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(%60/3+ 6.!-(3#% *# %#;)(40*!, *# *(.!E8+ ! *+ 3!)4(*+ 6#.# 3!.!% -+%& 10 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2 1 !" !"#$%"&#" '("&#'fi*$($+, -++$+ ./$&("0$+ +12 *3$%$($+ ./$&("0$+ 4"#2/'$'+ 25 +'% !"+%"&#" 4"#2/"+, 6"/7 85" ("5 $/$ $.59$/ 5% 2585'&32 $ +5$ *5/'2+'($(": ;5"/ +$<"/ %$'+: - 4"&3$ *2&2+*2 $/$ $ &2++$ /'%"'/$ "#$ $, !" #$%&'()%&*+ ,-.%' *-fi&(01-' -+#$ $5!$ "+#7 +".%"&#$($ "% (5$+ $/#"+, ="+#$ /'%"'/$> 4$%2+ $ /"+"&#$/ $/$ 42*?> *$/2 $!5&2> $!.5%$+ ("fi&'9@"+ 85" +"/12 A5&($%"&#$'+ $/$ $ *2% /""&+12 ($ "#$ $ +".5'&#", 3!fi+(56# 2727 !"#$ %&'"(#$)$ * "'+%, B%$ /"#$ r C 2/'"&#$($ 85$&(2 +" fiD$ &"!$ 5% +"&#'(2 (" "/*5/+2 *2&+'("/$(2 8#%("(1# " '&('*$(2 2/ 5%$ +"#$, E +"&#'(2 2 2+#2 C +!9)"(1#, B%$ /"#$ 2/'"&#$($ C ("&2%'&$($ !(:#, 3!fi+(56# 27;7 -"./"(#% %&'"(#$)%, B% +".%" 2/'"&#$(2 C ("#"/%'&$(2 2/ 5% $/ 2/("&$(2 (" 2+> 2 /'%"'/2 *3$%$(2 #$(9!' (2 +".%" " 2 +".5&(2> !:"$!'(<)<!, E +".%" 2/'"&#$(2 (" 2/'."% A " "D#/"%'($(" B +"/7 /"F /"+"&#$(2 2/ AB " ."2%"#/'*$%"&#" '&('*$(2 2/ 5%$ +"#$ 85" *$/$*#"/'0$ (" A2/%$ 4'+5$! 2 +"&#'(2 (2 +".%" G4"/ fi.5/$ H,IJ, 3!fi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u $*:$*8*,-% . 4.+:$!+*,-. #,!-B$!.' !"#$%&'() &* C%D ?8 8*"+*,-.8 ,#3.8 -E+ 4.+:$!+*,-.8 !"#%3 % F*$.' CGD AB = BA' 12 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2 1 !" !"# #$%&$'(!# !)"$'(*+!# ',! '-.!#/ AB $ CD/ (0& * &$#&* +")$1,! #$ *# )$(*# #-2!)($# +$##$# #$%&$'(!# #,! 2*)*.$.*# !- 3!"'4 3"+$'($#5 6"%-)* 7589 :$%&$'(!# !)"$'4 (*+!# +$ &$#&* +")$1,!5 6"%-)* 75;9 :$%&$'(!# !)"$'4 (*+!# !2!#(!#5 <# 2)=>"&*# fi%-)*# ".-#()*& #$%&$'(!# !)"$'(*+!# @-$ #,! 3!"'3"+$'($# A"#(! B / *&C!# !# #$%&$'(!# $#(,! '* &$#&* )$(*D5 6"%-)* 75E9 6"%-)* 75F9 !" #$%&$'()* $+,-.)/$'($* 3!fi+(56# 2787 !"# #$%&$'(!# !)"$'(*+!# AB $ CD #,! !9:(; <#,!+"!% @-*'+! (0& * &$#&* +")$1,!/ ! &$#&! #$'("+! $ ! &$#&! 3!&2)"&$'(! AG$H* '*# fi%-)*# A75ID $ A75JDD5 :$&2)$ @-$ !# #$%&$'(!# AB $ CD K!)$& $@L"2!.$'($#/ #$),! )$2)$#$'(*+!# 2!) AB ∼ CD5 M*)* @-$ ! #$%&$'(! AB #$H* $@L"2!.$'($ * CD A'* fi%-)* 13 !"#$!% &!#'("$)*#% !"#$% &'() !"#$% &'*) +,-., /%-01 0- -,"2 3,4.0- AB , CD 450 6,$.,42 /,3 7 3,-3% $,.%' 8&'*991 : 4,/,--;$!0 <#, AB//CD , ABCD =0$3,3 #3 6%$%>,>02 "$%30' !"! #$%&$'()*)(+ )* (,-'&%./01'* ?"0$% <#, @0/A B; -%C, 0 <#, : #3 -,"3,4.0 ,<D!60>,4.,1 @%30- %6$,-,4.%$2>E, %- -#%- 6$06$!,F%F,-' 8!9 AB ∼ AB 8$,fl,H!@%9' 8!!9 I, AB ∼ CD1 ,4.50 CD ∼ AB 8-!3:.$!/%9' 8!!!9 I, AB ∼ CD , CD ∼ EF 1 ,4.50 AB ∼ EF 8.$%4-!.!@%9' 8!@9 J%F0 #3 -,"3,4.0 0$!,4.%F0 AB , #3 604.0 C1 ,H!-., #3 K4!/0 604.0 D1 .%> <#, AB ∼ CD' !" #$%&'$( L,.0$ F,.,$3!4%F0 60$ #3 -,"3,4.0 0$!,4.%F0 AB : 0 /04B#4.0 F, .0F0- 0- -,"3,4.0- 0$!,4.%F0- ,<D!60>,4.,- % AB' M--, /04B#4.0 : !4F!/%F0 60$ ~v ' 14 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2 1 !" !"#$% &"#"%'()*&$ +$% AB , &")$#*&$ +$%- −−→ AB $. B − A $. ~v/ !"#$%&'() 0* 1.*23."% !"#$% −−→ AB , .' %"+%"4")#*)#" &$ 5$)6.)#$ !"#$%"4 &"4&" 3." #")7* * '"4'* &(%"89$: '"4'$ 4")#(&$ " 5$'; +%('")#$ &" AB/ <)&(5*'$4 $ '=&.2$ >$. '*?)(#.&"@ &" ~v +$% |~v|/ #$%$# &'() '*+,-') ./fi1(23/) !"#$!% (34)(% 5 A$(4 !"#$%"4 −−→ AB " −−→ CD 49$ (?.*(4 4": " 4$'")#" 4": AB ∼ CD/ !"#$ +4,# 5 4 4"?'")#$4 ).2$4: +$% 4"%"' "3B(+$2")#"4 ")#%" 4(: &"#"%'()*' .' C)(5$ !"#$%: 57*'*&$ &" !"#$% ).2$ $. !"#$% D"%$: ()&(5*&$ +$% ~0/ !"#$!% #6#%"#% 5 A*&$ ~v = −−→ AB: $ !"#$% −−→ BA , $ $+$4#$ &" −−→ AB " $ ()&(5*'$4 +$% − −−→ AB $. −~v/ !"#$ 4+("7$(# 5 ~v , .)(#E%($ 4" |~v| = 1/ !$%#$ 5 !"%4$% &" .' !"#$% )9$ ).2$ ~v , $ !"#$% .)(#E%($ &" '"4'* &(%"89$ " '"4'$ 4")#(&$ &" ~v/ >F"6* * fi?.%* >H/I@/@ 15 !"#$!% &!#'("$)*#% !"#$% &'() ~u1 * ~u2 +,- #.!/ 01$!-+2 3%+ ~u1 0*3 % 3*+3% 4!$*5,- 4* ~v' 6-$0%.0-2 ~u1 7 8*$+-$ 4* ~v' !"#$% &'&9) :*+0* ;%+-2 ~u2 ~v * ~w <*$0*.;*3 %- <=%.- π' !"#$% &'&&) ~u2 ~v * ~w .,- +,- ;-<=%.%$*+' !"#$% &'&>) ~s = ~u+ ~v !"#$!% *#+),!-$!% . ~u * ~v +,- ;-.+!4*$%4-+ 8*0-$*+ *#+),!-$!% +* 0!8*$*3 % 3*+3% 4!$*5,-' ?3 -#0$%+ <%=%8$%+2 ~u * ~v +,- ;-=!.*%$*+ +* 0!8*$*3 $*<$*+*.0%.0*+ AB * CD <*$0*.;*.0*+ @ 3*+3% $*0% -# *3 $*0%+ <%$%=*=%+' !"#$!% *#/+-,-$!% . +* -+ 8*0-$*+ .,- .#=-+ ~u,~v * ~w 0A3 $*/ <$*+*.0%.0*+ AB2 CD * EF <*$0*.;*.0*+ %- 3*+3- <=%.-2 4!B*3-+ C#* +,- *#/+-,-$!%' DE*F% % fi"#$% D&'&9HH' 16 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2 1 !" !" #$%&'()%* +,- .%/,&%* !"#$ %&' (")* +, '-., /$0- '1.#"-$2" )"/ " )"1.'32" 2' 1"--$ $&4$5 6"0- +, )"17')'& $4!&1- )"1)'0."- 0/6"#.$1.'-5 6"2'/"- $($18 9$# &/ 6"&%&017" /$0-: ;$/"- $6#'-'1.$# 6$#$ (")*5 1'-.$ -'8 !&12$ 6$#.' 2' 1"--$ $&4$5 "- /')$10-/"- 6$#$ '<'.&$# $- "6'#$8 9='- )"/ ('."#'-: #$%$# &'()* 3!fi+(56# 2787 >'+$/ "- ('."#'- ~u ' ~v #'6#'-'1.$2"- 6'4"- -'!8 /'1."- "#0'1.$2"- AB ' BC: ?- 6"1."- A ' C 2'.'#/01$/ &/ ('."# ~s5 %&' @ $ -"/$ 2"- ('."#'- ~u ' ~v5 "& -'+$5 ~s = ~u+ ~v. ;'+$ $ fi!&#$ BC:CDE: F0!&#$ C:CDG ~u = ~AB5 ~v = ~BC ' ~s = ~AC: #$%$+ ,-*.-'/&0&/1 &0 0&'()* >'+$/ ~u5 ~v ' ~w ('."#'- %&$0-%&'#5 ($47$/G !"#$%$&'% ( ~u+ ~v = ~v + ~u )**!+&%$&'% ( (~u+ ~v) + ~w = ~u+ (~v + ~w) 17 !"#$!% &!#'("$)*#% !"#"$%& '"(%)& * !"#$% &' %(%'%)$* ~0+ $,( -&% ~v +~0 = ~0 + ~v = ~v, ∀~v. +$,")-& ./0%0,& * .,/, $*0* 1%$*/ ~v %!"#$% &' 2)"3* 1%$*/ −~v 41%$*/ *5*#$* 0% ~v)+ $,( -&%~v + (−~v) = (−~v) + ~v = ~0. !"!# $%&'(')*+ ,' -'./('0 +!fi-)./# 0121 6"7%'*# -&% ~d 8 , 3)4!$!-.5 0% 0*"# 1%$*/%# ~u % ~v #% ~d = ~u− ~v+ *& #%9,+ ~d = ~u+ (−~v). :,# fi<&/,# 4=>=?@ % 4=>=A@ %#$B* /%5/%#%)$,0*# *# 1%$*/%# ~u % ~v+ /%#5%3$"1,'%)$%+ 5%(*# #%<'%)$*# */"%)$,0*# AB % AC> ABCD 8 &' 5,/,(%(*</,'* 3&9,# 0",<*),"# AD % BC /%5/%#%)$,'+ /%#5%3C $"1,'%)$%+ ~s % ~d 4#*', % 0"D%/%)E,@> F"<&/, =>=?G ~s = ~u+ ~v F"<&/, =>=AG 3 !"!1 234.%54%6+*7/ 5/( 38 )98'(/ ('+4 +!fi-)./# 0161 6,0*# &' 1%$*/ ~v 6= ~0 % &' )2'%/* /%,( k 6= 0+ 3H,','*# 0% 5/*0&$* 0* %#3,(,/ k 5%(* 1%$*/ ~v * 1%$*/ ~p = k~v+ $,( -&%G 18 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2 1 !" ! !"#$%&' |~p| = |k~v| = |k||~v|" #! ()*+,-&' $ %&'%$ (& ~v" )! .+/0)#&' * %&'%* (& ~v '& k > 0, * +*,-./.0* (& ~v '& k < 0. • 1& k = 0 *2 ~v = ~03 * 4.*(2-* 5 ~0! • 1&6$ 2% 7&-*. k~v3 &% 82& ~v 6= ~0! 9$:&,(* +*% 82& k 7$.0& '*;.& R <* +*,62,-* (*' ,=%&.*' .&$0'>3 *;-&%*' *' 0,fi,0@ -*' 7&-*.&' +*A0,&$.&' $ ~v <$A5% (& '&.&% -$%;5% +*A0,&$.&' &,-.& '0>! B*. *2-.* A$(*3 4$.$ 82$0'82&. (*0' 7&-*.&' ~u & ~v3 +*A0,&$.&'3 '&%4.& &C0'-& 2% k ∈ R3 -$A 82& ~u = k ~v. • D 7&.'*. (& ~v 6= ~0 5 * 7&-*. 2,0-/.0* ~u = 1 |~v| ~v *2 ~u = ~v |~v| ! E&6$ 82& |~u| = ∣∣∣∣ ~v |~v| ∣∣∣∣ = |~v| |~v| = 1, 4$.$ -*(* ~v 6= ~0! F''0%3 -&%*' 82& ~v = |~v| ~u3 *2 '&6$3 -*(* 7&-*. 5 * 4.*(2-* (& '&2 %G(2A* 4&A* 7&-*. 2,0-/.0* (& %&'%$ (0.&HI* & %&'%* '&,-0(* 82& ~v! 19 !"#$!% &!#'("$)*#% !"!# $%&'%()*+*), *+ -./0('/(1+23& '&% .- 45-)%& %)+/ !"#$ ~u ! ~v %!&'(!) *+#,)*+!( ! a ! b -.$!(') (!#,) /&#$01$ 2'-3!2,4') 2'$' !)2#5#(!)67 8)),$9 &!$') #) )!:+,-&!) ;(';(,!4#< 4!)= !!"#$%&$'%( a(b~v) = (ab)~v> )*+,&$*%*+( 1~v = ~v; -$!&.$/0&$'$*%*+ +1 .+2%34" %"! +!#%2%.+!( (a + b)~v = a~v + b~v> -$!&.$/0&$'$*%*+ +1 .+2%34" %"! '+&".+!( a(~v+~u) = a~v+ a~u7 ? 0'$ *+! %'2@ #&!-&! ;#(# ') !A!$;5') *+! 53! #;(!)!-&#$')9 ;',) )B' C+-4#$!-&#,) ;#(# #+A,5,D<5' -# (!)'5+EB' 4') !A!(2F2,') #' fi-#5 4!)&# #+5#7 +,!'-.# /010/0 H#4') ') %!&'(!) ~u, ~v ! ~w9 2'$' -# fi:+(# # )!:+,(9 %#$') 2'-)&(+,( ' %!&'( 2~u− 3~v + 1 2 ~w = ~s7 20 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2 1 !" !"#$% &'&() *+,#-.+/ ~s = 2~u− 3~v + 1 2 ~w' !" #$%&'() *+ *(,) -+.(/+) 3!fi+(56# 2787 0 9+:;,# <! <#(% 1!"#$!% ~u 1 ~v 2.+ 2#,+3 4 + 52"#,+ θ 6+$7%8+ 91,%3 317!:$1;%3 OA 1 OB/ <+7+ 2% fi"#$% >&'?@/ 1 ;%, A#1 0 ≤ θ ≤ π' • *1 θ = π/ ~u 1 ~v ;B7 % 7137% 8!$1-.+ 1 312;!8+3 <+2;$C$!+3' • *1 θ = 0/ ~u 1 ~v ;B7 % 7137% 8!$1-.+ 1 7137+ 312;!8+' 21 !"#$!% &!#'("$)*#% • ! θ = π 2 " ~u ! ~v #$% %&'%(%)*+# ,+#'% -" #$% .!&.!)/+012*&!#3" ! /!)%'*4%# .%& ~u⊥~v5 6!#'! 0*#%" '!4%# 71! |~u + ~v|2 = |~u|2 + |~v|25 • 8 9!'%& ~0 - 0%)#+/!&*/% %&'%(%)*2 * 71*271!& 9!'%&5 • ! ~u - %&'%(%)*2 * ~v ! m 14 ):4!&% &!*2 71*271!&" ~u - %&'%(%)*2 * m~v5 !" #$%&'( 6!#'* *12*" 9%0; *.&!)/!1 71! % #!(4!)'% %&+!)'*/% )% .2*)% &!< .&!#!)'* 14 %=>!'% (!%4-'&+0%? % 9!'%&" 71! .%& #1* 9!@ .%/! #!& &!.&!#!)'*/% *2(!=&+0*4!)'! !" 0%4% 0%)#!7A;)0+*" .%##+=+2+'* /!< fi)+& %.!&*CD!# 0%4% */+C$%" /+E!&!)C* ! .&%/1'% .%& 14 !#0*2*&5 22 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2 1 !" !"# $%&&'( )'*+ ,-./0$/1 21/ /3%&4/ 1# 5061!' /04./ $'%& )/4'./&( ,%0$, 21/ &1,& /34./#%$,$/& 07' *'%0*%$,#8 !" #$%&%'(')* 98 :/*%$, &/ " )/.$,$/%., '1 ;,!&, *,$, 1#, $,& ,fi.#,=>/& , &/61%.8 ?,@ A/ ~u = ~v( /047' |~u| = |~v|8 ?B@ A/ |~u| = |~v|( /047' ~u = ~v8 ?*@ A/ ~u ‖ ~v( /047' ~u = ~v8 ?$@ A/ ~u = ~v( /047' ~u ‖ ~v8 ?/@ A/ |~w| = |~u|+ |~v|( /047' ~u ~v / ~w &7' -,.,!/!'&8 ?;@ A/ −−→ AB = −−→ DC( /047' ABCD ?)".4%*/& 0/&4, '.$/#@ " -,.,!/!'6.,#'8 C8 :,$'& '& )/4'./& ~u / ~v $, fi61.,( #'&4.,. 1# ./-./&/04,04/ $' )/4'. ,4.,)"& $/ 1# 6.Dfi*'E ?,@ ~u− ~v ?B@ ~v − ~u ?*@ −~v − 2~u ?$@ 2~u− 3~v F8 :/4/.#%0/ ' )/4'. ~x /# ;10=7' $/ ~u / ~v 0,& fi61.,& , &/61%.8 23 !"#$!% &!#'("$)*#% !" #" $" %& '!()* +,-* .)/+)* A0 B0 C /1)2$)34/5!,5*0 $)6) /! fi89,! ! *5894,0 ,5.,5*5/+!, ) :5+), ~x /)* *5894/+5* $!*)*; !" ~x = −−→ BA+ 2 −−→ BC< #" ~x = 1 2 −→ CA+ 2 −−→ BA< $" ~x = −→ AC + −−→ CB − −−→ AB& =& >!#5/() ?95 ) @/893) 5/+,5 )* :5+),5* ~u 5 ~v A (5 30o0 (5+5,2 64/!, ) @/893) B),6!() .53)* :5+),5* ! *5894,& !" ~u 5 −~v #" −~u 5 2~v $" −~u 5 −~v #" 3~u 5 5~v ! " #$%&'()*+$ ,-. -(+/+,-,&. >5 :)$- $)/*58949 B!C5, ! !+4:4(!(5 D0 5/+1) 5/+5/(59 )* ,9(42 65/+)* ()* $)/$54+)* (5 6E(93) 5 :5+),5* .!,!353)*& F ?9!/+) G* !+4:4(!(5* H0I 5 % J K)/*58949 ,5*)3:-23!*J F/+1) LM 5/+5/(59 ! 4(A4! (5 *)6!0 (4B5,5/N! (5 :5+),5*0 !3A6 (5 693+4.34$!N1) .), 96 5*$!3!,& F ! !+4:4(!(5 =J >5 :)$- $)/*58949 ,5*)3:-23!0 !L9(! ! fiO!, ! 4(A4! () @/893) 5/+,5 :5+),5*& >5 !4/(! +4:5, (4fi$93(!(5*0 :)3+5 5 ,5:5L! $)6 $94(!() )* $)/$54+)* !.,5*5/+!()* /! !93!& P1) 5*?95N! ?95 QM +9+),5* ?95 ) !L9(!,1) ! 53464/!, !* *9!* (R:4(!*& '5*(5 LM0 356#,52*5 (5 (4*$9+4, )* $)/+5R()* $)6 *59* $)358!*& 24 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2 1 !" ! "#$#%&'()*+ !"#$%&'()* +,-./01 * !"#!$%&' ()'*+$&,'2 31 456,1* 758 9.1: %119;* <=>?2 @#7+* ",1: @5A/; * !"#!$%&' ()'*+$&,' ! -*.!/%' 0&)!'%2 &B1 0/ C5:/B.1* #74+* DEEF2 %G@H&#$#* C1;I @6BJ* 1-*.!/%' 0&)!'% 2 31 456,1* )5.K.5* <=>E2 25
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