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!"#$ %& '#$ & %" " & (%) * "%& "#$ #$ &(""# " "$ + !"# " ' , .& $ " . , # "(#'- & %""# ." O "(#' ". & %". /+% * .& A " & ' ' '' . x y , % #$ && % "# (& ' OAx OAy , % .& Ax Ay ! ." "%& ', 0""#$ ( .& A " & %""# . 1 "' " .& O & .& A '% + !"# "' " %"" * , x, y & %" + !"#* ("&, + 2 "' " + !" "' "- , & %" ' +" 3 !"#* ("&. 4 " A A +% .& A ( x , y ) 1 2 1 1 1 * ' -0 '#. ' # 5 % A2 ( x2 , y2 ) , 65 -0* '% A A = ( x − x ) 2 + ( y − y )2 . 2 1 2 1 7 " (& A 1A 2 ' (%"" " 8" λ "(#' & 1 2 & * .& A( x, y) " ", %" -0* .& A A , & 1 2 % ' ' " 8"- A A AA = λ , 2 & %"# & * 1 2 .& - '#+" % x = ( x1 + λx2 ) (1 + λ ) y = ( y1 + λy2 ) (1 + λ ) . 9" 8" λ + :#! 3 !"# . , ' 2 . .& A + ( % (& A A " ", 1 2 %" -0* .& A A . 1 2 * * '"' 4 F ( x, y ) = 0 , % F ' "& 5 "&3 & %" * ( x, y ) .& & , % " '" '"" &' F " & . 1 %' &'# F1 F2 (%"# '"" F1 ( x, y) = 0 F2 ( x, y ) = 0 , $ .& ." % - && 8" #, 0* ( %' $ %""#$ '""*. 1 &+% ( & %" ' * t , "(#' * , ' 5 "&3* "& * "" (%" '"" &' * x = x(t ) , y = y(t ) ' .& * 5 . 6 +" .& A( x, y) " & + :#! &+ " * . ( ρ ,ϕ ) , ' (""#$ %& '# & %" " 8" x = ρ cosϕ y = ρ sin ϕ , "(#'#$ "# & %" .& A 1.
3
3 !" ' ." ρ %' : * " .& A % ". & %", ϕ (%) +% "' " OA !- Ox , & ' " * "(#' " * !-. ϕ :#." (%) ' "' [0,2π ] [−π , π ]. /' ' "#$ & %"$ + :#! (%" && ' '" '% ρ = ρ (ϕ ) , & ' " '" * 5 F ( ρ ,ϕ ) = 0 , ' .& * 5 ρ = ρ (t ) , ϕ = ϕ (t ) .
(.1).
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'"" %"$ $ + .&$ :& ' '# % -(" ' ("#$ %& '#$ $ & %", ( ."# ". .)
"' " *. 7 0" '""* " - ( ."# ( '" & %". : # . " # : :( '" %& '#$ & %" " * & #$ & %" ( x, y ) & " '# ( x' , y ' ) ' !"# " ". & %" x = x'+ x0
y = y '+ y0
' .& ( x , y ) , &+ ' & %""#$ *, % + !"#* 0 0 ϕ '. "' "- '0" ' . ' * & (.2):
x = x' cos ϕ − y ' sin ϕ y = x' sin ϕ + y ' cos ϕ .
9:"#* $ % & # & %" +" (' , (8' (""# '#8 '"' " !" ( x' , y ' ) && 5 "&3* ( x, y ) .
4
.2.
7 &(!, . !"& '8" P(−2,−1) , Q(6,1) , R(3,4) ' !"#. . #. % "# " %"" !"&: 1.
PQ = (6 − (−2))2 + (1 − (−1))2 = 68 ,
PR = (3 − (−2)) 2 + (4 − (−1))2 = 50 , QR = (3 − 6)2 + (4 − 1) 2 = 18 .
*%""# % "# % ' ' - '- PQ 2 = PR 2 + QR 2 , .. !"& PQR ' !"# & PR , QR " ( * PQ . 2. 7"# .& A(3,−1) B(2,1) . 9% ! & %"# .& M , ." * .& A " !" .& B , & %"# .& N , ." * .& B " !" .& A (.3).
.3. . .& B % ( & AM ' " 8" λ = AB BM = 1, 2 '"" xB = ( xA + λxM ) (1 + λ ) " & %" xM . 48 , "$ % xM = 1. " ." '"" % & %"# yM %) ("." yM = 3, .. & ' .& ! M (1,3) . 4 +% $ + :( , % .& N . " 8" λ = NA AB = 1 '"" x A = ( xN + λxB ) (1 + λ ) , & % xN = 4 . " ." '"" ' % & ("."- yN = −3 , 2 ' .& ! N (4,−3) . 3. ! " ! ", #$% &' #' # &' ! " M 1 (−2,4) M 2 (6,8) . 5
. " !" M , #$ # $, " # & ( x, y ) . $! #$ $ " !" & $ # M1M = M 2 M , ( x + 2) 2 + ( y − 4)2 = ( x − 6)2 + ( y − 8) 2 . # "# $ # # & $ & , $!
! # " M1M 2 . 2 x + y + 10 = 0 , ' # #"$ . 4. !" M " v # ON , " " !" O ( !$ " # ) $ " ω ( .4). " !" M $" ( ρ ,ϕ ) & $ ' #. % $ &' " # '.
.4.
. t , ! , # !$ &'# !" M !$ " # . # #$ # !$ " # & ρ (t ) = vt , #$ $ $ #$ ϕ (t ) = ωt . "$! ' t , $! $ &' " # ' $ ' # # ρ (ϕ ) = aϕ , # a = v ω . 1.1.
& " # & $ " ABC : A(1, − 3) , B(3, − 5) , C (−5, 7) . ! # $ " # & # .
6
& " # & A(1, − 2) B(3, 2) $$ $
ABCD , " !" N (5, − 1) ! # $ . " # & #' # ' C D . " # & # " , ' # ! !" 1.3. O(0, 0) , M (3, − 1) N (8, 4) . !" ! $ , # &' L : 1.4. 1 1.2.
1.5.
1.6.
x 2 + y 2 = 25 L2 : x + 7 y − 25 = 0 . !' " ! " $ ! " Ox , " # & " &' #$ : 1) x > 2 ; 2) x − 3 ≤ 0 ; 3) 12 − x < 0 ; 4) 1 < x < 3 ; 2−x 2x −1 > 0 ; 6) > 1; 7) x 2 − 8 x + 15 ≤ 0 . 5) x− 1" # & x −! "2, ! &' $ " & " # $ y = − x $ # !" :
B(−4, 3) ; 3) C (7, − 2) . & !" A(1, − 1) , B(3, 3) C (4, 5) , $ # . ! # $ λ , " "# ! " # $ ", !" . ! & # #
! ", # $ & !" M 1 (−6, 7) M 2 (−2, 3) , # $ ! & & !. " # & ! " # $ L , M N . "" !" P #$ " M1M 2 , ! & #$ ? $!$
" & " & ! " $% &' "
1) 1.7.
1.8.
A(3, 5) ;
2)
# # , # # !" M (1, 8) . & # & & "# A(2;−1) B(− 1;3) . ! # $ 1.10. # # &.
1.11. " " " # & X = 1 , Y = − 3 , " , " $ Ox $ Θ = 2π . 3
!" P (2;2)
$ " & " " " & # A B , 1.12. ! # Q (1;5) # $% & !. 1.13. ! # $ $ & " # & ! ", ! &' $ $ !" M (3, π 4) , M (2,− π 2) , M (3,− π 3) , 1 2 3
"
& $ # . M 4 (1, 2) M 5 (5, − 1) , # 1.9.
7
$ " # # & # & A(3, − 4π 9)
B(5, 3π 14) $$ $ ABCD , !" ! # $ " # $ . ! # $ $ & " # & #' # ' $$ $ . 1.15. $ " # # & !" A(8, − 2π 3) B (6, π 3) . &!$ $ & " # & # & ", # !" A B. π 1.16. &!$ $ # $ ", & " A 3; , 8 1.14.
7π B 8;
# & $ &' " # '. 8 " , '#
$ &' " # ' ! !$ " # $ # a .
$ " # $ , # $% ρ (ϕ ) = 1 sin ϕ , " # & ! ", " &' !$ " # &: 1) 1; 2) 2 ; 3) 2 . " $ # $ # & ? % ! .
#"$ $ " "
OM = 3 . $ &' " # '. ! " ! ", # " &' # #' # &' ! " F1 F2 $! a 2 , # a = F F 2 . !% "" 1 2 " &' " $ &' " ' #$ ", $$ # (# " »). « " ! # R " "$ & Ox . ! " $ x = x(t ) , y = y (t ) , # t , ( x, y ) #$ " # & !" , '# t = 0 !$ " # (# " " «"$ #»). 24
1.17. 1.18.
1.19. 1.20.
1.21.
5π C 6;
8
2- 3- $ $ A &' $ " $ " &' ! $, # m " n $ , & #" m × n $ aij , # i = 1, m j = 1, n . m = n , !"# $% &# ' ( $( ) %'& n . *+,-+ - A = aij . - ./ #$0$#$ $ & )$& $ 1 α , ) 2 #$ 2$ $ " " - . +% 2 1: (αA)ij = α aij . 3 " ' &#"4 ) %'&# . & '"# /, ) 2 ( A + B )ij = aij + bij . % &# ' ( " A . ##$ ) %$ ) $'$$%, 5 '$$ , ) $' #%+0$6 ,( 1#-+ 7- & + $8 2$ $ # ,! 1 $ 6 det A . 5% " )$ #6 ) %'& A = a11 ) $'$$/, ,! 1 $ "( # 1$ ( " ' ( ) % ( 1$ (, #$ $' #$ - $+0$ -% 2$ $ -, .$. det A ≡ A = a11 . 5% " # 6 ) %'& 2.
A=
a11 a12 a21 a22
) $'$$/ #"1%$% ) $'-+0$ - ) #-: det A = a11a22 − a12a21 ,
-' !- $'-$, 1 ) $'$$/ $ %$ ! & ) )$ $ #&$ & & , #, " ' &-6.'$ -6 #$ - + , $ '#$ & '# , , . ) ) #, &.$ '% -'% 1 % -'$#( & 9 $ , 1 /' " $ #( $( +,6 $+ $ ) $ $ $ ) %'& . 5% " $/$6 ) %'& a11 a12 A = a21 a22
a13 a23
a31
a33
a32
) $'$$/ #" . $% 1$ $! ) $'$$ # 6 ) %'& & & det A = a11
a22
a23
a32
a33
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9
a21 a23 a31
a33
+ a13
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,
6'$ '$$ " # 6 ) %'& & "# +% ) ) - #" $ ) #. # "$ #(# $ $( "4 - # $ ( ) $ $ 9 " # !# " '#-4 '#% $ $ ( $ $ ( x1 , x2 ). $ , 1 " !' !-/ " $'" $ $ % ! ' 1 -% ) $. -.' , $ $ $/ $ '% $ $/$6, ,$$ #"&4 ) %'&#. $ ( $( "4 - # $ ( # 6 ) %'& '#- % $!#$ " !"# $% $ #$ # $'-+0$6 #' : a11x1 + a12 x2 = b1 , a21 x1 + a22 x2 = b2
6'$ &277 $ " a , !-+ &# ' -+ - A ' ( $ ", ij ,$ (b1 , b2 ) !"# $% , #,' "4 1$ #. -1 $, &6' 2 ! -$(, 6# % , ' ' ( $ $, $ .$ ' "( ,$ !"# $% ',$ ''$ .9' 4'% %," ' # $ 6'-$#(# 2$ $ , $ $ (. $ $ $ , )&/&- $$ & & - ' $ $ $ ( x1, x2 ) = (0,0) . $ 6, ) det A = 0 ' ( $ " - # $ % ) ) / " ' -6 ' -6-, .$. $$% ' '! $ # $ - # $ $ '% '#-4 $!#$ "4. ! 1 $, 1 ' $!#$ $ $( #" . $% 1$ $! # $, ) $ , x2 = − a11x1 a12 , 1 6# ,$& $1 1$ $ $ (. $' ' % $ &# ' ( $( A ) det A ≠ 0 $$ $' #$ $ $ $ $, #" . $ $ 7 - $ ∆1 ∆ , x2 = 2 , ∆ ∆ ∆1, 2 ) 1 +% ! ∆ ! $
x1 =
6'$ ∆ ≡ det A , ) $'$$ )$ #6 # 6 , ,$ #,' "4 1$ #: ∆1 =
b1 b2
a12 a22
,
∆2 =
a11 b1 a21 b2
( #$#$
.
det A = 0 , 4% ," ' ! ∆1, 2 1$ -%, $ $ ( $ '$$% # " -+, - # $ % $ " $ # $ . " .' $-6#$' ) $ -6- $ / $! # " %#%$% / $ ) ) / ' . ! # 4. & # -1 $ ' ' ( $ ", !'$/ $$% ,$& $1 $ $ $ $ (.
10
4 −2 4 1. "1/ ) $'$$/ $/$6 ) %'& 10 2 12 . 1
2
2
. )/!-% ) #$'$ -+ #" $ 7 -- '% & "% '$$ $/$6 ) %'& , ) 0/+ $) $'#$ 6 #"1$ % )-1 $ 4 −2 4 2 12 10 12 10 2 10 2 12 = 4 ⋅ − (−2) ⋅ + 4⋅ = 2 2 1 2 1 2 1 2 2
= 4 ⋅ (−20) + 2 ⋅ 8 + 4 ⋅ 18 = −80 + 16 + 72 = 8.
2. ) / #" .$ $
0 cos α
0 sin β
1 1.
sin α
cos β 1
. & "# % ' "( ) $'$$/, )-1 $ ' $ # $ -+ 6 $ $ ∆ = 1⋅
cos α
sin β
sin α
cos β
= cos α ⋅ cos β − sin β ⋅ sin α = cos(α + β ).
3. $ / $ - - # $ ( 4 x1 + x2 = 5
3 x1 − 2 x2 = 12
.
4 1 . 9) $'$$/ " $ " ∆= = −8 − 3 = −11 ≠ 0 , 3 −2 )2 - -0$#-$ $' #$ $ $ $ $ x = ∆ ∆ , 6'$ ) $'$$ 1, 2
∆1 =
5
1
= −10 − 12 = −22 ∆ 2 =
4
1, 2
5
= 48 − 15 = 33 , &-'
12 − 2 3 12 4' x1 = − 22 (−11) = 2 x2 = 33 (−11) = −3. 9,%! $/ (
#$ &( - ,$.' $ %, 1 ('$ "$ ! 1$ % $!#$ "4 , 0 + ) - # $ % $ " # .'$# .
11
4. $ / $ - - # $
2 x + 4 x2 + x3 = 4 1 ( 3x1 + 6 x2 + 2 x3 = 4 . 4 x1 − x2 − 3 x3 = 1
. 9) $'$$/ " $ " 2 3
4 6
1 2 = 2 ⋅ (6 ⋅ (−3) − 2 ⋅ ( −1)) − 4 ⋅ (3 ⋅ (−3) − 2 ⋅ 4) +
4 −1 − 3
,
+ 1 ⋅ (3 ⋅ (−1) − 6 ⋅ 4) = 2 ⋅ (−16) − 4 ⋅ (−17) + 1 ⋅ (−27) = 9 ≠ 0
.$. - $ " $$% $' #$ $ $ $ $. 9) $'$%% $6 ) 7 - $ x1, 2,3 = ∆1, 2,3 ∆ , 4' 4 ∆1 = 4
4 6
2 4 4 4 1 4 2 = 27 , ∆ 3 = 3 6 4 = −36 , 4 1 −3 4 −1 1
2 1 2 = −18 , ∆ 2 = 3
1 −1 − 3
&-' x1 = −2 , x2 = 3 , x3 = −4 . #$ &( - ,$.' $ %, 1 ('$ "$ ! 1$ % $!#$ "4 , 0 + - # $ % $ " # .'$# . 5. $ / $ - - # $
x1 − 3 x2 + 2 x3 = 0 ( x1 − x2 + x3 = 0 . 2 x1 + x2 − 3 x3 = 0
. "1%$ ) $'$$/ ' ( $ ": 1 −3 2 - ! - '- -0 # # ∆ = 1 − 1 1 = −7 ≠ 0 , )2 7 $ $ $ $ $ 2
1
−3
$' #$ 6 -$#6 ( # / 6) $ $ % x1 = x2 = x3 = 0 .
6. $ / $ - - # $ ( 2 x1 − 3x2 + 4 x3 = 0 . 3 x1 + x2 − 2 x3 = 0 . ! " #$%&, " ' $(. ) % * $ + , #$' $' ,$ ( -(, * + .%, * , x3 $ * $'/ " :
12
2 x1 − 3 x2 = −4 x3
.
3 x1 + x2 = 2 x3 ' #$ . " $, . - $ %& " $ . * % (− 4 x3, 2 x3 ). 2 −3 ∆= = 2 − (−9) = 11 ≠ 0 , *' * , #" x3 3 1 '$' $ !. , ( ' % , &
1 − 4 x3 − 3 1 2 x1 = = ⋅ (− 4 x3 ⋅ 1 − (−3) ⋅ 2 x3 ) = x3 , 1 11 ∆ 2 x3 11 1 2 − 4 x3 1 16 x2 = = ⋅ (2 ⋅ 2 x3 − (−4 x3 ) ⋅ 3) = x3 , ∆ 3 2 x3 11 11 " $ ! %. # ( * $ ,( '$ $, " #$%& (2 x 11, 16 x 11, x ) * 3 3 3 / #" x ' $ % $ $. 3
x1 + x2 + x3 = 0 ! ' ' $(
3 x1 − x2 + 2 x3 = 0 . 7. x1 − 3 x2 = 0 . * ( % ∆ = 0 , " $ $' #$ +& ' $( '. '.. * $% $ ' $: &, "$, # $ '. , '.' ' " , #$%. % * '" $'/ ', '/ # & $'& ' $(: x1 + x2 + x3 = 0 . 3 x1 − x2 + 2 x3 = 0 $ %, ( $ * %' * . #$'/ x $ * $'/ " , * '" * ' 3 x1 = 3 x3 (−4) x2 = x3 ( −4) , . * x3 * / % #". $ ,( ' , " (% #" #$%& / $ ' $ &( % $ $.
13
.
* . * ,: 2 1 3 3 2 1 1 −3 2
2.1. %" 1)
5 3 2;
2)
2 5 3;
3)
1 4 3 3 4 2 3 − 38 4 1 6) cos α 5) 5 − 35 2 ; cos β
2 − 49 3 2.2.
1
−1 1 = 0;
4
2
2.5.
;
4)
−3
2 1 cosα 1
6
−5 1 ;
−2 −7
cos β cos(α + β ) .
cos(α + β )
1
1
2)
x 1 = 0.
1
1 1
x
' (%& ' $( $ . * ,: !
2 x1 + x2 = 10 3 x1 + 5 x2 = 2
1) 2.4.
1
3 4
* #$ x # ' $: x2 x 1 x 1 1 1)
2.3.
−1
1
2 −1
x1 + x2 = 17
;
2)
5 x1 + 9 x2 = 4
.
!" #$ %$ &'(!: , 0 )2 x + x − x = 2 - x + 3 x = −1 1 2 3 2 3 + / 2) -2 x1 + 3 x2 + 5 x3 = 3 . 1) )3 x1 + x2 − 2 x3 = 3 ; * .
x1 + x3 = 3
3 x1 + 5 x2 + 7 x3 = 6
2 x1 + 4 x2 + 3 x3 = 0
2 x1 − 3 x2 + x3 = 0
1 '" !" & 1" : 4 7 3 x1 + 3 x2 + 2 x3 = 0 65 x1 − 8 x2 + 3 x3 = 0 1) 2 ; 2) 5 .
14
2 . ! "# $ %&'()*)+ ,-./0-&(12 ,-3*-04&,,/5 )(*&.)', 1)&67,28975 60& ():'7 0 3*)1(*-,1(0&, ,- 34)1')1(7, 747 ,- 3*2+)5. %&'()* 1 ,-:-4)+ 0 ():'& A 7 '),;)+ 0 ():'& B )<).,-:-&(12 '-' AB 47<) '-' )6,- <='0- 3)4=>7*,)?) @*7A(-, ,-3*7+&*, a . B)6=4&+ 0&'()* - | a |≡ a ,-./0-&(12 647,- )(*&.'AB . %&'()*, ,-:-4) 7 '),&; ')()*)?) 1)03-6-8(, ,-./0-&(12 ,=4&0/+ 7 )<).,-:-&(12 '-' 0 . %&'()*/ a 7 b ,-./0-8(12 ')447,&-*,/+7, &147 1=9&1(0=&( 3*2+-2, ')()*)5 ),7 3-*-44 &4C,/, :() )<).,-:-&(12 '-' a || b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a = AB 7 b = BC . JG 1=++)5 c = a + b ,-./0-&(12 0&'()* AC , 3)4=:&,,/5 3) 3*-074= (*&=?)4C,7'- (*71.5). 3.
KLM.5. I*)7.0&6&,7&+ 0&'()*- a ,- 0&9&1(0&,,)& :714) α ,-./0-&(12 0&'()* b = α a , 647,- ')()*)?) *-0,- | α | ⋅ | a | , - ,-3*-04&,7& 1)03-6-&( 1 3*-04&,7&+ a 3*7 α > 0 7 3*)(70)3)4)>,) &+= 3*7 α < 0 . ,-)14 I &6)0-(&4C,)& 3*7+&,&,7& )3&*-;75 14)>&,72 7 =+,)>&,72 ,- :714) 3).0)42&( 1)1(-042(C 47,&5,/& ')+<7,-;77 0&'()*)0. %&'()*/ a , a , …, a 1 2 n ./0 5 . 0717+/+7 5 8(12 47 ) 147 7G 47 2 ')+<7 72 ,- ,& , ,& ,& ,,-; , & 01 * 0 / 47@C ()?6 ')?6 :714 & α1a1 + α 2a2 + N + α nan = 0 , α i - , ,=48. O- 3*2+)5 47,&5,) ,&.-0717+/+ 2042 &(12 ()4C') )67, ,&,=4&0)5 0&'()*, ,34)1')1(7 – 48
0 6-,,)+ 3*)1(*-,1(0& ,-./0-&(12 *-.+&*,)1(C8 D()?) 3*)1(*-,1(0-. 4&6)0-(&4C,), 3*2+-2 2042&(12 )6,)+&*,/+ 3*)1(*-,1(0)+, 34)1')1(C – 60=+&*,/+, - 3*)1(*-,1(0) 1 6&'-*()0/+7 '))*67,-(-+7 ( x, y, z ) (*&G+&*,). 8 () ?)0)*2( :() 3) 171( &, &+& x = x1a1 + x2a 2 + + xna n , , x 0&'()*)0 {a , a }. 7,&5,) ,&.-0717+-2 171(&+- 0&'()*)0 {e , e } 1 n 1 n 0 D()?) 6-,,)?) 3*)1(* 1(0 3) ')()*)5 48<)5 7. 3*)1(* 1(0 +)> ) '()* -, -, , & , *-.4)>7(C, ,-./0-&(12 <-.71)+ 6-,,)?) 47,&5,)?) 3*)1(*-,1(0-, - ,-<)* :71&4 ( x1, xn ) ,-./0-&(12 '))*67,-(-+7 0&'()*- x 0 6-,,)+ <-.71&. I*7+&*)+ <-.71- 2042&(12 (*)5'- 0.-7+,) 3&*3&,67'=42*,/G &67,7:,/G 0&'()*)0 {i, j, k} 6&'-*()0)?) <-.71-. -.71 +)>&( & 0&'()* x 7+&&( *-.,/& 1()4<;/ '))*67,-( 0 *-.,/G <-.71-G. *7 3)+)97 1()4<;)0 '))*67,-( 01& 47,&5,/& )3&*-;77 1 0&'()*-+7 +)?=( I
a b a1 b1 = ! = an bn = λ , " λ # $ ' % . & " ϕ " () e1' ,e'2 ) $ ' " () {e1, e 2 } ', "%' ' ' ) '* ' ' ', '' .1 (.2): , e1' = e1 cos ϕ + e 2 sin ϕ . +' e 2 = −e1 sin ϕ + e 2 cos ϕ
{
}
& ' () ' * ' . .' , " " * a )' () )' " ': (a1 , a2 ) → (a1' , a2' ) , %' )' ' * , % 1 %, )' ' .1: 0 a1 = a1' cos ϕ − a2' sin ϕ / a2 = a1' sin ϕ + a2' cos ϕ . 16
(# ' , % () () ' ' {e1 , e 2 } → {e1 , e 2 } * ' * () , ' ' ' (a1 , a2 ) → (a1 , a2 ) . ( ( () (); ( ( " ' ) " )" ). ' )' (a, b ) a b ) % , | a | ⋅ | b | ⋅ cos ϕ , " ϕ # " '* a b . , ' # '* (# * %) ) | a |2 = (a, a) . , (a, b ) = 0 , ) " #', () {e1 , en }, " (ei , e j ) = 0 i ≠ j ' | e |= 1, ) ' ' ()'. &'' i ' " () '* *# () {i, j, k} ' . '' () $ ) * %) ' n n n ) (a,b ) = a b , % (" ( (a, b ) = a b (e , e ) . )
∑ i =1
∑∑
i i
i =1 j =1
i j
i
j
, % ' ' () '* ' " ) ai = (a, ei ). 1. e1 = (2, − 3) e 2 = (1, 2 ). a = (9, 4 ) {e ,e }. 1 2 ! # & e1 e 2 &&(& . " , $% , ' , ), #$ %) % ' % . *%$& ' α1, 2 a = α1 e1 + α 2 e 2 . + # ' ,
- % ) ' ) a = (9, 4 ) , $(. (α1 ⋅ 2 + α 2 ⋅ 1, α1 ⋅ ( −3) + α 2 ⋅ 2). "
& , $' $ $- $ $ & : 1 0 2α1 + α 2 = 9 . / − 3α1 + 2α 2 = 4 23 & - ,$ $, α1 = 2 α 2 = 5 , ' & . 17
- a, b, c , l = 2a − b − c , m = 2b − c − a n = 2c − a − b . &(& l, m, n ? , $ ), & & & ) $ $.$. . + ) - ' , ' . & , ), &&(& ) ' $( ) % .
)# & , & ,- - % , %$ ) $ ), , % $ &$ $. # . '&& ), - - 2.
%
, , ' 2 −1 −1 −1 2 −1 = 0, −1 −1 2 . . &&(& , ), . 2 & $ - % - , ), ' % ) $ # )#, & & $, , $ & & & ) l = −(m + n) .
3. " ), ' ( %- - a, b, c
a
d = (a, c)b − (a, b)c &&(& $& . . "$&) ' $( &# &. & (a, d) , - , (a, d) = (a, ((a, c)b − (a, b)c) ) = (a, c) ⋅ (a, b) − (a, b) ⋅ (a, c) = 0 ( %- - a, b, c , ' % ) ).
'
e1 e 2 , ' | e1 |= 2 , | e 2 |= 1 , $# $ ϕ = π 4 . # , # a b (.6), (. % {e ,e } a = (2, 2) b = (− 1, 4) . 1 2 # $# ,# # . 4.
.6. 18
. " & # ) & % $ $
- # , & # ) – $( ) ,- , . . d1 = a + b d 2 = a − b . ), % {e1,e 2 } # ( d1 = (1, 6) d 2 = (3, − 2) . %) .)( &# &: | d1, 2 |= (d1, 2 , d1, 2 ) . " ( & & # , $' | d |2 = (e + 6e , e + 6e ) =| e |2 +36 | e |2 +12(e , e ) 1
1
2
1
2
1
2
1
2
| d 2 |2 = (3e1 − 2e 2 , 3e1 − 2e 2 ) = 9 | e1 |2 +4 | e2 |2 −12(e1 , e 2 ) . -&. & # $& ', & '& ( (e1 , e 2 ) =| e1 || e 2 | cos ϕ . $) $' | d1 |= 5 2 | d 2 |= 10 . # α $ # ) .)( &# & cos α = (a, b) (| a || b |) , # ) & a b % $ {e1,e 2 } ' &# &, #' ' $ & d d . $) 1 2 # $' , ' $
α = π 4, π − π 4 = 3π 4 . 3.1. 3.2. 3.3.
3.4.
) ' M , $- &
& $#, $) $- . ), - '&- α β a(− 2, 3, β ) b(α , − 6, 2 ) %$$ . c % $ {i, j, k}: c = 16i − 15j + 12k . ) , $ % $ d || c , , | d |= 75 . # ABCD % a = AB , b = AD . , % & BD , CO KD # K ) BC , O - ' '& # .
19
3.5.
) 3$#) ABCDEF & AB AF ( % . , % BC ,
CD , DE , EF , BD , CF CE .
c a b : 1) a = (4, − 2 ) , b = (3, 5) , c = (1, − 7 ) ; 2) a = (5, 4 ) , b = (− 3, 0 ) , c = (19, 8); 3) a = (− 6, 2 ) , b = (4, 7 ) , c = (9, − 3). 3.7. OA OB . " ), ' ' C $ AB # ) # , OC = α OA + β OB , # α ≥ 0 , β ≥ 0 α + b = 1. 3.8. a = (1, − 1, 1) , b = (5, 1, 1) c = (0, 3, − 2 ) . ') ' $(.- : 1) b(a, c ) − c(a, b ); 2 2 2) | a | + | c | −(a, b )(b, c ). + ( &)
$$ $ 3.9. p q , '% p + q % # $ p − q ? 3.10. ' a , b c , $ &(. $ ( a + b + c = 0 . ') '$ (a, b) + (b, c) + (c, a) . 3.11. a , b c % $( $# $# $#, - 600. &, ' | a |= 4 , | b |= 2 | c |= 6 , ) $) p = a + b + c . 3.12. ) # ' # x , # ' -& ' A x $& $( (x,a) = α , # a - α '. 3.13. x , &, ' $& a = (2, 3, 1) b = (1, − 2, 3) , $& $( 3.6.
3.14.
3.15.
(x, (2i − j + k )) = −6 . ( S = ( 2 , − 3, − 5) ), &(.$( & Ox , Oz $# α = 45° , γ = 60° , )( Oy $# β . , & & R = (1, − 8, − 7) , $& & , - 20
3.16.
3.17.
a = 2i + 2 j + k . &(.$( R a . x , &, ' $& a = (2, − 3, 1) b = (1, − 2, 3) $& $( (x, (i + 2 j − 7k ) ) = 10 . e1 e 2 , ' | e1 |= 4 , | e 2 |= 2 , $# $ ϕ = 2π 3 . $#) ABC , $- 3 # OA , OB OC . % {e1, e 2 } , ( OA = (− 2, 2), OB = (− 2, − 1) OC = (− 1, 0). # $# $#) ABC .
a, b, c , !" ' $ '' #' # $ $ , % $& , $' ' * a *+ ()'& * c ! b '$ )' ' # $ $ ' ('$.7 ), ' ' $+# ('$.7 ). 4.
,-..7.
/ # '( $ %'$ ' 0 +$ '( +( + ' *'(, )& )'& , # '), &) $) '& %'$ . 1 !2( 3 a, b, c $ ' 2 '), 3 " %+( ' ) . 4'$, 53+ )" ') $ % «($» $+# $ )(! ' *'' ' a, b, c ' ' %'$ , % $& $0 '% )' a, b, c ' 21
%# $& (a, b, c ) . $+#, ") ' *'& ' a, b, c ' ' *'' %'$ ', $0 '% )' ) + $$)' +% + 53+ $ % «'+ $». /' $ $ ' $0 '% )' & % , ' % *''#$ $ '%& $&: (a, b, c ) = (c, a, b ) = (b, c, a ) . 10 '% )' +(, $' ) ' & &( $& '. $' '% $ )' a, b, c ' %'$ {e , e , e }, $0 '% )' 1 2 3 $ ' !2( ) ' & 2" &): a1 a2 a3 (a, b, c) = b1 b2 b3 . c1 c2 c3 10 '% )' (a, b, c ) )$ ' 2 $& '% )' ) + (d, c ) , ") d % $& '% )' a ' b , %# $& [a, b] ' ) $)+(!'' $ $ ' ('$.8): [a, b] )'+& $$ ' a ' 1) , ' [a, b] & & $& b , +)+#' ' ' , # a b ; 2) )+2 [a, b ] ! )' " , $ " a ' b , . . [a, b ] =| a || b | sin ϕ , ") ϕ $ 2 +" )+ a ' b .
.8. % ' $ $ $ ) # '% ) ' ' ' + , + , . . $ ' ' ( )& # & + . 10 [a, b ] = −[b, a], + '% )' '%& $& ' % "+ ' $& '% )'' $ $ ' &) $) '& , . . 22
' %'$ (a, b, c ) = ([a, b], c) = (a, [b, c]). '% )' 2 ) #% )' 2 { e1 , e2 , e3} 2" &)
e1 e 2 [a, b] = a1 a2
b1
b2
e3 a3
.
b3
"'# + '% )'( 2 ) ) '% )' 3 a, b, c , %# [a, [b, c]]. ' $ 2 ( %'$, %+2 #" +# $& +) & " % '$2 ')
[a, [b, c]] = b(a, c) − c(a, b ). 1. % 2, # )& ( a ' b & $& )$
[a, b ] 2 + (a, b )2 =| a |2 | b |2 .
. )+2 " '% )'& | a || b | sin ϕ , ") ϕ $ 2 +" )+ ' a ' b , $& '% )' | a || b | cos ϕ . $ ' #$ ' 1) 2, ' # | a |2 | b |2 sin 2 ϕ + | a |2 | b |2 cos 2 ϕ =| a |2 | b |2 (sin 2 ϕ + cos 2 ϕ ), | a |2 | b |2 , $ 2 #$ ' $$ ' " )$ . 2. & 3 a, b, c & $& $ [a, b]+ [b, c] + [c, a] = 0 . /% 2, # a, b, c & &( $& '. /'' #$ '$2%+ ) %)# '3 . , %(#(!'$& $& + '' ' #$ " $ -' '% $ , ) $+#, '+, : a (a, [a, b]) + (a, [b, c]) + (a, [c, a ]) = (a,0) = 0 . / ' 2 $" #$ ' $2 ' )& ) + , $ ) (!' , ) (!' + 53 $ " ) $ ') . +#'' 1) 2, (a, [b, c]) = (a, b, c) = 0 , # " ' $ ' a, b, c . 23
3. $ x +) & + '( [x, a] = b , ") a ' b $ 2 '$' ? . '% )' '%' $&, $' )' '% " $ ' '%& 2 , # !)2 ", $ " x ' , '%&$2, $ x $3 & $ $& ) a $$ ' P , )'+& b . $' a $ 2 $ ' ", " $ +) *'& x &+(, )'+&+( a ' !+( $$ ' P . $ ' + *'( '%, ')', # $ +$ '& +) &( $ , !' $$ ' P &, 2 + a ' $ &! " $$ &'' | b | | a | ('$.9).
.9.
4. % 2 )$ : ([a, b], [c, d]) = (a, c ) (a, d ) .
(b, c) (b, d )
. %#' [c, d] f , ") #$ ' )$ +) $ & 2 $0 '% )' 3 ([a, b], f ) = (a, b, f ) . $2% & ' ' $ 2 $ 0 " '% ) '& $' 2 " + + " $ " ' ' '% ) '& " & + , +#' (a, b, f ) = ([a, b], f ) = (a, [b, f ]), $) $ )$ ' f = [c, d] ' $ ) '% )':
(a, [b, f ]) = (a, [b, [c, d ]]) = (a, {c, (b, d ) − d(b, c)}) = , = (a, c )(b, d ) − (a, d )(b, c ) 24
# %#'( )' & #$ ' )$ $ " $ '&. . 4.1. 4.2. 4.3.
4.4.
4.5.
4.6.
4.7.
4.8.
4.9. 4.10.
) %'$ {i, j, k} #'$' 2 '% )'& [i, j], [j, k ], [k, i ], [j, i ], [k, j], [i, k ]. ' ! )2 +"2' ABC $ 0' ' A(− 1, 0, 2) , B(1, − 2, 5) , C (3, 0, − 4) . )' 2, +( ' *'( ' a, b, c $ ' ) %'$ a = (1, 1, 2 ) , b = (2, 1, 1) , c = (1, − 2, 3) . ' 53 ) ABCD , 0' " ) %'$ '( )' A( x , y , z ), B( x2 , y2 , z2 ) , 1 1 1 C ( x3 , y3 , z3 ) ' D( x0 , y0 , z0 ) . $' F1 = (2, 4, 6 ) , F2 = (1, − 2, 3) ' F3 = (1, 1, − 7 ), ' ) # A(3, − 4, 8). )' 2 '#'+ ' ' %+2 '+(! ' $' $' 2 #' B(4, − 2, 6 ). '% 2 : a, b, c , x . % 2, # F1 = [a, x] , F2 = [b, x] F3 = [c, x] . 0' +"2' A(1; − 1; 2 ) , B(5; − 6; 2) ' C (1; 3; − 1) . #'$' 2 )'+ " $ , +! '% 0' B $ + AC . % $ )' 0' ) ABCD : A(0, − 2, 5), B(6, 6, 0 ) , C (3, − 3, 6 ) ' D(2, − 1, 3) . ' )'+ $ " ), +!+( '% 0' C . 0' ): A(2, 3, 1), B(4, 1, − 2) , C (6, 3, 7 ) , D(− 5, − 4, 8). ' )'+ " $ , +! '% 0' D . 53 ) V = 5 , ' " 0' )& $& # A(2, 1, − 1), B(3, 0, 1) , C (2, − 1, 3) . ' )' # 3 0' D , $' '% $ , # ' $' Oy 25
4.11.
4.12.
4.13.
4.14.
% 2, # # #' A(1, 2, − 1) , B(0; 1; 5), C (− 1, 2, 1) , D(2, 1, 3) ) $$ '. 0' +"2' A(2, − 1, − 3), B(1, 2, − 4) ' C (3, − 1, − 2 ) . #'$' 2 )' x , ' " $ " '% 0' A ' +( $ +, ' $ , +! +$ '', # x %+ $ $2( Oy + +" ' # " )+2 2 34 . ' a, b, c ' ) !+ #+. % 2, # $$ 2, )&!& #% * ' , )'+& + [a, b] + [b, c] + [c, a ]. '% $ x +) & $ )+(! $'$ + ' , ") a, b, c ' #'$ p, q, s $#' ( $& '% $ ':
(x , a ) = p (x, b ) = q (x, c ) = s.
4.15.
%' 2 x #% a, b, c ' #'$ p, q, s . % 2 )$ :
(a, b, c)2 + [[a, b], c] 2 = [a, b] 2 ⋅ c 2 ; 2) [[a, b ], [c, d ]] = c ⋅ (a, b, d ) − d ⋅ (a, b, c ) ; 2 3) ([a, b ], [b, c], [c, a]) = (a, b, c ) ; 4) [a,[b, c] ] + [b,[c, a ] ] + [c,[a, b ] ] = 0 ; 3) [a,[b, [c, d ]] ] = [a, c] ⋅ (b, d ) − [a, d ] ⋅ (b, c ) ; 2 2 2 2 2 4) [a, b ] ⋅ [a, c] − ([a, b ], [a, c]) = a ⋅ (a, b, c ) . 1)
26
3
5.
!" #$%$ L %" &#'()'(*$ + ,-)" *'+'. ($(*-!- )'' ,$%"* ( x, y ) /","0*( + '12-! +$,- 3 "+%-%$-! &- +'. (*-&-%$ Ax + By + C = 0 , 4,)'566$7$-%*8 A $ B %- "+%8 %3#9 ',%'+ -!-%%'. : "+%-%$- '12-4' +$," &3*0! .5#-!-%*" %8; & -'1 "/'+"%$. !'<-* 18*= & -,(*"+#-%' )") 3 "+%-%$& !' ( 34#'+8! )'566$7$-%* '! y = kx + b $#$ )") 3 "+%-%$- & !'. + '* -/)"; x a + y b = 1, 4,- a $ b -(*= "#4-1 "$>-()$-, *. -. ( 3>0*'! /%")", 8 '* -/)'+, '*(-)"-!8; ,"%%'. & !'. %" )'' ,$%"*%8; '( ; (?$(.10"). ,#$% @ $(&'#=/'+"%$-! "&&" "*" +-)*' %'. "#4-1 8 3 "+%-%$- & !'. %" &#'()'(*$ !'<-* 18*= /"&$("%' + &" "!-* $>-()'! +$,- r = r0 + a t , 4,- r = ( x, y ) '&$(8+"-* ",$3(-+-)* ' *'>)$ %" & !'., r0 = ( x0 , y0 ) -(*= ,+3!- %8. +-)*' , /","92$. %">"#=%39 *'>)3 %" & !'., a = (a x , a y ) -(*= %"& "+# 92$. +-)*' & !'., " &" "!-* t & '1-4"-* +(9 >$(#'+39 '(= (?$(.10 1).
ABC.10
D()#9>" &" "!-* t $/ 3 "+%-%$ & !'., -4' !'<%' /"&$("*= + )"%'%$>-()'! +$,G F x − x0 y − y0 = . E ax
ay
27
(#$ $/+-(*%8 )'' ,$%"*8 ,+3; *'>-) r1 = ( x1 , y1 ) $ r2 = ( x2 , y2 ) , >- -/ )'*' 8- & ';',$* & !" , *' -0 %"& "+# 92$. +-)*' !'<%' /","*= + +$,. . a = r2 − r1 , >*' &'/+'# -* /"&$("*= 3 "+%-%$- & !' , & '; ', 2- >- -/ ,+,"%%8- *'>)$, + +$, x − x0 y − y0 = .
x2 − x1
y2 − y1
'!- *'4', & $ $(&'#=/'+"%$$ %' !"#=%'4' +-)*' " n ) ,"%%'. & !'. -0 3 "+%-%$- + +-)*' %'! +$,- !'<-* 18*= /"&$("%' )") ((r − r0 ), n ) = 0 , %"/8+"-!'! 3 "+%-%$-! & !'. + %' !"#=%'! +$,-. '()'#=)3 (r0 , n ) = D -(*= &'(*' %%'- >$(#', 3 "+%-%$- & !'. + %' !"#=%'! +$,- *")"- 34'# !-<,3 & !8!$ %";',$! + +$,tan ϕ = ( A1B2 − A2 B1 ) ( A1 A2 + B1B2 ) . *(9," (#-,3-*, >*' 3(#'+$-! ; & !8;, 5)+$+"#-%*%'- '1 "2-%$9 + %3#= >$(#$*-# &" "##-#=%'(*$8 ,+3 &'(#-,%-4' + "<-%$ , +# -*( & '&' 7$'%"#=%'(*= )'566$7$-%* '+: A1 A2 = B1 B2 , " 3(#'+$- &- &-%,$)3# %'(*$ & !8; +8*-)"-* $/ '1 "2-%$ + %3#= /%"!-%"*-# $ $!--* +$, A1 A2 + B1B2 = 0 . (#$ /"6$)($ '+"*= * '>)3 & !'. ( x0 , y0 ) , *' + 3 "+%-%$$ & !'. ( 34#'+8! )'566$7$-%*'! y − y0 = k ( x − x0 ) !'<%' $/!-% *= &" "!-* k , '& -,-# 92$. *"%4-%( %")#'%" & !'. ) '($ "1(7$((. 5*'! (#3>"- 4'+' * ' &3>)- & !8;, & ';', 2$; >- -/ 7-%* &3>)" ( x0 , y0 ) . (#$ ,"%8 ,+&- -(-)"92$-( & !8- ( 3 "+%-%$ !$ ( A1,2 x + B1,2 y + C1,2 = 0 , *' &3>') & !8;, & ';', 2$; >- -/ *'>)3 $; &- -(->-%$ , !'<-* 18*= /","% ( &'!'2=9 ,+3; &" "!-* '+ α $ β )")
α ( A1x + B1 y + C1 ) + β ( A2 x + B2 y + C2 ) = 0 .
?"((*' %$- '* *'>)$ M ( x1, y1 ) ,' & !'. L , /","%%'. 3 "+%-%$-! '12-4' +$," Ax + By + C = 0 , '& -,-# -*( &' 6' !3#d ML =
Ax1 + By1 + C A +B 2
28
2
,
+#)* 92-.( )'' ,$%"*%'. /"&$(=9 -/3#=*"*" +8>$(#-%$ 5* '4' "((*' %$ + +- ' %'! +$,d ML =
(r1 − r0 ), n
,
n 4,- & !" L /","%" + +$,- ((r − r0 ), n ) = 0 , " r1 -(*= ",$3(-+-)*' *'>)$ M ( $(.11).
.11
1. L(−6, 0) N (0, 8) . , LN ! Ox " # , Oy . $%& . ' x a + y b = 1 " a b . ( ! a = 3b , ) , ! ! M (−3, 4) , !
LN . * , − 3 3b + 4 b = 1 b , b = 3 a = 9 . + x 9 + y 3 = 1, x + 3 y − 9 = 0 . 2. ( "
L1 L2 ,
r = r1 + a1 t r = r2 + a 2 t ) ! ; ") , !; ) !. $%& . , ) " ! a a , ) 1 2
( .12 ). , ) ,
" ! , -
29
, " ! , r − r ! ) a = a = a 1 2 1 2 )
" ( .12 "). ,
! r − r ! a = a = a ( .12). 1
2
1
2
.12
3. # ) PQR : P(0, 5) ; Q(−3, 1) ; R(1, − 2) . ) , # R . $%&
. R(1, − 2) L ,
P (0, 5) Q (−3, 1) ,
x−0 y−5 = , − 3 − 0 1− 5
4 x − 3 y + 15 = 0 . ! #
,
d RL = . .
h = 5.
4 ⋅ 1 − 3 ⋅ (−2) + 15 4 + (−3) 2
2
= 5,
4.
d ML M - r0
L , r, n = D . $%& . M ! L , ! M . , " 1
MM 1 = d ML ( . .11). M 1 - r1 , ! ! r , n = D , " 1 ! . !
(
(
30
)
)
d ML =
(r0 − r1 ), n n
=
M (r0 )
(r0 , n ) − (r1 , n ) n
.
=
((r − r1 ), n ) = 0 ,
(r0 , n ) − D n
,
5.1. ( 3 x − 4 y − 12 = 0 . ) !
, 3 x − 4 y − 12 > 0 3 x − 4 y − 12 < 0 ? 5.2. ( C , 2 x + 3 y + C = 0 , Oy b1 = 4 b2 = −6 ? 5.3. ) , 6 x + 8 y + 5 = 0
2x − 4 y − 3 = 0.
5.4. ) ,
r = r2 + a 2 t .
5.5.
) ,
(r, n 2 ) = D2 .
(
r = r1 + a1 t
(r, n1 ) = D1
)
5.6. ! P − 6, 4 ! 4 x − 5 y + 3 = 0 . 5.7. ' ) , ! ) 2 x + 7 y − 14 = 0 . 5.8. ' ) , ) L ,
L3 , ! : x − 3 y + 11 = 0 ;
L2 L1 : L2 : 5 x + 2 y − 13 = 0 ; L3 : 9 x + 7 y − 3 = 0 .
1
5.9. , M (1, − 2) ) L : 4 x + 7 y − 3 = 0 ; ") - . 5.10. ( L1 r, n = D , L2 r = r + a t , n, a ≠ 0 . - 0
L1 L2 .
(
31
(
)
)
5.11. - rn M - r0
! L , ! r, n = D . 5.12. , )
! x − y + 3 = 0 .
(
)
(
) (
)
(
)
5.13. , # ) A 1, − 1 , B − 2, 1 C 3, 5 . * , ) # ,
! # '. 5.14. * , P 3, 5
A − 7, 3 B 11, − 15 . 5.15. x − 2 y + 5 = 0 . , 3 x − 2 y + 7 = 0 , . * , . 5.16. , a , a + 2 x + a 2 − 9 y + 3a 2 − 8a + 5 = 0 , ": ) "
; ") ; ) . ' . 5.17. , m n , mx + 8 y + n = 0 2 x + my − 1 = 0 ) ; ") !; ) . 5.18. M 4, 3 , ! ) ) ) , ) 3 . .
- . 5.19. , ) 3 x − 2 y − 5 = 0 ,
(
(
(
)
(
)
(
)
(
)
)
)
) # A(−2, 1) . ' - ) ) . 5.20. , : 10 x + 15 y − 3 = 0 , 2 x + 3 y + 5 = 0 , 2 x + 3 y − 9 = 0 . + , ) , #,
. 5.21. , M ( 2, 3) N (5, − 1) , ) , " :
2x + 3y + 7 = 0
x − 3y − 5 = 0, 2x + 9 y − 2 = 0; ") 2 x + 7 y − 5 = 0 , x + 3 y + 7 = 0 ; ) 12 x + y − 1 = 0 , 13x + 2 y − 5 = 0 . )
32
5.22.
*
) "
0 , 6 x − 4 y + 7 = 0 , ) ) ,
2x − 3 y − 5 = C ( 2, − 1) .
5.23. , M ( 4, − 5) . * - ) " !, ) ! L : 2 x − 3 y + 6 = 0 ; ") ! L . 5.24. , L1 L2 L3 ,
! :
L1 : 5 x − y + 10 = 0 ; L2 : 8 x + 4 y + 9 = 0 ; L3 : x + 3 y = 0 .
5.25.
"
!
A2 x + B2 y + C2 = 0 .
) ,
"
A1x + B1 y + C1 = 0
5.26. , α (2 x + 5 y + 4 ) + β (3 x − 2 y + 25 ) = 0 . ! - ) , ! ! ( ). 5.27. * ,
3 x + y − 5 = 0 , x − 2 y + 10 = 0 C − 1, − 2
d = 5 . # , . 5.28.
Ax + By + C = 0
(
)
1
Ax + By + C2 = 0 .
5.29.
M - r0 L , r = r1 + at . 5.30. , α 3 x − 2 y − 1 + β 4 x − 5 y + 8 = 0 . ! - ) , !
! )
x + 2y + 4 = 0, 2x + 3y + 5 = 0 x + 7 y − 1 = 0 .
(
33
)
(
)
6.
(
)
' xyz
Ax + By + Cz + D = 0 . ) ! , " )" a, b, c , ! , x a + y b + z c = 1. ! " n = A, B, C - " r ( x , y , z ) , 0 0 0 0 ! . ' - " A( x − x0 ) + B ( y − y0 ) + C ( z − z0 ) = 0 , - (r − r0 ), n = 0 ( .13). " , - , . '
" u v , "
a, b ! r = r0 + au + bv . " " n = a, b . , "
# )
(r − r0 ), a, b = 0 , ! ! " ) , ) , " ( .13).
(
(
(
)
)
)
{ }
[
(
]
)
.13 ( )
d M P , MP (r − r ), n = 0 ( .13),
(
0
)
34
d MP =
(r1 − r0 ), n n
,
#
d MP =
Ax1 + By1 + Cz1 + D A +B +C 2
2
2
.
1. , M ( x , y z ) , M ( x , y z ) M ( x , y z ) . 1 1, 1 2 2 2, 2 3 3 3, 3 .1 M ( x, y, z ) . !" # $ ((r − r0 ), a, b ) = 0 , " r = ( x, y, z ) , r0 = ( x1, y1 , z1 ) , $ .13 a = M M b = M M . % 1 2 1 3 &
, : x − x1 x2 − x1
y − y1 y2 − y1
z − z1 z2 − z1 = 0 .
x3 − x1
y3 − y1
z3 − z1
2. ! M ( x , y , z ) # P , 0 0 0 Ax + By + Cz + D = 0 , MM 1 ( A, B, C ) . ' , " ( M ( x , y , z ) .14 # # 1 1 1 1 P , .. Ax1 + By1 + Cz1 + D > 0 .
)*+.14 35
. x − x = A , 1 0 #
M y1 − y0 = B z1 − z0 = C , Ax0 + By0 + Cz0 + D = 0 , $# , Ax1 + By1 + Cz1 + D = A( x0 + A) + B( y0 + B) + C ( z0 + C ) + D = Ax + By + Cz + D + A + B + C = 0 + A + B + C > 0 2
2
2
2
2
2
,
0 0 0
$ . 3. & ABCD : A(2, − 1, 3) , B(1, − 3, 5) , C (6, 2, 5) D(3, − 2, − 5) . , & D " ABC . . # D P , A, B, C . % & P ,
1:
x − 2 y +1 z − 3 1 − 2 − 3 + 1 5 − 3 = 0, 6−2
2 +1 5 − 3 P 2 x − 2 y − z − 3 = 0 .
, d DP =
2 ⋅ 3 + ( −2) ⋅ (−2) + ( −1) ⋅ (−5) − 3 2 + (−2) + (−1) 2
2
.
2
= 4.
% , M ( 4, − 3, 5) . 6.2. , M 1 (1, − 2, 6) , M 2 (5, − 4, − 2) , Ox Oy . 6.3. , M 1 ( x1 , y1, z1 ) M 2 ( x2 , y2, z2 ) a(a1, a2 , a3 ) , a M 1M 2 . 6.1.
36
6.4. 6.5.
6.6.
( "$ r = r0 + au + bv (r, n ) = D . , A, B, C , : ) A(1, 1, 2) , B(2, 3, 3) , C (−1, − 3, 0) ; $) A(2, 1, 3) , B(−1, 2, 5) , C (3, 0, 1) . %
x =1+ u − v y = 2 + u + 2v
z = −1 − u + 2v
,
Ax + By + Cz + D = 0 . 6.7. % $ 2 x − 3 y + z + 1 = 0 ,
r = r0 + au + bv . 6.8. Ox , A(9, − 2, 2) P : 3x − 6 y + 2 z − 3 = 0 . 6.9. Oz , M (1, − 2, 0 ) P : 3x − 2 y + 6 z − 9 = 0 . 6.10. , Oz c = −5 n = (− 2, 1, 3) . ,
6.11.
"
", $ : 3x − 4 y − z + 5 = 0 , 4 x − 3 y + z + 5 = 0 . 6.12. M - r1 P , (r, n ) = D . 6.13. #
P1 P2 , (r, n ) = D (r, n ) = D . 1 2 a b 6.14. , 2 x − y + 3 z − 1 = 0 , x + 2 y − z + b = 0 , x + ay − 6 z + 10 = 0 : ) $ ; ; $ ) ) .
37
, , # $ r = r0 + a t ( .10 $), " r = ( x, y, z ) - , r = ( x , y , z ) , 0 0 0 0 , a = (a , a , a ) x y z
"
. t , "t $
, # x − x0 y − y0 z − z0 = = . ax ay az
r = ( x , y , z ) r = ( x , y , z ) , 1 1 1 1 2 2 2 2
, #
a = r − r , , 21 , x − x0 y − y0 z − z0 = = . x2 − x1 y2 − y1 z2 − z1 " # ,
r − r0 " ,
: [(r − r0 ), a] = 0 ' ( .15). # $ # [r, a ] = b , " a , b - . 7.
.15 38
P1 P2 ,
A x + B y + C z + D = 0 , 1, 2 1, 2 1, 2 1, 2
L = P1 ∩ P2 , : A1x + B1 y + C1 z + D1 = 0 , A2 x + B2 y + C2 z + D2 = 0
a = [n1 ,n 2 ] (.16), ( x0 , y0 , z0 )
.
.16 ,
, , , , :
α ( A1 x + B1 y + C1z + D1 ) + β ( A2 x + B2 y + C2 z + D2 ) = 0 !
α β
"
" .
39
! M - r1 L , [(r − r ), a] = 0 , 0
d ML =
[(r1 − r0 ), a] |a|
,
" , " (.17).
.17
, . .
L1 L2 ,
r = r1 + a1 t r = r2 + a 2 t ,
d12 =
((r2 − r1 ), a1 , a 2 ) , [a1, a2 ]
, " .17.
1. L , M (5, − 1, − 4) , : 1) L A ,
x = 3 + 6t , y = 2 − 4t , z = 7 − t ; 2) L Ox ; 3) L P : x + 2 y + 3 z − 5 = 0 . L A , . 1) , "
40
A , a(6, − 4, − 1) . ,
L x = 5 + 6t , y = −1 − 4t , z = −4 − t . 2) L Ox ,
" ,
Ox , , a(1, 0, 0 ) . , L
x = 5 + t , y = −1 ,
z = −4 . 3) L P ,
"
, n(1, 2, 3). L
x = 5 + t , y = −1 + 2t , z = −4 + 3t .
2. L , A ,
x − 2 y +1 z − 5 = = , 6 −5 4
P ,
x − 3 y + 2 z − 7 = 0 (.18).
.18 . L
P P1 , " P , P1 A M (2,−1,5)
, "
0
41
P . M ( x, y, z ) " : P1 .
M 0 M = ( x − 2, y + 1, z − 5) , a(6, − 5, 4 ) ,
P n(1, − 3, 2 ) , P . 1 ,
, P : 1
x−2 6
y +1 z − 5 −5 4 = 0,
1 −3 2 2 x − 8 y − 13z + 53 = 0 . , " P , : x − 3y + 2z − 7 = 0 L: . 2 x − 8 y − 13 z + 53 = 0
L1 L2 , 3.
[r , a] = b [r, a ] = b . 1 2 "#
#
M r1 L1 , $ r1 .! % & [r , a] = b . ' 1 1
# # "# L1 L2 (
M
L2 ( .17), # ( # " "# #
r0 . ) *, " :
d12 =
[(r1 − r0 ), a] [r1, a] − [r0 , a] a
=
a
=
b1 − b 2 a
,
" # . +,-,. -/0 ,2 20 /4 252 0 1 13 3 . 7.1.
6 # " # *, ( %7 * $ $ ABCD , 8 # $ * "# *
A(2, 4, 6) , B(−3, 5, 4) C (8, − 6, 2) .
42
7.2.
6 , * 7 " "# A(1, − 1, 2) # # #
P , :
) x − 3 y + 2 z + 1 = 0 ;
7.3.
()
x = 4−u +v y = 2 + u + 2v
.
z = −1 + 7u + 3v
# r = r0 + a t (r, n ) = D . ! # #
(* "
:
7 "#
) # # % ( 7( * "# ); # ()
( ( ); ) # .
Ox # # " * # . 7.4. 6
$ %
#
7.5. A B ,
?
x = 1 − 2t x = −4 + 3t
) A : y = 2 + 5t , B : y = 2 − t ; z = −3 + 4t z = 5 − 2t x = 6 + 3t x = 1 + 2t () A : y = 7 + t , B : y = −1 − 2t . z = 3 + 4t z = −2 + t
7.6.
6 , * 7 " "# M ( 2,−3,5)
# A B , # " # x −1 y − 3 z + 5 x − 2 y +1 z + 7
A:
−1
=
2
=
2
B:
6
=
3
=
−2
.
7.7. 6 # , * 7 " "# M ( 4,−1,−1)
% L , % # # % " * # :
7.8.
7.9.
2 x − 3 y + 5 z − 7 = 0 L:
. 4x + 2 y − 6z − 5 = 0
6 # L , " # # # x = x0 + axt , y = y0 + a yt , z = z0 + a z t , # P , Ax + By + Cz + D = 0 . 6 # P , * 7 " % L
Oy , L # # " * # :
43
L:
7.10.
3 x − y + 2 z + 9 = 0
.
x+ z −3= 0
# A : r = r0 + a t P : (r, n ) = D , ( . "# M A
# P ρ . - # "# M .
# , # "# # α (10 x − 8 y − 15 z + 56) + β (4 x + y + 3z − 1) = 0
"# C (3, − 2, − 3) d = 7 . # , # %7 % 7.12. 6 3 x + 2 y − z − 1 = 0, #
x + 2 y + 3z − 5 = 0 . 7.11.
2x − 3 y + 2z − 2 = 0
7.13.
7.14.
7.15.
7.16
7.17.
8 $ # A(3, − 1, − 1) , B(1, 2, − 7 ) C (− 5, 14, − 3) . 6 # " # ( # $ $ $ 8 B . 2 x − y + 3 z + 1 = 0,
{i, j, k} 3 x + y − z − 2 = 0. !" a . #$ "%
{i, j, k} !" & % .
' % , ( ) )
a = (6, − 2, − 3) M 1 (− 1, 2, − 3) x −1 y +1 z − 3 %! = = . 3 2 − 5
' % , ( ) ) x +1 y + 3 z − 2 %$ M 1 (− 4, − 5, 3) = = , 3 −2 −1 x − 2 y +1 z −1 = = . 2 3 −
5 " % ' ) ( %$(, $( % x = 3t − 7, y = −2t + 4, z = 3t + 4 x = t + 1, y = 2t − 8, z = −t − 12 .
44
4 8.
,
% % % $
$ % %
) $ ! ) $ % " ) ( ) . .19 $ %$ ) ( ) ,
, ) % % % % ) & ) , $ & ( .19 ; ) $ ), ) , !" , & .19 .
!".19 % )( ) ,
.20 $ $ ) ) , $ # . " ( $ % (.20 ),
!" %$, & .20( ).
45
!".20 % ,
, % ! . , & $ , ) , " # ) $ % ( .21 ), %$ , ( " ) ( .21( )).
!".21 46
# ) ) , $ .19-21, ! % "% %: ! ) M , " ),
MF ) F , " !" ) ,
! MM 1 % δ , " & , ) :
MF =ε, MM 1
% ε $ & %, ) F $ %, $ . ) ) % $ % δ $ % , & .22 % $. ) , !, %" )
%% ) , .
!".22 % ) ) $ , , MF MM ! ! 1 % ρ (ϕ ) , , & .22, %
ρ (ϕ ) =
p , 1 − ε cosϕ
47
% p & % ε ) . ! ( %( ) ) ) . % )
) % ) , ) ,
) !" ! , ) , , % % ) ) , % ( ) , $ %) ) ( . .22).
# "% ) %)%
Ax 2 + 2 Bxy + Cy 2 + 2 Dx + 2 Ey + F = 0 . %
, % ) " $ , % )( , !"( $ .
% ( % $ ! $ % . , % 2a 2b ) $ ) % ) % % ) % ( .23), $ % x2 y 2 + = 1. a2 b2
!".23 % $ a ≥ b $ ! % !
! & , ) $( .
) (± a, 0) (0, ± b) $ ! % & . %
$ % ) % ), %%
48
( ( ( % ( % $ $ ) , $ $ & , ) : r1 + r2 = 2a , r , $ $, $ 2
1, & (.23), $ ) % )
! ) ( F1, 2 (± c,0) % % c = a 2 − b 2 . ε = c a < 1 $ & % & %
! ) # ) « $ ». ,
! ( % & .
$% %$% x = ± a ε , $ %$% % . ' & % ) ) %$ % " ! % ) % ), $( ) ( ) % ( $) ) & , . . .23 $ MF1 MM 1 = ε . %( % ( ( ) & % % ( .24) 2 a 2 b $ ) % % x2 y2 − 2 = 1. 2 a b
.24 49
) (± a, 0) $ ! % $.
x 2 a 2 − y 2 b 2 = −1 $ %! ! , Ox Oy % % %. % $ % ) % ),
( ( ( % ( % $ $ ) , $ $ & , ) : r1 − r2 = 2a , r , $ $, $ 1, 2 % ! ( ) ) ( .24), $
% % c = a 2 + b 2 . ε = c a >1 F1, 2 (± c,0) $ & % $, $ x = ± a ε ! % % % $ ) ( .21). $ $ & $ ! $% % $ ) , . . r1, 2 =| a ± ε x | . ) $ ! %$% %, %!"% y = bx a y = − bx a $ (.23). %$ $ ! % % $, ,
% % % % ( . , ) , & , % % % ) % ), $( $ ) , $ % %, % , $ % . .22, ! % MF MM 1 $. # ) % $ y 2 = 2 px ,
& % ) F ( p 2 , 0) , % x = − p 2 . & $, $
% $ $ ) ) & ) ) r = x + p 2. ,
) $ ε = 1. % p ) ) $( ( % a b % p = b 2 a . y = f ( x )
% , %!" "! ) x0 ,
! % )
50
f ' ( x0 ) . $% " % ( , % $ % , %!" !"(
) ) % L . %,
% ) % % ) ( x0 y0 ) ,
C ) ! ( , "% , % , !"( ) ) C % L Ax 2 + 2 Bxy + Cy 2 + 2 Dx + 2 Ey + F = 0 x = x0 + a xt
y = y0 + a y t
) % ) (
% t , $ !"( ) ) % ( % t , & $ ) $ ) ( .25). ) , ) % ! ( )$( ) (, . . ( . , ! ) ! )
( x0 y0 ) , % , .
.25
51
1. ,
L(3 2 , 2 2 ) N (6, 0) . !. " # x 2 + y 2 = 1 # 2 2 a b #
. $ % , ( a b : '18 a 2 + 8 b 2 = 1 . & 2 2 36 a + 0 b = 1 )* , a = 6 b = 4 , .. x 2 36 + y 2 16 = 1. 2. + , #, , %
. !. - # y = ± bx a , bx ± ay = 0 . . / ( x, y )
, # d1, 2 =
bx ± ay
a +b
, , d1d 2 =
2
2
,
b2 x2 − a 2 y 2 a 2 + b2
.
$ , # % #, 0 a 2b 2 , , 0 , a 2b 2 . 1 , d1d 2 ,
. 3. 2 , ρ (ϕ ) =
5 1 1 − cos ϕ 2
, % : . !. 1
, , , 52
. ε = 0.5 , . . , p = b2 a = 5 .
0 ε = c a = a 2 − b 2 a , a b . )* , a = 20 3 b = 100 3 , % 9 x 2 400 + 3 y 2 100 = 1.
4. $ , ,
, ( x0 , y0 ) , xx yy 20 + 20 = 1. a b !. , # / y = y ( x) , // # % , . , y − y = y' ( x) ⋅ ( x − x ) . . 0 0
0 0 y ' ( x) , // , x , 2 x a 2 + 2 y ( x) ⋅ y ( x )' b 2 = 0 , y ' ( x) = − x ⋅ b 2 (a 2 ⋅ y ( x)) . + 0 y ' ( x) , , 0 , y b 2 . . x 2 a 2 + y 2 b 2 , . 5. 0 x 2 + y 2 + 4 x − 8 y + 2 = 0 , . !. $ , , 0 , * y = k x . , , , x2
+ y2 + 4x − 8 y + 2 = 0 . y=kx
,
x , $ // . - , , , *
, .. # *
. # : 53
(4 − 8k ) 2 − 8(1 + k 2 ) = 0 .
+
, # , ,
// k . $0 , , # , ,,
# .
0 , # k = 1 1
k2 = 1 7 , y = x y = x 7,
# #
0 . , , , , 0
. !
!.
+ 9 x 2 + 5 y 2 = 45 . : 1) ; 2) / ; 3) ; 4) . 2 2 8.2. + , 16 x − 9 y = −144 . : 1) a b; 2) / ; 3) ; 4) ; 5) . , : 8.3. 2 0 1) / 60 ; 2) 0 / * ; 0 , * 0 3) ; / ,
4) , * . 8.4. M (3, − 1) , / 0 y + 6 = 0 . 1 , ε = 2 2 . 2 2 8.5. , # / x 25 + y 9 = 1 . 1 ,, % ε = 2 . 2 2 8.6. + x 15 + y 6 = 1. ,, * / , / –
. * (, ) 8.7. x 2 12 + y 2 6 = 1,
,
. 8.1.
54
8.8.
8.9.
+ ,
x 2 25 + y 2 16 = 1, , 0 ,) / ; . # ) / 2 , : 12 10 1 ) ρ (ϕ ) = . ; ,) ρ (ϕ ) = ; ) ρ (ϕ ) = 2 − cosϕ
3 1 − cos ϕ 2
8.10. 8.11.
8.12.
sin 2
ϕ
2
1 , % F (0, 3) y = −5 . + , y 2 = 6 x . , N ( 4, 1) , , ,
B C , , BC , N . ,,
Oy x + y = 0 0 x2 + y2 + 8 y = 0 .
8.13.
$ , ,
( x0 , y0 ) , xx yy ) 20 − 20 = 1 ,, a b ,) y y = p( x + x ) ,. 0
0
x2 y2 .
P(− 16, 9) + = 1. 8.14. 4 3
# d P ,
. x2 y2 . / + = 1 α Ox 8.15. 45 20
. .
, tgα = −2 . + ,
. 1 , 0
0%
.
55
8.16. 8.17. 8.18. 8.19.
8.20.
$ , x 2 16 − y 2 8 = −1 2 x + 4 y − 5 = 0 d 0 .
, ,, # , / , # + . 1 , , y 2 = 8 x 2 x + 2 y − 3 = 0 . 1 , # , , y 2 = 16 x , ) A(1, − 2) ; ,) B(1, 4) ; ) C (1, 5) . # , 0
, – ,
%:
x − y + 2 = 0 , y 2 = 8x ; 2 2) 8 x + 3 y − 15 = 0 , x = −3 y ; 2 3) 5 x − y − 15 = 0 , y = −5 x . 1)
$ , Ax + By + C = 0 , y 2 = 2 px ? 8.22. 1 6 xy + 8 y 2 − 12 x − 26 y + 11 = 0 ,
) 6 x + 17 y − 4 = 0 ; ,) 41x − 24 y + 3 = 0 ; ) y = 2 . 8.21.
, % 0 F ( x, y, z ) = 0 , F , / .
/ ,
, , , . " , %
0 * # / . (ρ ,ϕ , z ) ( .26) ,, % , ,
Oz ,
( ρ ,ϕ ) . 9.
56
/ # :
x = ρ cos ϕ y = ρ sin ϕ . z=z
.26
(r ,θ ,ϕ ) ( .26( )) M
r = OM O , θ OM z , ! ϕ "
. $
OM ! ( xy ) !# x #% & & : * 'x = r sin θ cos ϕ ) 'y = r sin θ sin ϕ . ( z = r cosθ +
- , , z , & & , #% F ( x, y ) = 0 , " & ! z . , & ! ! %!# F ( x, y ) = 0 , #% , , ! Oz #% .
57
" & " , .27. + & "
ϕ F ( ρ , z ) = 0 , & % !# % , ! Oz . , & ! ! & & ! % . & % #
F ( x, z ) = 0 , % Oz & ! % . ! & " , & ρ = x 2 + y 2 , ! & % F ( x 2 + y 2 , z ) = 0 . .27( ) & ! % , #% x 2 + y 2 = 2z 4 . + & ! #% , &% #% # F ( x, y ) = 0 # M ( x , y , z ) , & 0 0 0 M . &
, .27 .
.27 , & # & & ! . & #% :
58
2 2 2 x + y + z = 1 ( .28); a 2 b2 c 2 x 2 + y 2 − z 2 = 1 ( .28 ); a2 b2 c2 2 y2 z2 x
.28); + − = − ( 1 a2 b2 c2 2 y2 z2 x
29); + − = ( 0 a 2 b2 c 2 2 2 x + y = 2 z ( .29( )); a2 b2 2 y2 x
.29). 2 − = z ( a2 b2
, & & & a = b , & , ,
! & % . & a = b = c = R R .
.28
59
.29
2 2 x a + z 2 c 2 = 1, % y = 0 . ! % , % Oz . . , #% F ( x, z ) = 0 , x → x 2 + y 2 , # 1.
2 2 z2
% x + y + 2 = 1. 2 2. a L c " : x = 9 − t , y = 4 − 2t ,
z = 7 + 2t , M 0 (1, − 2, 3) , % & . ! " & & . . " ! , & , #% !# "
( .30).
.30 60
! M ( x, y, z ) & " . ,
d ML = d M 0 L ! & ( x, y, z ) , " .
[(r − r ), a] d ML = 1 0 , |a|
r = ( x, y, z ), ! 1 #% r0 = (9, 4, 7 ) a = (− 1, − 2, 2 ).
%!# ! , , 1 d ML = (2 y + 2 z − 22) 2 + (25 − 2 x − z ) 2 + (14 − 2 x + y ) 2 . 3 & ( x, y, z ) M , & 0 "
" R = d
, = 10 . , M 0L 1 (2 y + 2 z − 22) 2 + ( 25 − 2 x − z ) 2 + (14 − 2 x + y ) 2 = 10 . 3 , & .
" , 8 x 2 + 5 y 2 + 5 z 2 − 4 xy + 4 xz + 8 yz − 156 x − 60 y − 138 z + 405 = 0 . ! , & , ! & & . + ! " Oz ' , Ox' Oy '
! , " ( x' ) 2 + ( y ' ) 2 = 100 .
3. ! M (r ) !# r = r + a t , , 0 #% !# 0 0α . # & , . , #% ,
α . !, ! r − r , r = ( x, y, z ) , a : 0 (r − r0 ) =| r | ⋅ | r0 | ⋅ | cosα | ,
61
α & r − r & . ! 0 & ,
r − r0 a , , . & & , ! , ! & &, #% & & ! .
9.1. 9.2.
9.3. 9.4.
9.5.
( x − 4)2 + ( y + 5) 2 + ( z − 6) 2 = 81
x = 3 − t , y = −3 + 2t , z = 4 − 2t . ! % , % x 2 a 2 − z 2 c 2 = 1, % y = 0 , Oz . #% x 2 9 + z 2 25 = 1, y = 0 , & S (0, − 3, 4) . ! . !
,
x 2 + y 2 + z 2 = 4 (. . #% ), & S (0, 0, 8) . !, #% ,
x = ϕ (t )
y = ψ (t ) z = χ (t ) ' , ! "# $ % & & u v + # (x = uϕ (v) * ()y = uψ (v) . z = uχ (v) 9.6. , & - % &. /0 $ / , !1 &% $# & λ . 2& $ & &- & % !#- λ : x2 + y2 + z 2 = λ ; 2 2 2 3) λx + y + z = λ ; 1)
λx 2 + y 2 + z 2 = 1 ; 2 2 2 4) x + y − z = λ ; 62
2)
5) 7) 9) 9.7.
9.8.
9.9.
9.10. 9.11.
x2 − y2 − z 2 = λ ; x 2 + y 2 = λz ; x2 + y2 = λ ;
2& $, !
x 2 + λ ( y 2 + z 2 ) = 1; 2 2 8) x + λy = λz + 1 ; 2 2 10) x − y = λ . M (1, 1, 1) / 6)
&
x 2 + 2 y 2 + 3z 2 = 4 . 2$ Oz & -. 2& $, ! M (1, 1, 1) # ! &" x 2 + 2 y 2 = 2 z . ,. &- x 2 + 2 y 2 − 3z 2 − 1 = 0 & . x = 0 , x = 1 , x = 2 & # & $ Oyz . 2& $ %"% $
& . . . . &. /- &- 0 &. & #. & / $ . ?. /- &- 0 &. & .. & / . 0 ? , 0 &. ? " #. &
10.
"1 / . 0 &. & % . / Ax 2 + 2 Bxy + Cy 2 + 2 Dx + 2 Ey + F = 0 , &- $ . & &# . / - / /#, 0 0 & " . . # $ z % !# . x y , 1 &# &. . . " . &- "- & . $ & . & .1 /. &- & " / % # #, & - . /. % & &- 0 &. $ 0 %&$ / ! $ 0 # , % , " #- , &. - ! - / , &#. # / ! &% , #- ! %" $. & $# & , 0" .%# &# . & $# # / & / 1 "% #! & & 0 #- . '# ! % % 0! . ! / 0// # &%
63
, "- &% $ &"% & . . # 0 , !1- & , & ! &- &. & , 0 /0 ϕ & . . #! tg 2ϕ =
2B . A−C
0 #!. / , & '' A C % - / % # / 0 0 $0 '' B & &- /0 ϕ = π 4 , # # .%# # & ' / x = ( x'− y') 2 y = ( x'+ y ') 2 . . . /. &- / / & 0 & 0 ! "# $ & 0# 0 & % #. ! 0 #- & & , !1 $ , % ! & & 0! % $ , #- "/ % !. , " & % - 0 &. 0/ "# $ # #- &'' & # & % # - 0 /., . . . & . #- " - '' % # &#% . & . "# & .# &"% . , . #- 0 %. . . & . δ=
A B
= AC − B 2 ,
B C ' . % & % $ / 0 &. : δ > 0 - . & 0 &; δ < 0 - . 0& " 0 &; δ = 0 - . &" 0 &. # & 0 0&" 0 & 0 %# . $# # . # ( x0 , y0 ) ! & $ % - 0 /. , . / /
Ax0 + By0 + D = 0 . Bx0 + Cy0 + E = 0
$ / & # . . . δ , / # $ . . . . $ $ & δ ≠ 0 , .. & & ! & / 0&" / &/.
64
. &- 0 &. , & . #- - & ( x1, x2 , x3 ) /
∑ aij xi x j + 2∑ ak 4 xk + a44 = 0 , 3
3
i , j =1
k =1
0 aij = a ji , ! /1 / . . " !# % . . . & $ $0 &. I3 = det (aij ), i, j = 1, 2, 3 .
& $ / ., % / &- 0 0 , # 0 ( x01, x02 , x03 ) & . ., / 0 &. , /
∑ aij x0 j + ai 4 = 0 , 3
i = 1, 2, 3 ,
j =1
$ I3 . . $ &- / ! "# $ %& λ1x12 + λ2 x22 + λ3 x32 + η = 0 ,
0 '' # λi η , & . 1 0 & &- , 0/ "# $ # & & 1 & % - 0 /. &- . , '' # λi . . . . 0" 0 /. $0 &. $ λ det (aij − λ ⋅ δ ij ) = 0 ,
0 i, j = 1, 2, 3, δ ij = 1 & i = j δ ij = 0 & i ≠ j %# . & . # #. / $0 &. %# . - / . # aij , 0 λ1, 2,3 - " # %. . , $ # aij = a ji 0 / , . . . . 1 # . η ! ! "# $ % # & ' / 65
η=
I4 , I3
0 I 4 $ & $ 0 &. , ' # % - '' /. &- , . # #: I 4 = det (aij ), i, j = 1, 2, 3, 4 .
" " #- % - .%$ #- ' ! . / 0 "#.
2& $ & & ! 4 x 2 − 9 y 2 − 8 x − 36 y − 68 = 0 & . . # . %
1.
δ=
A B
=
4
0
= −36 < 0 ,
B C 0 −9 0 & ! 0&" / &/. $ / / / / 0 # &% , & / / / ! & 1$ $ $#- &, .. % # & 0 . & % . / & , & / : 4( x 2 − 2 x + 1) − 9( y 2 + 4 y + 4) − 4 + 36 − 68 = 0 , 4( x − 1) 2 − 9( y + 2) 2 − 36 = 0 .
% . &"% & $0 &
x − 1 = x' , y + 2 = y'
( x' ) 2 ( y ' ) 2 − = 1, 9 4 ! " # a = 3 b = 2 , $ # (1, − 2) ( x, y ) .
66
2. 2 x 2 − 2 3 xy + 9 = 0 . . δ = A B = 2 − 3 = −3 , B C − 3 0 # ! " . # ! , " " x = x' cosϕ − y ' sin ϕ y = y ' sin ϕ + x' cosϕ ! ϕ , ! ! # # tg 2ϕ =
2B 2 ⋅ ( − 3) = = − 3, A−C 2−0
ϕ = − π 6 , " : 3 1 x = x ' + y' 2 2 . y = − 1 x'+ 3 y' 2 2 # # ,
## 3 1 2 3 1 1 3 2 x'+ y ' − 2 3 x'+ y ' − x'+ y ' + 9 = 0 , 2 2 2 2 2 2 # # " " " ! :
3( x' ) 2 − ( y ' )2 + 9 = 0 ,
( x' ) 2 ( y ' ) 2 − = −1, 2 2 ( 3) (3)
# # ! " # a = 3 b = 3 . !" 3. , z = xy # ! " " . #"$"% ". & ! , , xy , ! " # " # 67
(Oxy) . $ x 2 y 2 " ( " ), # # ! ϕ = π 4 , # x = ( x'− y ') 2 y = ( x '+ y ') 2 . ## # , ( x' ) 2 − ( y ' ) 2 = z, 2 "
# # # ! ! " a = b = 1. !" 4. , 5 x 2 + 11y 2 + 2 z 2 − 16 xy + 20 xz + 4 yz = 18 . #"$"% ". ! # I : 3 − 8 10
5 I3 = − 8
11
2 = −1458 = −18 ⋅ 81,
10 2 2 .. # # # $ " λ1x12 + λ2 x22 + λ3 x32 + η = 0 . # $ λ1, 2,3 −8 5−λ 10 3 2 , λ − 18 λ − 81λ + 18 ⋅ 81 = 0 . −8 10
11 − λ 2
2 =0 2−λ
" $
λ1 = 18 , λ2 = 9
λ3 = −9 . η = I 4 I 3 , ! ! # 5 I4 =
− 8 10
− 8 11 10 2 0
0
2 2 0
0 0 , = −18 ⋅ I 3 0 − 18
η = −18
# # # Oz .
18( x' ) 2 + 9( y ' ) 2 − 9( z ' )2 − 18 = 0 , ( y ' )2 ( z ' ) 2 2 ( x' ) + − = 1, 2 2 "
! ! 68
. 10.1. , ! ! " # #$! % ## % & # #$!%' !!(% # : 1)
xy − 2 x − y + 6 = 0 ;
2)
y=
9−x ; x−3
6 x 2 + 6 y 2 + 6 x − 2 y − 1 = 0 ; 4) 9 x 2 − 16 y 2 − 6 x + 8 y − 144 = 0 ; 5) 2 x 2 + 6 xy + 10 y 2 − 121 = 0 ; 6) 9 xy + 4 = 0 . 3)
, $ ## ! *# # . 10.2. ) + # , # &- % ## $# * ! # ( . # # $ ! ! # *#:
x 2 − 8 xy + 17 y 2 + 8 x − 38 y + 24 = 0 ; 2 2) 5 x + xy − 4 x − y − 1 = 0 ; 2 2 3) 8 x − 24 xy + 16 y + 3 x − 7 y − 2 = 0 . 1)
(( - $! - ! 10.3. / & , $ (# 0! *# % ! , - ! ! & # $, - ! ( # . 10.4. 1# ! ! " % ## # #$! (% %:
9 x 2 + 16 y 2 + 36 z 2 − 18 x + 64 y − 216 z + 253 = 0 ; 2 2 2) x + 4 y − 6 x + 16 y + 9 = 0 ; 2 2 2 3) 4 x − 9 y + 36 z − 16 x − 54 y − 72 z − 65 = 0 ; 2 2 2 2 4) z = xy ; 5) 3 x − 4 y + 6 z + 24 x + 8 y − 36 z + 122 = 0 ; 2 2 6) 4 x − 9 y − 8 x − 36 y − 72 z + 184 = 0 ; 2 7) y − 4 x − 6 y + 17 = 0 ; 2 2 2 8) 2 x − 6 y + 3 z − 12 x − 24 y − 24 z + 30 = 0 . 1)
10.5. 1# ! ( . # # :
2 x 2 + 5 y 2 + 2 z 2 − 2 xy + 2 yz − 4 xz + 2 x − 10 y − 2 z − 1 = 0 ; 2 2 2 2) 7 x + 6 y + 5 z − 4 xy − 4 yz − 6 x − 24 y + 18 z + 30 = 0 ; 2 2 2 3) 2 x + 2 y + 3 z + 4 xy + 2 xz + 2 yz − 4 x + 6 y − 2 z + 3 = 0 ; 2 2 2 4) x + 4 y + 9 z − 4 xy + 6 xz − 12 yz − x + 2 y − 3 z − 6 = 0 . 1)
10.6. 1# ! & !( ! (
α: x12 − 2 x22 − 3 x32 − 4 x1x2 − 6 x1 x3 − 2 x1 + 4 x2 + 6 x3 + α = 0 . 69
( , ,
1. /. . 2. 3. 4. 5.
6. 7.
8.
9.
., 2000. . . #, . . ) &# , ! " " , ., 1981. .. ) , ! " " , ., 1968. + .. #$( , , ., 2002. %. . ( , .&. ) $, . . '%- , ( )* + , ., 2003. /.. ,#, ( )* + , ., 1969. + . . -%--, .* + / " + , ., 1970. . . %! , (+ 0 + 12 )*. ! " " " , #!, 1998. ( )* + , * .3. 4# , #!, 1999.
70
1. ) ! & $ # $! ( # ! ! 1.
2. !!(, # #,1 % ## 2- 3-
3. 3
3 9
2. 3. %# #, * # (. 3 # &# 4. # !( ## &# 3.
5. ) ( # # ! !
15 15 21 27 27
6. ) ! ! ! #!
34
7. ) ( # ! ! ! # !
4. 8. !, - -
9. !# #, ! ! 1# ! # #$! % ## ,1 1# ! 10.
,
38 45 45 56 63
70
71
! " #$%$ &'(!) *&!+&%,$$ -./012/ 324205/
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72
