B> B@8 ?@O<V EMBED Equation.3, EMBED Equation.3, EMBED Equation.3 ?5@5B8=0NBLAO 2 >4=V9 B>GFV. 828. =09B8 B>G:C ?5@5B8=C ?@O<>W  EMBED Equation.3  V ?;>I8=8 3x + 5y  z  2 = 0. 830. !:;0AB8 @V2=O==O ?@O<>W, I> ?@>E>48BL G5@57 B>G:8 ?5@5B8=C ?;>I8=8 2x + y  3z +1 = 0 7 ?@O<8<8  EMBED Equation.3  EMBED Equation.3 . 832.@8 O:8E 7=0G5==OE :>5DVFVT=B0  V  ?;>I8=0 Ax + y  6z - 7 = 0 ?5@?5=48:C;O@=0 4> ?@O<>W  EMBED Equation.3 ? 834. '5@57 ?>G0B>: :>>@48=0B ?@>25AB8 ?;>I8=C, ?5@?5=48:C;O@=C 4> ?@O<>W  EMBED Equation.3 . 836. 5@52V@8B8 G8 ;568BL ?@O<0  EMBED Equation.3 =0 ?;>I8=V 4x + 3y  z + 3 = 0;  EMBED Equation.3  =0 ?;>I8=V 5x  8y  2z  1 = 0;  EMBED Equation.3  =0 ?;>I8=V 3x  2y  z + 15 = 0 838.'5@57 ?@O<C EMBED Equation.3?@>25AB8 ?;>I8=C, ?5@?5=48:C;O@=C 4> EMBED Equation.3?;>I8=8 EMBED Equation.3EMBED Equation.3 840. 5@52V@8B8, I> ?@O<V EMBED Equation.3 V EMBED Equation.3 ?5@5B8=0NBLAO, V =0?8A0B8 @V2=O==O ?;>I8=8, I> ?@>E>48BL G5@57 =8E. 842. 0?8A0B8 @V2=O==O ?;>I8=8, I> ?@>E>48BL G5@57 =0ABC?=V 42V ?0@0;5;L=V ?@O<V:EMBED Equation.3 V EMBED Equation.3. 844. 0?8A0B8 @V2=O==O ?;>I8=8 , I> ?@>E>48BL G5@57 ?@O<C EMBED Equation.3 V ?0@0;5;L=0 ?@O<V9 EMBED Equation.3. 846. '8 <>6=0 G5@57 ?@O<C EMBED Equation.3 ?@>25AB8 ?;>I8=C ?0@0;5;L=> ?;>I8=V EMBED Equation.3? EMBED Equation.3 848.=09B8 2V4AB0=L 2V4 B>G:8 EMBED Equation.3 4> ?@O<>W EMBED Equation.3. 850. 0 ?@O<V9  EMBED Equation.3  7=09B8 B>G:C, >4=0:>2> 2V440;5=C 2V4 42>E 40=8E B>G>: EMBED Equation.3 V EMBED Equation.3 851. =09B8 B>G:C, A8<5B@8G=C 7 B>G:>N EMBED Equation.3 2V4=>A=> ?@O<>W EMBED Equation.3. 852. =09B8 2V4AB0=L <V6 42><0 ?0@0;5;L=8<8 ?@O<8<8 EMBED Equation.3 V EMBED Equation.3. 853. =09B8 =09:>@>BHC 2V4AB0=L <V6 42><0 ?@O<8<8, O:V =5 ?5@5B8=0NBLAO EMBED Equation.3 V EMBED Equation.3. 854. 1G8A;VBL 2V4AB0=L <V6 ?@O<8<8: EMBED Equation.3 V EMBED Equation.3. 855.0=> 25@H8=8 B@8:CB=8:0 EMBED Equation.3, EMBED Equation.3, EMBED Equation.3. !:;0AB8 @V2=O==O 9>3> 28A>B8, >?CI5=>W 7 25@H8=8  =0 ?@>B8;56=C AB>@>=C. 856.!:;0AB8 @V2=O==O 1VA5:B@8A :CB0  B@8:CB=8:0 40=>3> 2 7040GV 855. 857.'5@57 AB>@>=C B@8:CB=8:0 7040GV 855 ?@>25AB8 ?;>I8=C, ?5@?5=48:C;O@=C 4> ?;>I8=8 B@8:CB=8:0. 858. 0=> :C1, @51@> O:>3> 4>@V2=NT >48=8FV. 1G8A;8B8 2V4AB0=L <V6 25@H8=>N :C10 V 9>3> 4V03>=0;;N, O:0 =5 ?@>E>48BL G5@57 FN 25@H8=C. 859.5@52V@8B8, I> ?;>I8=0, O:0 T ?5@?5=48:C;O@=>N 4> 4V03>=0;V :C10 V ?@>E>48BL G5@57 WW A5@548=C, ?5@5B8=0T :C1 ?> ?@028;L=><C H5AB8:CB=8:C. !:;0AB8 @V2=O==O ?;>I8=8, I> ?@>E>48BL G5@57 B>G:8 L ( 0; 0; +1) V N (3;0;0) V CB2>@NT :CB (/3 7 ?;>I8=>N (EC). @8 O:><C 7=0G5==V :>5DVFVT=B0  ?;>I8=0 x + 3C  5z + 1 = 0 1C45 ?0@0;5;L=0 ?@O<V9 E  1 = C  2 = z ? 4 3 1 (824).  ?>G0B:C :>>@48=0B >?CAB8B8 ?5@?5=48:C;O@ 4> ?@O<>W E  5 = C  2 = z + 1 . 4 3  2 7. (825). '5@57 B>G:C A( +4; 0;  1 ) ?@>25AB8 ?@O<C B0:, I>1 2>=0 ?5@5B8=0;0 42V 40=V ?@O<V: E  1 = C + 3 = z  5 V E = C  2 = z + 1 . 2 4 3 5  1 2 8. (851). =09B8 B>G:C, I> A8<5B@8G=0 B>GFV  ( +4; +3; +10 ) 2V4=>A=> ?@O<>W E  1 = C  2 = z  3 . 2 4 5 (854). 1G8A;8B8 2V4AB0=L <V6 ?@O<8<8: E + 3 = C  6 = z  3 V E  4 = C + 1 = z + 7 . 4  1 2 8  3 3 87=0G8B8 :CB8, CB2>@5=V ?@O<8<8: E  1 = C + 2 = z  5 V E = C  3 = z +1 3 6 2 2 9 6 @8 O:><C 7=0G5==V :>5DVFVT=B0  ?;>I8=0 x + 3C  5z + 1 = 0 1C45 ?0@0;5;L=0 ?@O<V9 E  1 = C  2 = z ? 4 3 1 !:;0AB8 @V2=O==O ?;>I8=8, I> ?@>E>48BL G5@57 ?>G0B>: :>>@48=0B V ?5@?5=48:C;O@=0 4> 42>E ?;>I8=: 2x  C + 5z + 3 = 0; x + 3C  z  7 = 0;  P p 0 x y $t "$&pr\^   : < > @  T V 뿴먝댁ujj.CJEHUmH jv? CJUVmHj CJEHUmH jϮv? CJUVmHCJmH"jCJEHUmH jԭv? CJUVmHjCJEHUmH jbv? CJUVmHjCJEHUmH jv? CJUVmHjCJUmH CJmH CJ56CJmH CJmH")$&t6  *ZL2$$&t6  *ZL2  : !"@%& ())))<*H++P,./0L12344D4H69:;>?@@@A =.Y0     [ AV | ~ & ( N P R T j l , . T V X Z b d zoj>CJEHUmH CJmH"jCJEHUmH j!w? 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