21
Æ
! !"#$%&'()*+, ./0123456...
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21
Æ
! !"#$%&'()*+, ./012345678, 9 :;?9!@"A#BCDE5FG, H$ %&'() -5I(-5J*HB+KL ,FG, 20
-./012 34
56IM, NOP-4QR ST %&J U, V,5 W, & J X, 7YQ Z '[\8 ] ^ 9:; ? JKLMPNOP (CIP)
/
—
2003.9
21 ISBN 7-03-011617-8 ···
···
-
CIP
(2003)
张克忠 /
-
051897
/
4Q f _`abcd_e QRghi ST jUklmnV opWqXrsY tt Zu [ l vwx Zuyz\] \_x abP c\^x `_xdef {+Sgh|ijÆR}k~! l 16
100717
http://www.sciencep.com
2003 2005
9 6
*
B5 (720×1000) 11 1/2 214 000
7 001—9 001
20.00
(
(
)
)
O18
20
Klein
ii
1999 2000
5 2003
§1.1
§1.2 §1.3 §1.4 §1.5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Desargues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
§2.1
§2.2 §2.3 §2.4 §2.5 §2.6
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
§3.1
§3.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
iv
§3.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
§4.1 §4.2 §4.3 §4.4 §4.5 §4.6 §4.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .110 Pascal Brianchon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .116 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .122 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .127 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .135 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .143 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Pappus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Gauss, Riemann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Cantor Poincar´e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
§5.1 Euclid §5.2 §5.3 §5.4 §5.5 §5.6
§5.7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
Desargues !" # " " ! " 1. 1.1 l, l O " l, l $! l l . " O l P OP l P , P P l O ( 1.1). OP . §1.1
! 1.1
P l P l , P P . P P l " l l X, #
!% #" X = l × l l l X. X , . " # l U , OU//l , $ OU l Æ U l U l . $ l ! V . $ ! $ . 2. 1.2 π, π O $! π π . π P , " OP π P , P P π O . OP ( 1.2). 2
1)
! 1.2
P P , P P . $ π × π = x( x π π ), x X , x . " O # π u, U O OU π , U ! u π . $ π ! v . $! $
! !! # $ Æ
%$ &% $ &% 1) &$ (injection) '&$ (surjection) ($&'$ (bijection), %&&&'
' 3 )( !
(i) % (ii) % % " 1.1 (1) $ ! #. (2) $ "& $ "& (3) $ "& ! §1.1
#
' $ ! %. P "& $ ! %. l "& "& "& !. "& % $ "& &% $
& $ "& $ "&
1.3 %"& &; $ "& "& & ! ; $ "& & ! . 1.1 (1) ! (); (2) ! (). )* " #( 1.1 ) %) % "
&& !'( &"#) .
∞
∞
!% #"
4
" (1) 1.3 *$ * $ $$ "& % % $ !$$ (*) $ 1.
! 1.3
(2) $%&*. %+* #Æ #Æ ' $!$) % ( 1.4, C, D A, B. 1.5, C, D A, B.
! 1.4
! 1.5
$ A, B # ÆÆ &
C, D A, B #Æ C, D A, B #Æ $ 2. $$ "&
' 5 " (1) # '+ (Æ'+ , +!!); # '+ ( 1.6). §1.1
! 1.6
! 1.7
# '+ # '+ 1.7, l , l #"#$%Æ ) "#$%'+ $ (2) $ $!) 1.8, )!! π *) O " ) ( ") #" 1
! 1.8
2
!% #" & ) * ) $!) $ $ ( ). $$ $ " " $ $! #&)$$ ( 1.9), %'& ) $ #"& &) "& $ 6
! 1.9 $!!* $$ &)$'!" ,, * !! * $ ! 1.10 * $ A, A; B, B ABAB # '!", ( 1.11), "- M¨obius %
! 1.10
! 1.11
' 7 !+ - (1) '. (1) ,. (, ( ,, ( !. ( !. l & l(A, L (( L(a, B, C, · · · ) ! l(P ). b, c, · · · ) ! L(p). $ $ % %"&
. ( ) (&) $ &% %%&%$( &% & (2) ). (2) ). " (, " (, " !. " !.
"!" " π !" π
- !+, - !+. ' &%# (3) +. (3) +. " " ' ! ! ' ' ! ABC abc !( §1.1
. /
1.1
/ ϕ * π ( π ) (#"(!) f * π +)"*)) π +!) p, q . f $* V , p, q / π +)+%** p , q . ,"+ p //q .
1.
!% #" 2. ,!& (#"() π +)-!0,,+#* π +)!&-0,,) 3. / ϕ * π ( π ) (#"(!) f * π +)"*)( P, Q *!) f + )&+*( R * π ##/ f +)-!*),"+ ∠P RQ / π +)+*+#) 4. .') π +)!,+!) p - O *+# (+#( π +,!) p . ,"+O -.1(!) p "(!&+*) 5. ./ !,"( (#"$))/.!,%*23!. (1) 0//*,/ (2) !*/*,/ (3) -,/ (4) 0,,/ (5) 0/ (6) 1%0)/ (7) %",,/ (8) ,-!)/ (9) ,$!!)/ (10) * *)4) 6. 5 . $0!,/ (#"/2#-) (&-*/%!,) . 8
12&& * 0 $ &%( %3 #++( —— ('3 %"& #+ % 3 #4 ( 3 n-01+, R R n Æ §1.2
n
Rn = {(x1 , x2 , · · · , xn )|xi ∈ R},
0 ∈ R n $ R \ {0} ∼: x, y ∈ R \ {0}. x y $ ∼ %% ρ ∈ R(ρ = 0) x = ρy. ∼ R \ {0} "% n
n
n
n
RP n−1 = (Rn \ {0})/ ∼
(n 2).
RP n $ x ∈ RP RP Æ R x = (x , x , · · · , x ) &Æ n x. x ' " n−1
n−1
1
2
n
n−1
n
'+)'% -1 9 RP '*% , %Æ x = [x , x , · · · , x ]. $ n §1.2
n−1
1
2
n
(RP n−1 )∗ = (Rn \ {0})/ ∼
(n 2),
! R % ) RP (RP ) (
.2/ 1. ( ) 1.4 % P , 3 x, x /x = x( x = 0) *( (x , x ) P .2/; "& P , (x , 0)(x = 0)
3 0.2/. 1.4, "& ( Æ" (x , x ) P ρ = 0, (ρx , ρx ) P ( (0, 1), "& ( (1, 0). 1.4 #+ % 1.4 RP $ ϕ, ϕ # 2. ( ) P l , l ( ! l , l && ), " P & l ∦ l ; ""& P , l //l . , ( n
n−1
n−1 ∗
1
2
1
1
2
2
∞
1
1
1
2
2
1
1
1
2
x=
%
B1
C1
B2
C2
A1
B1
A2
B2
1
1
2
i = 1, 2.
& P 3 P (x, y),
∦ l2 , P C1 C2 , y = A1 A2
B1 x1 = B2
1
∞
li : Ai x + Bi y + Ci = 0,
" l
2
C1 , C2
A1 A2 . B1 B2
C1 x2 = C2
A1 A2
A1 , A2
B1 = 0 ( B2
l
A1 x3 = A2
1
B1 , B2
∦ l2 ).
2
!% #" x = x /x , y = x /x , Æ x : y : 1 = x : x : x . (x, y, 1) *( (x , x , x ) P " l //l , "& P , % P l , l "& x = A B = 0. # & (x , x , 0) *( A B P $ l = l x , x $) x = 0, ! x /x A A A B = = , = = . B B A B A A = 0(l ∦ x )) ! B B = 0(l ∦ y )), ) B B = 0, 10
1
3
2
3
1
1
1
2
∞
1
1
2
2
1
1
2
∞
1
∞
1
3
3
2
3
2
2
2
1
2
1
1
2
2
2
2
2
1
1
1
2
2
1
i
1
i
1
C2 A2 C1 A2 C1 A1 C2 − − x2 A2 C1 − A1 C2 B2 B1 B2 B1 B2 B1 B2 = = =− B2 C1 C2 B1 C2 C1 x1 B1 C2 − B2 C1 − − B1 B2 B1 B2 B1 B2
2
=−
A2 A1 =− . B2 B1
x /x l 1' (x , x , 0) λ = x /x "& (x , x , 0) 0 (1, λ, 0). λ = 0(Æ x = 0) x )" & y )"& (0, x , 0), (x = 0). ( 1.5 % P , "3( (x, y), *( (x , x , x ) .2/, x = 0 % x /x = x, x /x = y. y )"& (x , x , 0), x = 0 % x /x %1' y )"& (0, x , 0), 2
1
1
i
1
2
2
2
1
2
2
1
2
3
3
1
1
2
2
3
2
3
1
2
2
1
%3 0.2/. $ 1.5, "& ( "& ( Æ" (x , x , x ) P ρ = 0, (ρx , ρx , ρx ) P # P P (ρx , ρx , ρx )(ρ = 0), P (x , x , x ). $ (0, 0, 0) (0, 0, 1), x )"& y ) "& (1, 0, 0) (0, 1, 0). % 1.5 #+ "& 1.5 #+ %1' 1.5 RP $ x2 = 0.
1
1
2
1
2
3
3
2
3
1
2
3
2
'+)'% -1 11 ϕ, ϕ # x, y, a, b, · · · (x , x , x ), (y , y , y ), (a , a , a ), (b , b , b ), · · · . 1 1.1 (1) §1.2
1
2
P1 (0, 0),
3
1
P2 (1, 0),
2
3
1
P3 (0, 1),
2
3
P4 2,
1
5 3
2
3
.
3x − 4y + 1 = 0 "& 2 (1)
P (0, 0, x ), x = 0, P (0, 0, 1). P (ρ, 0, ρ), ρ = 0, P (1, 0, 1). P (0, ρ = 0, P (0, 1, 1). ρ, ρ), 5 P 2ρ, ρ, ρ , ρ = 0, P (6, 5, 3). 3 (2) (1') λ = 3/4, "& 3 1, 4 , 0 , ! (4, 3, 0).
.2/34
1.2 (2) 1
3
3
1
2
2
3
3
4
4
3
ui xi = 0.
(1.1)
(1.1) 5 (1.1) .34. 35 % l, 3( i=1
l : Ax + By + C = 0 (A2 + B 2 = 0),
(1)
x = x /x , y = x /x ( 1
3
2
3
x1 x2 +B + C = 0, x3 x3
(2)
Ax1 + Bx2 + Cx3 = 0.
(3)
A
Æ
(3) Æ (1) $ "& (B, −A, 0) 5 (1.1). "& l , * x = 0, Æ (1.1) u = u = 0, u = 0 (1.1) " u + u = 0, $ u x + u y + u = 0 $ "& (u , −u , 0) " u = u = 0, u = 0, Æ x = 0, "& ∞
3
2 1
2
3
3
2
1
1
2
3
2 2
1
1
2
3
12
4
!% #" 1.1
u1 x1 + u2 x2 = 0.
$ & "
.2/ + &% 2 , &%
(,3(), Æ
. ( u1 x1 + u2 x2 + u3 x3 = 0,
% ρ = 0, ρu1 x1 + ρu2 x2 + ρu3 x3 = 0
2(!$ ($ 1. 1.6 l u x + u x + u x = 0. ( l .2/, [u , u , u ]. $ 1.6 !
(1) & $*( [u , u , u ] $*( [u , u , u ], ! u x = 0. ρ = 0, [ρu , ρu , ρu ] ≡ [u , u , u ], % [0, 0, 0] (2) "& u x = 0(u = 0) ! x = 0, [0, 0, u ] ! [0, 0, 1]; u x + u x = 0, [u , u , 0](u , u $); y ) x = 0, [1, 0, 0]; x ) x = 0, [0, 1, 0]. &' '%*% $ (RP ) $ ψ, ψ # 1 1
1
2
2 2
3 3
3
1
1
1
2
3
2
1
2
2 ∗
3
3
1
3 3
1 1
2
2 2
2
3
i=1
i i
3
3
3
3
1
2
1
2
'+)'% -1 13 - 3( 15 65.2/. $ # ψ $ # ϕ ! 1 u,v, · · · 2 [u , u , u ], [v , v , v ], · · · . 2. 1.3 x u ⇐⇒ §1.2
1
3
2
3
1
2
3
ui xi = 0.
(1.2)
,$ " %% ! .34 i=1
1.7
[u1 , u2 , u3 ]
P
P
.
1.4
a = (a1 , a2 , a3 )
a1 u1 + a2 u2 + a3 u3 = 0.
(1.3)
$) [u , u , u ] " 35 1.3, u = [u , u , u ] a = (a , a , a ) (& a u + a u + a u = 0. 1.7, (1.3) a $) " (1.3), a , a , a $" (1.3) # (1.3) a = (a , a , a ). $ 1.3 1.4 x 1
2
3
1
1 1
2 2
2
3
1
2
3
3 3
1
1
2
2
3
3
3
xi ui = 0.
(1.4)
!$ x *# x u ) 5 5 ' u x ) * (1.2) 5 !- $ $" a = (a , a , a ), (1.3) 5 a u ) Æ 1 1.2
(1) (3, 1, 5); (2) (3) x )"& (4) x + 2x + x = 0 "& 2 (1) 3u + u + 5u = 0; i=1
i
i
i
i
1
i
i
1
1
2
3
2
3
2
3
!% #" (2) (0, 0, 1), u = 0; (3) x )"& (1, 0, 0), # u = 0; 1 (4) (1') λ = − , "& 2 (1, λ, 0), Æ (2, −1, 0), 2u − u = 0. % x = 0, x : x , Æ x + 2x + x = 0 "& (2, −1, 0), 0 "& 5 . 0.2/ 1.8 u = [u , u , u ], " u = 0, [U, V ], 14
3
1
1
3
1
1
U=
2
2
3
u1 , u3
2
1
2
3
3
V =
u2 . u3
$ 1.8 [u , u , 0] 1
2
U x + V y + 1 = 0.
(1.5)
"& &
$%Æ"& 40 67 " (x , y ) x U + y V + 1 = 0; " [U , V ] "& U x + V y + 1 = 0. ) "& & $ 2' 1 & + ) ,3( 5(& &, (Æ3() + ) , "+ 3( (Æ, *). 7&6.2/ !8 - 0
0
0
0
0
0
0
0
§1.2
'+)'% -1 1.5 a, b ⇐⇒
/
a1
a2
a3
b1
b2
b3
/
= 1.
35
15
1.5 a 1 b1
a, b ⇐⇒
a2
a3
b2
b3
= 1.
0 1.5. a, b ⇐⇒ a, b RP &% &% ⇐⇒ 1/2 2, a, b ∈ (R \ {0}), # 1/2 1, / 1. 1.6 a, b 1.6 a, b 2
3
a 2 b2
35
x1 a1 b1
x2 a2 b2
a3 a3 , b3 b3
x3 a3 = 0. b3 a1 a1 , b1 b1
a2 . b2
a 2 b2
0 1.6. a, b u x
1 1
u x + u2 x2 + u3 x3 = 0 1 1 u 1 a1 + u 2 a2 + u 3 a3 = 0 u b +u b +u b =0 1 1
2 2
u1 a1 b1
2
a2 b2
a3 a3 , b3 b3
u3 a3 = 0. b3
a1 a1 , b1 b1
a2 . b2
+ u2 x2 + u3 x3 = 0 ⇐⇒
u , u , u $ 1
u2
3
3 3
x1 ⇐⇒ a1 b1
x2 a2 b2
x3 a3 = 0. b3
( $ #- 1.7 a, b, c 1.7 a, b, c
⇐⇒
a1
a2
1
b2
c1
c2
/ b
35
a3
b3 = 2. c3
⇐⇒
a1
a2
1
b2
c1
c2
/ b
a3
b3 = 2. c3
1.6, /2 3 # / 2.
!% #" 1.8 a, b 1.8 a, b & %% (( la + mb(l + m = 0). %% la + mb(l + m = 0). 35 0 1.8. c = la + mb 16
2
2
2
a1 b1 c1
a2 b2 c2
2
a3 b3 = 0. c3
# c a, b & c a, b & a1 b1 c1
a2 b2 c2
a3 b3 = 0. c3
Æ a, b # a, b " c = la + mb, % l + m = 0. $ 1.8, c = la + mb a, b . m 798. " λ = l , 0.798 c = a + λb. x x = ∞(x ∈ R, x = 0), = 0(x ∈ R, x = 0), l = 0 λ = ∞, %2 0 ∞ ∞ = 1. R = R ∪ {∞} R &07:, $ c = a + λb ∞ R $ $ λ = 0 c = a; λ = 1 c = a + b; λ = ∞ c = b. * $ 1.8 $(**( * $ 1.9 a, b, c 1.9 a, b, c ⇐⇒ p, q, r(pqr = 0), ⇐⇒ p, q, r(pqr = 0), 2
2
pa + qb + rc = 0.
pa + qb + rc = 0.
- $ $ -5 2() 0$ 5
'+)'% -1 1 1.3 a = (3, 1, 1), b = (7, 5, 1), c = (6, 4, 1). (1) 30 (2) l, m, c la + mb 5 λ, c = a + λb. 2 (1) . (2) c = la + mb. Æ
§1.2
6
7
3
17
l 1 + m 5 = 4 , 1 1 1
l = 41 , m = 34 , Æ c=
3 1 a + b. 4 4
6# a, b, c c = a + 3b, Æ λ = 3. % 6 3 7 ρ 4 = 1 + λ 5 1 1 1
.
λ = 3. # a, b, c c = a + 3b. 1 1.4 (1) a = (a , a , a ), b = (b , b , b ), c = (c , c , c ) 0
!
ρ,
1
2
3
1
lai + mbi + nci
2
3
1
2
3
(i = 1, 2, 3),
l, m, n $ (2) d = (d , d , d ) % a, b, c, d " 0 $ p, q, r, s, 1
2
3
pai + qbi + rci + sdi = 0
(i = 1, 2, 3).
$26#
(a , a , a ), (b , b , b ), (c , c , c ), (d , d , d ), 1
2
2
3
1
2
3
1
2
3
1
3
35
ai + bi + ci + di = 0
(1)
(i = 1, 2, 3).
d 2 d a, b, c
18
$
!% #" 1. d a, b, c 5% l, m, n
2. d a, b, c ( d & 5 l, m, n $ 3. d "Æ a, b, c, d t a, d b, c 1.8 di = lai + f ti
(i = 1, 2, 3).
d = a, t, lf = 0. ti = gbi + hci
(i = 1, 2, 3),
gh = 0. - 5 di = lai + mbi + nci (2)
(i = 1, 2, 3).
$ (1) pai + qbi + rci + sdi = 0,
a, b, c, d "# pqrs = 0. a c = rc , d = sd , ( i
i
i
i
= pai , bi = qbi ,
i
ai + bi + ci + di = 0
(i = 1, 2, 3)
1.4 )" *
, +61 $ " & ( 3 A , A , A 23 x, y )"& 2( (1, 0, 0), (0, 1, 0), (0, 0, 1), x 1
2
3
x = (x1 , x2 , x3 ) = x1 (1, 0, 0) + x2 (0, 1, 0) + x3 (0, 0, 1).
A , A ! A + 52 x, 7 I, % 1
2
3
I = (1, 1, 1) = (1, 0, 0) + (0, 1, 0) + (0, 0, 1).
'+)'% -1 19 A , A , A , I (A A A |I) 65.2 /*, A A A 2/ +, I ;9. 6 I ' # 1 1.5 A A A 3 §1.2
1
2
3
1
1
2
3
1
2
2
3
3
P1 (p, g, h), P2 (f, q, h), P3 (f, g, r).
0 P P A A P P A A P P A A (2) " f gh = pqr, 0 A P , A P , A P A P , A P , A P 35 A , A , A A (1, 0, 0), A (0, 1, 0), A (0, 0, 1). (1) 1.6, P P (1)
2 3
2
3
3 1
1
1
2
2
2 3
3
1
3 1
3
1 2
1 3
1
2 1
2
1
2
3 2
3
2 3
x1 f f
Æ A A 2
x2 q g
x3 h = 0, r
(qr − gh)x1 + (f h − f r)x2 + (f g − f q)x3 = 0.
3
x1 = 0.
1.6 , P P A A (0, g − q, r − h). P P A A (p − f, 0, h − r). P P A A (f − p, q − g, 0). 30
2 3
3 1
2
3
3
1
0 p−f f −p
1 2
g−q 0
q−g
r−h h − r = 0, 0
# 1.7, x1 p−f f −p
Æ
x2 0 q−g
h − r = 0, 0 x3
x2 x3 x1 + + = 0. p−f q−g r−h
1
2
20
A P hx − qx = 0; A P gx − px = 0. ⇐⇒
(2) A3 P1
1 2
1
2
2
h 0 −r 0 g −p
3
2 3
!% #" f x − rx = 0; 3
1
−q f = 0, 0
Æ f gh = pqr. 0 A P , A P , A P (& f gh = pqr. 1 1.6 ABC ! BC, CA, AB α = 0, β = 0, γ = 0, A, B, C 0 ⇐⇒ qβ−rγ = 0, rγ − pα = 0, pα − qβ = 0, p, q, r 35 ⇐= =⇒ ) ! A qβ − rγ = 0. B 6# p, rγ − pα = 0. C 6# q , pα − q β = 0. %! l, m, n(lmn = 0) 1 3
2 1
3 2
l(qβ − rγ) + m(rγ − pα) + n(pα − q γ) = 0,
Æ
(np − mp)α + (lq − nq )β + (nr − lr)γ = 0.
α = 0, β = 0, γ = 0 Æ q = q .
. /
1.
1.2
,/ -*)'% - 1(34+4324-4!2)
(1)
(2, 0),
(1, 3),
(4, 1),
5 ,2 ; 3
- 34 *.5)635*/ (3) !) 3x + y = 0 +)635*) 2. .',/(,/ -*)/'% - 1) √ √ P (2, 4, −1), Q( 10, − 6, 2), R(0, 1, 0), S(0, 4, 3), T (1, 4, 0), U (1, 0, 4). 3. .'%0-(--544(. (±1, ±1, ±1) (-78/&76)*.4+5,-,+, 7.)/'%-1) 4. ,/ -!))'% - 1) (1) x 3/ (2) y 3/ (3) 635!)/ (4) x − 2x − 3x = 0; (5) 6(6*(8-* 2 )!)/ (6) * (0, 1, 0) . (1, 0, 1) ).)) (2)
1
2
3
'+)'% -1 5. .',/(, 4 . -!))/'%) - 1) 6. ,/ -!)+)635*)
§1.2
(1)
7.
x1 + x2 − 4x3 = 0;
(2)
x1 + 2x2 = 0;
x2 − 3x3 = 0;
(4)
x1 + 5x3 = 0.
,/ -)-1478)!)..)
(3)
21
(1)
[0, 1, 1];
(2)
[1, 1, −1];
(3)
[1, 0, 1];
(4)
[1, −1, 0];
(5)
[2, −1, 5];
(6)
[0, 0, 1].
8. (1)
u1 = 0;
(2)
u3 = 0;
(3)
u2 − u3 = 0;
(4)
2u1 + u2 = 0;
(5)
u1 + u2 + 2u3 = 0;
(6)
u21 − 5u1 u2 + 4u22 = 0;
u21 − 3u1 u2 + 2u23 = 0;
(8)
au1 + bu2 = 0.
/ )-1..-783!!,.
,/ -*)'%) -1..) (1) x 3+)635*/ (2) y 3+)635*/ 1 )!)+)635*/ (3) 8-* − 2 (7)
9.
(4)
(0, 0, 1);
(5)
(2, 4, −3).
.',/(,+. -*)/'%)- 1..) 11. 9,- 2x + x + x = 0, 3x − 4x + 2x = 0, 4x + x − 3x = 0 *,)/*,) 8*-1) 12. 9,- A(1, 2, 3), B(2, 2, 1), C(3, 4, 3) *8*)/),)/,..) 13. ,/ !))0* - 1...) (1) x + 2x − 4x = 0 . 2x − x + x = 0; (2) x − 2x = 0 . 4x − 5x + x = 0. 14. 74/* P (1, 4, −3), P (0, 2, 5), P (3, 8, −19). (1) "-+ P , P , P /*$)/ (2) ,4/!)../ (3) 44 P )'% - 1:, P = P + λP . 15. //*, A B C . A B C //!#( B C . B C 0 X, C A . C A 0 Y , A B . A B 0 Z, 95 X, Y, Z /*$)),"+/!) A A , B B , C C $ *) (78+//*, A B C . A B C ), B C , C A , A B . B C , C A , A B ) '%..8%* α = 0,β = 0,γ = 0; . α = 0,β = 0,γ = 0, 4/ X, Y, Z 4/!)'% ..* δ = 0, 6)07814;)) 16. / O(f, g, h), A A A * - 1/*,( A O, A O, A O %*. A A , A A , A A 0 P, Q, R; QR, RP, P Q %*. A A , A A , A A 0 L, M, N . ,"+ L, M, N /*$ )(-,+/!))..) 10.
1
1
2
1
2
2
3
1
3
1
1
1
1
2
2
3
2
1
1
2
2
1
2
3
3
3
3
1
3
3
2
3
1
2
1
2
2
1
2
2
1
1
2
2
1
1
1
1
1
2
1
1
2
2
1
1
2
1
2
2
3
3
3
1
1
1
1
1
2
2
1
1
1
2
2
2
2
1
2
2
2
2
2
1
2
1
2
2
2
3
2
3
3
1
1
2
!% #" 17. / O *+!) x − kx = 0 +).*( A A A *:1)/=)D()7@#")-94;(?/83C# "+ ) 0 3 :: &0)
J7@A ) ) #+ Æ
u x = 0 ! F5 &% 5F) A
A
A
A
A
A
3
i=1
i i
Aξ1 + Bξ2 + Cξ3 = 0.
(1)
/86< 35
(ξ , ξ , ξ )
[ξ , ξ , ξ ] ) (1) ) (1) [A, B, C] (A, B, C) ( §1.4
1
2
3
1
2
3
A1 ξ1 + B1 ξ2 + C1 ξ3 = 0,
(2)
A ξ + B ξ + C ξ = 0. 2 1 2 2 2 3
( (2) ( (2) (A, B, C). [A, B, C]. - &% . ! $ ) (1) (A, B, C) (1) [A, B, C] Au + Bu + Cu = 0. Ax + Bx + Cx = 0. (2) (0, 0, 1) (2) "& [0, 0, 1] u = 0. x = 0. (3) "& (A, B, 0) (3) [A, B, 0] Au + Bu = 0. Ax + Bx = 0. (4)∼(8) 1.5∼ 1.9. (4) ∼ (8) 1.5 ∼ 1.9 . 1)
1
2
3
1
2
3
3
3
1
1.
D+/
2
1
. /
!,)/8!,)
! 1.19
2
1.4
! 1.20
G48CDG;H?8& RP H (RP ) >@&>9>($IH8? RP H (RP ) > @:5AI