x0x1x2x3
x0x1x2 x3x4x5x6
x1 x2 1–1
1,x8x9,xaxbxcxdxexfx10ABCD?A1B1C1D1,Ex0 F x11x12x13x14BCx0 C1D1x15x16x17,x18?!AB =?!a,
!AD =?!b,!AA
1 =?!c, x19x1a?!a,
!b,?!c x1bx1cx1dx1ex1fx20:
(1)!AC1; (2)!BD1; (3)?!AF; (4)?!EF.
x5,(1)x21x22
!BC =?!AD;!CC
1 =
!AA
1;
!AC
1 =
!AB +?!BC +!CC
1;
x23x24
!AC
1 =?!a +
!b +?!c,
(2)x21x22!BD1 =?!BD +!DD1,x25
!BD =?!AD!AB =?!b!a ;!DD
1 =
!AA
1:
x23x24
!BD
1 =
!b!a +?!c,
(3)?!AF =?!AD +!DD1 +!D1F,x25
!DD
1 =
!AA
1;
!D
1F =
1
2
!D
1C1 =
1
2
!AB;
x23x24
!AF = 1
2
!a +?!b +?!c,
(4)?!EF =?!AF!AE =?!AF?(?!AB +?!BE) =?!AF?(?!AB +?!BE) =?!AF?
!
AB + 12?!BC
·
= 12?!a +
!b +?!c!a? 1
2
!b =? 1
2
!a + 1
2
!b +?!c,
x75x75x75x8x1cx1dx1c
x1dx1cx1d
x17x3x8x75x75x75
x3x17x1dx1c
x1dx1cx1d
x1c
x1x13xcxc
xcxcxc
xcxcxc
xcxcxc
x75x75x75x8x1cx1dx1c
x1dx1cx1d
x17x3xcxc
xcxcxc
xcxcxc
xcxcxc
x13x1
xcxcxc
xcxcxc
xcxcxc
xcxcx13x1
x7x1fx3ax9x21x46
x1x12x6x2x3x13
x3x2x14x5x27
x76x2exfx0x10x2 x7dx2fxfx1x12x8x3x2x14x5x27 x3x1cx1cx2e
x3ex17 x3ex17
x3fx17 x3ex17
x17x3
x31x11x31x10
x31x11x31x10
x31x11x31x10
x31x11x31x10
x31x11x31x10x3fx17x17x3fx18x3e
x17x3fx18x3e
x17x3fx18x3e
x3fx13 x36x13
x36x13 x36x13
x36x13 x36x13
x36x13 x36x13
x6ex2fx6ex2f
x6ex2fx6ex2f
x6fx2fx6ex2f
x6ex2fx6ex2f
x6fx2fx6ex2f
x6ex2fx6ex2f
x43x1a x43x1a
x43x19 x43x1a
x43x1a x44x19
x43x19 x43x1a
x43x1a x43x19
x43x1a x43x1a
x44x19 x43x19
x43x1a
x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x8
A B
D C
A1 B1
D1 C
1
F
E
!a
!c?!
b
x01x1x2
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6exex75x75x75x75x75x75x75x75x28x28x28x28x28x28
x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x44x1ax3fx1ex2fx30
x7x3bx3ax9x45x46
x67x63xdx76x75
!b!a?!b
!a
x03(1)x1x2
2,x26x27x1dx1ex28x29x2ax2b,x1fx20?!a ;?!b x2cx2dx2ex2fx30x31x32?
(1) j?!a +?!b j = j?!a j+j?!b j;
(2) j?!a +?!b j = j?!a j?j?!b j;
(3) j?!a!b j = j?!a j?j?!b j;
(4) j?!a!b j = j?!a j+j?!b j.
x5,(1) x33x1a“x34x35x36x37x38x39x3ax3bx3cx3dx34x38”x3exb,?!a ==?!b, x3fx26x27j?!a +?!b j = j?!a j+j?!b jx40x41,?!a
x42?!b x43x1f,x44?!a,?!b x16x45x46x47x48x220.
￠ 1 ￠
(2)x49?!c =?!a +?!b,x4a?!a =?!c!b,x4bx29x4cx22,j?!c!b j = j?!c j+j?!b j,x23x24?!b ==?!c x3fx4dx1f,x4e
x4fx3ex50,?!a ==?!b,x4dx1f,x3fj?!a j>j?!b j,x44?!b = 0.
(3)x49?!c =?!a!b,x4a?!a =?!b +?!c,x4bx29x4cx22,j?!b j+j?!c j = j?!b +?!c j,x4e(1)xb,?!b ==?!c x3fx43x1f.
x23x24?!a ==?!b x3fx43x1f,x51x21j?!a!b j> 0,x23x24j?!a j>j?!b j,x44?!b = 0.
(4)x49?!c =?!a!b,x4a?!a =?!b +?!c,x4bx29x4cx22,j?!b +?!c j = j?!c j?j?!b j,x4e(2)xb,?!c ==?!b x3fx4dx1f,
x44?!b = 0,x43x52,j?!c j>j?!b j,x23x24?!a ==?!b x3fx4dx1f,x44?!b = 0x44?!a = 0.
3,x53x54x1dx1ex55x56x29,x57x58x54x56x59x2fx30x52x5ax2ax2b.
(1) j?!b!a j>j?!a j?j?!b j;
(2) j?!a +?!b +?!c j6j?!a j+j?!b j+j?!c j.
x3x4,(1)x8x9,x33x1a“x34x35x36x37x38x39x5bx5cx3cx3dx34x38”x3ex50x5dx53x15x55x56x29.x56x29x2ax2bx15x31x32x3ex5ex5fx60
x612(3),?!a ==?!b,x43x1f,x3fj?!a j>j?!b j,x44?!b = 0.
(2)x49?!d =?!b +?!c, x4a,j?!a +?!b +?!c j = j?!a +?!d j6j?!a j+j?!d j = j?!a j+j?!b +?!c j6j?!a j+j?!b j+j?!c j.
x56x59x2ax2bx62x3fx63x62(i)?!a,?!b,?!c x64x65xcxdx3fx43x1f,x44(ii)?!a,?!b,?!c x16x45x46x37x66x220(x67x3ex68x2a(i)x15x69
x6a).
4,x6bx6cx38x36ABCDx16,?!AB =?!a + 2?!b,?!BC =?4?!a!b,?!CD =?5?!a?3?!b (?!a,?!b x6dx13x6ex6f
x1fx20),x53x54ABCDx22x70x36.
x3x4,x6?!AD =?!AB +?!BC +?!CD =?8?!a? 2?!b = 2?!BC,x7?!AD==?!BC,x71j?!ADj = 2j?!BCj,x23x24
ABCDx13x70x36.
5,x18ABCDEFx22x72xex38x36,x73?!AB +?!AC +?!AD +?!AE +?!AF.
x5,x6?!AD =?!AC +?!AF =?!AE +?!AB,x7?!AB +?!AC +?!AD +?!AE +?!AF = 3?!AD.
x12x13x12
x13x12x13
x12x13x12
x13x12x13
x12x13x12
x15x2x75x75x75x75x40x50x20x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x2bxe
x2x15x12x13
x12x13x12
x13x12x13
x12x13x12
x13x12x13
x12
x20x50x40x75x75x75x75
xex2bx6ex6d
x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x6dx6ex6d
x6e
x75x75x75x75x75x75x75x75x72x40x50x8x6x46x2ex2dx2ex2d
x2ex2dx2ex2e
x2dx2ex2dx2e
x2dx2ex2dx2e
x2dx2ex2dx2e
x2dx2ex2dx2e
x2dx2e
x52x53x52x53
x52x53x52x53
x52x53x52x53
x52x53x52x53
x52x53x52x52
x53x52x53x52
x53x52x5axa
xbxcx4x3dx1b
xcx63x65x75x74
x4x43x25x35x34
x4bx4cxcx5dx5b
x7x3bx3ax9x45x46A D
F
B
E
F
x05x1x2
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6exex75x75x75x75x75x75x75x75x28x28x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28
x28x44x1a
x2xcxdxd
xcxdxd
xcxdxd
xcxdxd
xcxdxd
xcxd
x36x36x36x36x36x36x36x36
x36x36x36x36x36x36x36x36
x36x36x36x37x36x36x36x36
x36x36x36x1dx7xax56x4dx4ex4dx4dx4dx4ex4dx4d
x4dx4ex4dx4dx4dx4e
x4dx4dx4dx4ex4dx4d
x6x6x1x13x3ax15x2
x2fx30x8x3cx3cx47x48x8x55x5ax25
B L C
N M
A
x06x1x2
6,x18L;M;Nx11x12x13x34x35x36ABCx15x34x38BC;CA;ABx15x16x17,x53x54x34x16x74x1fx20?!AL,!BM,!CNx3e
x24x75x2ax48x66x34x35x36.
x3x4,x21x22?!AL;!BM;!CNx3ex24x75x2ax48x66x34x35x36,x62x3fx63x62x76x77x34x66x1fx20x39x3ax22x6fx1fx20,x4e
!AL = 1
2 (
!AB +?!AC);!BM = 1
2 (
!BA+?!BC);!CN = 1
2 (
!CA+?!CB);
x3ex50,?!AL+!BM +!CN = 0.
7,x6bx34x35x36ABCx16x73x48x17O,x27
!OA+?!OB +?!OC = 0:
￠ 2 ￠
x5,x18Lx13BCx15x16x17,x4a?!AL = 12 (?!AB+?!AC),x6bx74x78ALx79x7aOx17,x27?!AO = 23?!AL = 13 (?!AB+?!AC),
x4aOx17x6b4ABCx15x7bx7c,x7dx53:
!OA+?!OB +?!OC =?!OA+(?!OA+?!AB)+(?!OA+?!AC) = 3?!OA+?!AB +?!AC
=?(?!AB +?!AC)+?!AB +?!AC = 0:
8,x18A;B;C;Dx13x48x66x6cxfx10x15x6cx66x7ex17,M;Nx11x12x13x38AB,CDx15x16x17,x53x54:
!MN = 1
2 (
!AD +?!BC):
x3x4,x8x9,
!CM = 1
2 (
!CA+?!CB);!CN = 1
2
!CD;
x23x24
!MN =!CN!CM = 1
2
!CD? 1
2 (
!CA+?!CB) = 1
2 (
!AD +?!BC):
x1ax28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x73x73
x73x73
x73x60
xcx1dx1dx1d
x1dx1dx1e
x1dx1dx1d
x1dx1dx1d
x1dx1dx1d
x3fx4
xax58x51x51x52x51
x51x51x51x51x51
x51
x1x11x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5
x5
x5x51x50x61x62
x5cx26 x5cx26
x5cx26 x5dx26
x5cx27 x5cx26
x5cx26 xbx4dx1e x4dx1f x4d
x1e x4dx1e x4cx1f x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f
x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f x4dx1e x4dx1f
B
D
C
A
M
N
x08x1x2
x30x31x30x30x31
x30x30x31x30x31
x30x30x31x30x30
x31x30x30x31x30
x30x31x30x31x30
x30x31x30x30x31
x30x30
x2x37xexf
xexfxf
xexfxe
xfxexf
xexfxe
xf
x8x20x50x75x75x75x75x75x75xfxexf
xexfxe
xfxexf
xexfxf
xexfxe
x37x2x75x75x75x75x75x75x50x20x8x27x5ex5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5c
x5
A
C
B
D
M
O
x09x1x2
9,x18Mx13xcxdx6cx38x36ABCDx15x16x7f,Ox13x0x1x48x17,x53x54:
!OA+?!OB +?!OC +?!OD = 4!OM:
x3x4,x8x9,x21x22
!OM = 1
2 (
!OA+?!OC);!OM = 1
2 (
!OB +?!OD);
x23x24
!OA+?!OB +?!OC +?!OD = 4!OM:
10,x18ABCDx13xcxdx6cx38x36,P;Qx11x12x13x38BC;CDx15x16x17,x53x54AP;AQx42x2x35x74BDx65x3x3c
E;F,x25x76BDx34x56x11.
x3x4,x18?!AB =?!a,?!AD =?!b,x4a
!BD =?!AD!AB =?!b!a ;
!AP =?!AB + 1
2
!BC =?!a + 1
2
!b ;
!AQ =?!AD + 1
2
!DC =?!b + 1
2
!a,
x51x18
!AE = k?!AP (k > 0);?!AF = m?!AQ (m > 0);
x4a
!AE = k?!a + k
2
!b ;?!AF = m?!b + m
2
!a,
x71x13
!AE =?!AB +t?!BD =?!a +t(?!b!a ) = (1?t)?!a +t?!b (t > 0):
￠ 3 ￠
x23x24
k?!a + k2?!b = (1?t)?!a +t?!b ;
x4:
(k +t?1)?!a =
t? k2
!
b ;
x4ex3c?!a x42?!b x55xcxd,x23x24 (
k +t?1 = 0
t? k2 = 0 x4
( k = 2
3
t = 13,
x43x5,x4e
!AF =?!AB +s?!BD = (1?s)?!b +s?!b (s > 0);
x3ex50,( m
2 +s?1 = 0
s?m = 0; x4:
( m = 2
3
s = 23,
x6x7x50x8:
!BF = 2
3
!BD;?!BE = 1
3
!BD;
x58x54E;Fx13x74x78BDx15x34x56x11x17.
11,Ox22x72x9x38x36A1A2￠￠￠Anx15x16x7f,x53x54:
!OA
1 +
!OA
2 +￠￠￠+
!OA
n = 0:
x3x4,xaxbxcnx22xdxex15xfx36,x4fx52.x10x11x47,!OA1 +￠￠￠+!OAn = 0,x12x68nx22x13xex15xfx36,x14x15x16
x17x48x18x7ex17B1;￠￠￠ ;Bnx27x4bx19x72nx38x36A1￠￠￠Anx2ax22,A1B1A2B2￠￠￠An?1Bn?1AnBn,x77x13x48x662nx38
x36,x23x24
!OA
1 +
!OB
1 +
!OA
2 +
!OB
2 +￠￠￠+
!OA
n +
!OB
n = 0:
x1ax1x8!OBix13x4e!OAix1bx1cx48x66x1dx35 x25x50x8(0 < < …),x1ex1f:
!p =!OA1 +￠￠￠+!OAn;
!q =!OB1 +￠￠￠+!OBn;
x20x30?!q x13x4e?!p x1bx1c x35x25x50x8,x4ex3c0 < < …,?!q x42?!p x55xcxd,x21?!p +?!q = 0x62x3fx63x62?!p =?!q = 0.
12,Ox22x72x9x38x36A1A2￠￠￠Anx15x16x7f,Px13x0x1x48x17,x53x54:
!PA
1 +
!PA
2 +￠￠￠+
!PA
n = n
!PO:
x3x4,x21x22
!PO =!PA
i +
!A
iO (i = 1;2;￠￠￠ ;n);
x23x24
n?!PO =!PA1 +￠￠￠+!PAn +(!A1O +￠￠￠+!AnO) =!PA1 +￠￠￠+!PAn
(x33x1ax3d11x61x15x22x23).
x1 x2 1–2
1,xaxbxcxdx6cx38x36ABCDx15x2x35x74x22ACx3aBD,x18?!AC =?!a,?!BD =?!b, x73?!AB,?!CD,?!DA.
x5,x8x9,
!AB =?!AO +?!OB = 1
2
!AC + 1
2
!DB = 1
2 (
!a!b );
￠ 4 ￠
!CD =!AB =? 1
2 (
!a!b );
!DA =!AD =?(?!AO +?!OD) =? 1
2 (
!a +?!b ):
x30x31x30x30x31
x30x30x31x30x31
x30x30x31x30x30
x31x30x30x31x30
x30x31x30x31x30
x30x31x30x30x31
x30x30
x2x37xexf
xexfxf
xexfxe
xfxexf
xexfxe
xf
x8x20x50x75x75x75x75x75x75xfxexf
xexfxe
xfxexf
xexfxf
xexfxe
x37x2x75x75x75x75x75x75x50x20x8x27x5ex5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5c
x1ax44x29x37x37x53x5ax25xax66x63xd
A
C
B
D
O
x01x1x2
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6exex75x75x75x75x75x75x75x75x28x28x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28
x28x28x44x1a
x17x16x17
x17x16x17
x17x16x17
x17x16x17
x17x16x17
x17x16x3
x1x2x3x3
x2x3x3
x2x3x3
x2x3x3
x2x3x3
x2x3
A BNM
C
x02x1x2
2,x6bx34x35x36ABCx16,x17M;Nx13ABx38x79x15x34x56x11x17,x18?!CA =?!a,?!CB =?!b, x73!CM,!CN.
x5,x8x9,x21x22
!AM = 1
3
!AB;?!AN = 2
3
!AB;
x23x24
!CM =?!CA+!AM =?!CA+ 1
3
!AB =?!CA+ 1
3 (
!CB!CA) = 1
3
!b + 2
3
!a ;
!CN =?!CA+?!AN =?!CA+ 2
3
!AB =?!CA+ 2
3 (
!CB!CA) = 2
3
!b + 1
3
!a,
3,x6bx6cxfx10O?ABCx16,x18x17Gx13x34x35x36ABCx15x24x7f,x1a?!OA,?!OB,?!OCx19x1bx1cx1fx20?!OG.
x5,x8x9,x4e
!OG =?!OA+?!AG;?!AG = 2
3
!AD;
!AD = 1
2 (
!AB +?!AC);
x3ex50?!AG = 13 (?!AB +?!AC),x25?!AB =?!OB!OA,?!AC =?!OC!OA,x23x24
!AG = 1
3 (
!OB +?!OC?2?!OA);?!OG = 1
3 (
!OA+?!OB +?!OC):
x1x10x2x2
x3x2x2
x2x2x2
x2x3x2
x2x2x2
x2x2x2
x3x2x2
x2
x25x59x51x52x52x52
x52x51x52x52x52
x51x52x52
x71x71
x71x71
x71x71
x71x18x6cx6cx6cx6cx6c
x6cx6cx6c
x6cx6cx6c
x2bxd
x1bx1cx1b
x1bx1cx1b
x1cx1bx1c
x1b
x70x0xcxb
x1fx4ex1fx4dx1fx4ex1fx4ex1fx4ex1fx4ex1fx4dx1fx4ex1fx4ex1fx4ex1fx4ex1fx4d
x1fx4ex1fx4ex8x1fx21x9 x51x21 x53x20 x51x22 x51x22 x51x22 x51x22 x51x22 x51x22 x51x22 x51x22
x51x22
x11x33x11x33
x12x33x12x33
x11x33x11x33
x12x33x11x33
x11x33x12x32
x1x33
B
D
C
G
O
A
x03x1x2
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6exex75x75x75x75x75x75x75x75x28x28x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x44x1a
x35x36x35x35x36x35x35
x36x35x35x36x35x35x36
x35x35x36x35x35x36x35
x35x36x35x35x36x35x7x6x1cx47x2fx7x3cx3b
xfx2ex6fx66x69
A
T
C
B
x04x1x2
4,x18ATx13x34x35x36ABCx16\Ax15xcx11x74(x42BCx3x3cTx17),x76?!ATx1a?!AB;?!ACx19x1bx25.
x5,x18?!BT = k?!BC,x4a?!TC = (1?k)?!BC,x4ex35xcx11x74x15x26x27x3exb,j?!ABj, j?!ACj = k, (1?k),x21x4f
k = j
!ABj
j?!ABj+j?!ACj, x3cx13
!AT =?!AB +?!BT =?!AB +k?!BC = (1?k)?!AB +k?!AC
= 1
j?!ABj+j?!ACj
(j?!ACj?!AB +j?!ABj?!AC):
5,xaxb?!a ;?!b x55x28x74,x4ax1fx20?!c = 3?!a +?!b x42?!d = 2?!a!b x13x29x74x26x65x2a?
￠ 5 ￠
x5,x18x47kx0 mx27,k?!c +m?!d = 0,x4
3k?!a +k?!b +2m?!a?m?!b = 0;
x2bx5x7x22
(3k +2m)?!a +(k?m)?!b = 0:
x4ex3c?!a ;?!b x55x28x74,x21?!a ;?!b x74x26x2cx2a,x23x24(
3k +2m = 0
k?m = 0 x2dx50k = m = 0;
x4?!c ;?!d x74x26x2cx2a.
6,x53x54x34x66x1fx20k1?!a?k2?!b,k2?!b?k3?!c,k3?!c?k1?!a x28xf.
x3x4,x4ex56x29
(k1?!a?k2?!b )+(k2?!b?k3?!c )+(k3?!c?k1?!a ) = 0;
x3exbx773x66x1fx20x74x26x65x2a,x23x24x28xf.
7,Ox13x48x66x1dx17,x53x54:x2x3cx55x6bx48x2ex74x79x153x66x17A;B;C,x17Mx2fx3cxcxfABCx79x15x30x11x40x26
x31x32x13x31x6bx32xek1;k2;k3,x27x50
!OM = k
1
!OA+k
2
!OB +k
3
!OC; x3fk
1 +k2 +k3 = 1.
x3x4,xaxbAx0 Bx0 C x34x17x55x28x74,x21?!AB;?!AC x74x26x2cx2a,x0x1x17M x2fx3cxcxfABC x79x62x3fx63x62
!AM;?!AB;?!ACx28xf,x4,!AM;?!AB;?!ACx74x26x65x2a,x62x3fx63x62x31x6bx55x33x220x15x32xem
1;m2;m3,x27
m1!AM +m2?!AB +m3?!AC = 0;
x62x3fx63x62x2x3cx1dx17Ox47:
m1(!OM!OA)+m2(?!OB!OA)+m3(?!OC!OA) = 0;
x62x3fx63x62
m1!OM = (m1 +m2 +m3)?!OA?m2?!OB?m3?!OC:
x10x11m1 6= 0,x55x11x42?!AB;?!ACx74x26x2cx2ax34x35,x21x4fx1ex1f:
k1 = 1m
1
(m1 +m2 +m3); k2 =? m2m
1;
k3 =? m3m
1;
x4a
!OM = k
1
!OA+k
2
!OB +k
3
!OC;
x3fk1 +k2 +k3 = 1.
8,O x13x48x66x1dx17,x53x54,x17 M x2fx3c4ABC x79 (x36x37x38x15x38) x15x30x11x40x26x31x32x13x31x6bx6ex39x32xe
k1;k2;k3,x27x50
!OM = k
1
!OA+k
2
!OB +k
3
!OC; x3fk
1 +k2 +k3 = 1.
x3x4,x3ax3bAM,x40x3ex3BCx3cDx17,x21x4f!AM = l?!AD,x3cx160 6 l 6 1,x4ex3cDx6bx74x78BCx79,x3dx3e
x6a2.1,x31x6bx32xem1;m2,x27x50
!OD = m
1
!OB +m
2
!OC; m
1 +m2 = 1;m1;m2 > 0:
x3cx13
!OM =?!OA+!AM = (1?l)?!OA+l?!OD = (1?l)?!OA+lm
1
!OB +lm
2
!OC:
￠ 6 ￠
x49k1 = 1?l,k2 = lm1,k3 = lm2,x4x50
!OM = k
1
!OA+k
2
!OB +k
3
!OC; k
1 +k2 +k3 = 1; k1;k2;k3 > 0:
x4dx39,x55x3fx18k1 6= 1,x2dx40x41x42
8>
<
>:
1?l = k1
lm1 = k2
lm2 = k3
x3ex50
8>
><
>>:
l = 1?k1;
m1 = k21?k
1;
m2 = k31?k
1;
x4ax47
m1 +m2 = 1; m1;m2 > 0; 0 < l 6 1:
x49
!OD = m
1
!OB +m
2
!OC;
x4aDx17x6bx74x78BCx79,x4e
!OM = (1?l)?!OA+l?!OD
x3ex24x50x25!AM = l?!AD,x21x4fMx6bx74x78ADx79,x43x25x6b4ABCx79.
9,x53x54:x0x1x55x43x15x34x17A;B;Cx28x74x15x30x11x40x26x31x32x13x31x6bx55x33x22x6fx15x32xek1;k2;k3,x27x50
0 = k1?!OA+k2?!OB +k3?!OC; x3fk1 +k2 +k3 = 0:
x3x4,Ax0 Bx0 Cx28x74,x62x3fx63x62l?!AB +m?!AC = 0 (l;mx6dx55x22x6f),x62x3fx63x62
l(?!OB!OA)+m(?!OC!OA) = 0;
x62x3fx63x62
(l +m)?!OA+l?!OB +m?!OC = 0:
x49k1 =?(l +m),k2 = l,k3 = m,x10x11x38x15x55x33x22x6f,x3f:
k1?!OA+k2?!OB +k3?!OC = 0; k1 +k2 +k3 = 0:
10,x53x54:x0x1x55x43x15x6cx17A;B;C;Dx28xfx15x30x11x40x26x31x32x13x31x6bx6cx66x55x33x22x6fx15x32xe,x27x50
0 = k1?!OA+k2?!OB +k3?!OC +k4?!OD; x3fk1 +k2 +k3 +k4 = 0:
x3x4,Ax0 Bx0 Cx0 Dx28xfx62x3fx63x62?!AB;?!AC;?!ADx74x26x65x2a,x62x3fx63x62x47x55x33x22x6fx15xel;m;nx27:
l?!AB +m?!AC +n?!AD = 0;
x62x3fx63x62
l(?!OB!OA)+m(?!OC!OA)+n(?!OD!OA) = 0;
x62x3fx63x62
(l +m+n)?!OA+l?!OB +m?!OC +n?!OD = 0:
x1fk1 =?(l +m+n),k2 = l,k3 = m,k4 = n,x10x11x38x15x55x33x22x6f,x27x50
k1?!OA+k2?!OB +k3?!OC +k4?!OD = 0; k1 +k2 +k3 +k4 = 0:
￠ 7 ￠
x13x12x13
x13x12x13
x13x12x13
x13x12x13
x13x12x13
x13x12x3x6fx6ex6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
xex75x75x75x75x75x75x75x75x28x28x28x28
x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x44
x1a
x35x34x35x35x35x34x35
x35x35x34x35x35x35x34
x35x35x35x34x35x35x35
x34x35x35x35x34x4bx1dx6xax9x56x4bx4cx4bx4cx4bx4bx4c
x4bx4cx4bx4cx4bx4cx4b
x4bx4cx4bx4cx4bx4cx4b
A F B
C
DE
M
x011x1x2
11,x1ax1fx20x15x40x44x53x54x45x46x1dx5,x1ex34x35x36ABCx15x34x31x38AB,BC,CAx47x48x49x11x4ax2aAF, FB =
k1, k2,BD, DC = k3, k1,CE, EA = k2, k3,x3cx16,k1;k2:k3x4bx22x72xe,x4ax34x35x36ABCx15x7ex17x42x38x2
x38x15x11x17x15x4cx74x3x3cx48x17M,x3fx2x3cx0x1x48x17Ox47
!OM = 1
k1 +k2 +k3(k2
!OA+k
1
!OB +k
3
!OC):
x3x4,x3dx3ex11x17Dx42Ex15x1dx4dx3ex50
!BD = k3
k1 +k3
!BC;?!AE = k3
k2 +k3
!AC:
x3cx13
!AD =?!AB +?!BD =?!AB + k3
k1 +k3
!BC =?!AB + k3
k1 +k3 (
!AC!AB)
= k1k
1 +k3
!AB + k3
k1 +k3
!AC;
!BE =?!AE!AB = k3
k2 +k3
!AC!AB:
x18ADx42BEx3x3cM,x4ax47
!AM = l?!AD;!BM = m?!BE:
x4ex4fxfx50x8x15x1bx50x29x51x52x24x1dx56x29,!AM =?!AB +!BM,x50x8
l
k
1
k1 +k3
!AB + k3
k1 +k3
!AC? =?!AB +m k3
k2 +k3
!AC!AB?:
x4ex3c?!ABx42?!ACx74x26x2cx2a,x4ex79x53x56x29x50x8x40x41x42:8
<
:
lk3
k1 +k3 =
mk3
k2 +k3
lk1
k1 +k3 = 1?m
x2dx50
8<
:
l = k1 +k3k
1 +k2 +k3
m = k2 +k3k
1 +k2 +k3
:
x4
!AM = k1 +k3
k1 +k2 +k3
!AD:
x51x18ADx42CFx65x3x3cM0,x43x5x3ex50
!AM0 = k1 +k3
k1 +k2 +k3
!AD;
x4Mx42M0x24x54,x21x4fAD;BE;CFx3x3cx43x48x17M.
x2x0x1x17O,x47
!OM =?!OA+!AM =?!OA+ k1 +k3
k1 +k2 +k3
k
2
k1 +k3
!AB + k3
k1 +k3
!AC?
=?!OA+ k1k
1 +k2 +k3
(?!OB!OA)+ k3k
1 +k2 +k3
(?!OC!OA)
= 1k
1 +k2 +k3
(k2?!OA+k1?!OB +k3?!OC):
￠ 8 ￠
x1 x2 1–3
1,x18P,Qx37x17x6bx55x56[O;?!e1;?!e2;?!e3]x1dx15x57x55x11x12x13(2;2;1),(?1;?1;3),x19x58x25P;Qx17x15x2fx59.
x5,x5fx5ax9.
x2ex2ex2fx2ex2e
x2ex1cx6x75x40
x2ex6cx78x78
x78x78x78
x5
x5
x26x43x5x35x35
x7x3bx3ax9x45x46
x72x72x7ex7exf
x6x1cx2ex2ex2ex2f
x2ex2e
x75x75x72
x2ex6cx78x78
x78x78x78
x47x1c x47x1c x47x1b
x47x1c x1cx6x8x20x20x50x20x50x20x50
x2ex6cx2ex6c
x2ex6cx2ex6c
x2ex6cx2ex6c
x2ex6cx2ex6c
x2ex6cx2ex6c
x2ex6c
O
!e1
!e2
!e3
P(2;2;1)
Q(?1;?1;3)
x01x1x2
x30x31x30x30x31
x30x30x31x30x31
x30x30x31x30x30
x31x30x30x31x30
x30x31x30x31x30
x30x31x30x30x31
x30x30
x2x37xexf
xexfxf
xexfxe
xfxexf
xexfxe
xf
x8x20x50x75x75x75x75x75x75xfxexf
xexfxe
xfxexf
xexfxf
xexfxe
x37x2x75x75x75x75x75x75x50x20x8x27x5ex5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5c
A
C
B
D
O
x02x1x2
2,x2x3cxcxdx6cx38x36ABCD,x73A;D,?!AD;?!DBx6bx55x56[C;?!AC;?!BD]x1dx15x57x55.
x5,?!CA =!AC = (?1)?!AC +0?!BD,x17Ax57x55x22(?1;0);
!CD = 1
2
!CA+ 1
2
!BD =? 1
2
!AC + 1
2
!BD;
x17Dx57x55x22
12 ; 12
·;
!AD = 1
2
!AC + 1
2
!BD;
!ADx57x55x22?1
2 ;
1
2
·;
!DB =!BD = 0?!AC +(?1)?!BD;
!DBx57x55x22(0;?1).
3,x18?!a,?!b,?!c x15x57x55x11x12x13(1;5;2),(0;?3;4),(?2;3;?1),x73x1fx202?!a +?!c,?3?!a +2?!b +4?!c
x15x57x55.
x5,2?!a +?!c = 2(1;5;2)+(?2;3;?1) = (2;10;4)+(?2;3;?1) = (0;13;3).
3?!a +2?!b +4?!c =?3(1;5;2)+2(0;?3;4)+4(?2;3;?1)
= (?3;?15;?6)+(0;?6;8)+(?8;12;?4) = (?11;?9;?2):
4,x53x54x34x35x36x15x34x31x16x74x3x3cx48x17(x24x7f).
x3x4,x18D;E;Fx11x12x13x38BC;CA;ABx79x15x16x17,ADx42BEx3x3cG,ADx42CFx3x3cG0,x4a
!AG = k?!AD = k 1
2
!AB + 1
2
!AC? = k
2
!AB + k
2
!AC:
x1ex5bx2bx5cx5dx55x56[A;?!AB;?!AC],x4ax17Gx57x55x22
k
2 ;
k
2
·
,x51
!BG = m?!BE = m 1
2
!BA+ 1
2
!BC? = m 1
2
!AB + 1
2 (
!AC!AB)?
= m2?!AC?m?!AB;
x23x24?!BGx57x55x22
m; m2
·
,x71?!AG =?!AB +?!BG,x23x24,
k
2 ;
k
2
= (1;0)+
m; m2
·;
￠ 9 ￠
x2dx40x41x42 ( k
2 = 1?m
k
2 =
m
2
x50k = m = 23, x23x24Gx15x57x55x22
1
3 ;
1
3
·
,x43x5,x3ex24x5ex50G0x15x57x55x22
1
3 ;
1
3
·
,x53x50G = G0.
xdxcxd
xdxcxd
xdxcxd
xdxcxd
xdxcxd
xdxcx2x6fx6ex6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
xex75x75x75x75x75x75x75x75x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28
x44x1a
x36x36x36x36x36x36x36x36
x36x36x36x36x36x36x36x36
x36x36x36x37x36x36x36x36
x36x36x36x1dx7xax56x4dx4ex4dx4dx4dx4ex4dx4d
x4dx4ex4dx4dx4dx4e
x4dx4dx4dx4ex4dx4d
A F B
C
DE
M
x04x1x2
x7x7x7
x7x7x7
x7x8x7
x7x7x7
x7x7x7
x7x7x1x6fx6ex6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
xex75x75x75x75x75x75x75x75x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28
x44x1a
x35x36x35x35x36x35x35
x36x35x35x36x35x35x36
x35x35x36x35x35x36x35
x35x36x35x35x36x35x7
xbx25x59x53x53x53
x53x53x53x53
x53x53x53x53
x53x53x53x53
x53x53
A F B
C
DE
M
x05x1x2
5,x53x54x34x35x36x15x34x31x35xcx11x74x3x3cx48x17.
x3x4,x184ABCx15x34x31x35xcx11x74x11x12x22AD;BEx3aCF,x3fx18ADx42BEx3x3cTx17,x49
!AT = k?!AD = k
j?!ABj+j?!ACj
(j?!ACj?!AB +j?!ABj?!AC):
x5bx2bx5cx5dx55x56[A;?!AB;?!AC],x3fx49?!AB =?!a,?!AC =?!b, x4aTx17x57x55
kj?!b j
j?!a j+j?!b j ;
kj?!a j
j?!a j+j?!b j
!
,x14x15x5f
xbx60
!BE = 1
j?!BAj+j?!BCj
(j?!BCj?!BA+j?!BAj?!BC);
x23x24
!BT = m?!BE = m
j?!a j+j?!b!a j
(?j?!b!a j?!a +j?!a j(?!b!a ))
=?m?!a + mj
!a j
j?!a j+j?!b!a j
!b,
x4ex3c?!AT =?!AB +?!BT,x23x24?
kj?!b j
j?!a j+j?!b j; kj
!a j
j?!a j+j?!b j
!
= (1;0)+
m; mj
!a j
j?!a j+j?!b!a j
!;
x4,8
>><
>>:
kj?!b j
j?!a j+j?!b j = 1?m
kj?!a j
j?!a j+j?!b j =
mj?!a j
j?!a j+j?!b!a jx2dx50:
k = j
!a j+j?!b j
j?!a j+j?!b j+j?!b!a j
:
x51x18ADx42CFx3x3cT0x17,
!AT0 = s?!AD = sj?!b j
j?!a j+j?!b j
!a + sj?!a j
j?!a j+j?!b j
!b,
x50T0x17x15x57x55x22
sj?!b j
j?!a j+j?!b j ;
sj?!a j
j?!a j+j?!b j
!
.
!CT0 = t?!CF = t
j?!CAj+j?!CBj
(j?!CBj?!CA+j?!CAj?!CB) = tj
!b j
j?!b j+j?!a!b j
!a?t?!b ;
￠ 10 ￠
x4e!AT0 =?!AC +!CT0,x50:?
sj?!b j
j?!a j+j?!b j; sj
!a j
j?!a j+j?!b j
!
= (0;1)+
tj?!b j
j?!b j+j?!a!b j;?t
!;
x4,8
>><
>>:
sj?!b j
j?!a j+j?!b j =
tj?!b j
j?!b j+j?!a!b j
sj?!a j
j?!a j+j?!b j = 1?tx2dx50:
s = j
!a j+j?!b j
j?!a j+j?!b j+j?!a!b j
:
x4ex4fx3ex5fs = k,x4T = T0.
6,xaxbx74x78ABx49x17C(2;0;2)x3aD(5;?2;0)x34x56x11,x19x73x25x77x74x78x15x37x66x61x17A;Bx15x57x55.
x5,x55x3fx18A;B;C;Dx6cx17x8x9x23x1c,x18A;Bx37x17x15x57x55x11x12x22(xA;yA;zA)x42(xB;yB;zB),x4a
!AC =?!CD,?!CD =?!DB,x23x24
(xC?xA;yC?yA;zC?zA) = (xD?xC;yD?yC;zD?zC);
x4,8
><
>:
xA = 2xC?xD =?1
yA = 2yC?yD = 2
zA = 2zC?zD = 4:
x43x5,8
><
>:
xB = 2xD?xC = 8
yB = 2yD?yC =?4
zB = 2zD?zC =?2:
x21x4fA;Bx37x17x15x57x55x11x12x22(?1;2;4)x42(8;?4;?2)(x37x62x3ex63).
7,xaxbAx0 Bx37x17x15x57x55x11x12x22(1;?2;3),(4;1;2).
(1)x19x64x1dx17Px15x57x55,x27x17Px11x74x78ABx2ax1dx653, 2;
(2)x19x64x1dx17Px15x57x55,x27x17Px11x74x78BAx2ax1dx65?2, 3.
x5,(1)x4ej?!APj, j?!PBj = 3, 2x3ex50?!AP = 32?!PB,x33x1ax6a3.1x15x1dx65x11x17x66x29,x7ak = 32,x3ex50Px17
x57x55
14
5 ;?
1
5 ;
12
5
·
.
(2)x4exaxbx31x32x3ex50?!BP =? 23?!PA,x1ax1dx65x11x17x66x29x67x50Px17x57x55(10;7;0).
x75x75x75x75x75x75x75x75x5 x5 x5 x5
A C BD
x06x1x2
x75x75x75x75x75x75x75x75x72x40x50x8x5 x5 x5 x5 x5 x5
A C BD E F
x09x1x2
8,ABCDx22xcxdx6cx38x36,xaxbA,Bx68x2x35x74x3x17x15x57x55x11x12x22(?3;1;5),(2;?3;4),(1;?1;2).
x19x64x1dx17C,Dx15x57x55.
x5,x18x2x35x74x3x17x22M,C;Dx15x57x55x11x12x22(xC;yC;zC);(xD;yD;zD),x4ex3cMx13A;Cx15x16x17,x21
x4f 8
>><
>>:
1
2 (?3+xC) = 1
1
2 (1+yC) =?1
1
2 (5+zC) = 2;x2dx50Cx17x57x55x22(5;?3;?1),x4ex3cMx67x13B;Dx15x16x17,x43x5x3ex50Dx17x57x55x22(0;1;0).
￠ 11 ￠
9,x18A;Bx37x17x15x57x55x11x12x22(?6;5;?8),(4;0;7),x19x64x1dx17C;D;E;F,x27C;D;E;F x76x74x78AB
x69x56x11.
x5,x55x3fx18A;B;C;D;E;Fx8x9,x23x24
!AC = 1
4
!CB;?!AD = 2
3
!DB;
!AE = 3
2
!EB;?!AF = 4?!FB:
x33x1ax1dx65x11x17x66x29x67x50Cx17x57x55x22(?4;4;?5),Dx17x57x55x22(?2;3;?2),Ex17x57x55x22(0;2;1),F x17x57x55
x22(2;1;4).
10,x8x9,xaxbxcxdxexfx10OABC?O1A1B1C1x16,x17Px6bx14AA1x79,x3fjAPj = 2jPA1j,x17Sx6bx14
CC1x79,x3fjCSj = 12 jSC1j,x17Q;Rx11x12x13x14O1C1;ABx15x16x17,x73x53:x2ex74PQx42x2ex74RSxcxd.
x3x4,x5bx2bx57x55x6a[O;?!OA;?!OC;!OO1],x21x22j?!APj = 2j!PA1j,x23x24
!AP = 2!PA
1 =
2
3
!AA
1;
!OP =?!OA+?!AP =?!OA+ 2
3
!AA
1 =
!OA+ 2
3
!OO
1;
!OQ =!OO
1 +
!O
1Q =
!OO
1 +
1
2
!OC;
x43x25
!PQ =?!OQ!OP =!OA+ 2
3
!OO
1 +
1
2
!OC:
x6bx6cx6d,
!OR =?!OA+?!AR =?!AB + 1
2
!OC;
!OS =?!OC +?!CS =?!OC + 2
3
!OO
1;
x23x24
!RS =?!OS!OR =!OA+ 2
3
!OO
1 +
1
2
!OC:
x77x6ex6fx47?!PQ =?!RS,x3cx13?!PQ==?!RS.
x10x70
x74
x74
x74
x75x75x75x75x75x75x75x50x71x71
x71x20
x40x74
x74
x74
x70x10x50x75x75x75x75x75x75x75
x40x20x71
x71x71
x75x75x75x75x75x75x75x50x71x71
x71x20
x40
x74
x74
x74
x70x10
x40x18 x40x18
x40x18 x40x18
x40x18 x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10
x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20x47x1b
x47x1b x47x1b x47x1b
x47x1b x47x1b x47x1c
x47x1c x47x1c x47x1c
x47x1c x47x1c x47x1b
x47x1b x47x1b
x47x1b x47x1b x47x1b
x47x1b x47x1b x47x1b
x47x1c x47x1c x47x1c
x47x1c x47x1c x47x1c
x47x1b x47x1b x47x1b
A B
CO
A1 B1
O1 C
1
P
Q
R
S
x010x1x2
x29x29x29x2a
x29x29x29x29
x29x29x2ax29
x29x29x29x6dx6dx6cx6dx6d
x6dx6dx6c
x6dx6dx66
xe
x38x10x10
x10x10x10
x10x10x10
xax50x50x50x50x50
x50x50x50x50x50
x50x50x50x50x50
x50x50
x5 x5
x5 x5
A
DB
C
E F
H G
x011x1x2
11,xaxbx70x71x6cx38x36ABCD,x76AB;AD;CDx68CBx24x65x43x65x11x39,x53x54x77x6cx66x11x17x75x2ax48x66xc
xdx6cx38x36.
x3x4,x8x9,x18x11x17x22E;F;G;H,x18
!AE = k?!EB = 1
1+k
!AB;?!AF = k?!FD = 1
1+k
!AD;
x4a
!EF =?!AF!AE = 1
1+k
!BD:
x6bx6cx6d,x4e!CH = k?!CBx24x68?!CG = k?!CDx3ex24x50x8!HG = 12?!BD,x21x4f?!EF =!HG,x53x54x72EFGHx13xc
xdx6cx38x36.
￠ 12 ￠
12,x53x54:x6cxfx10x15x6cx31x16x74x3x3cx48x17(x4x6cxfx10x15x24x7f),x3fx4fx3x17x76x73x48x31x16x74x11x2ax1dx65x223, 1
(x4ex7ex17x67x74)x15x37x7cx11,(x1a,x6cxfx10x15x16x74x4x6cxfx10x15x7ex17x8x3cx2xfx15x24x7fx15x4cx74)
x3x4,x5bx2bx5cx5dx57x55x6a[V;?!VA;?!VB;?!VC],Gx17x224ABCx15x24x7f,G1x224VBCx15x24x7f,x4ex60x611–2
x15x3d3x61xb:
!VG = 1
3 (
!VA+?!VB +?!VC) = 1
3 ;
1
3 ;
1
3
;
!AG
1 =
1
3 (
!AB +?!AC +?!AV) = 1
3 (
!VB!VA+?!VC!VA!VA)
= 13 (?3?!VA+?!VB +?!VC) =
1; 13 ; 13
:
x7ax4ex16x74VGx11x2a3, 1x15x11x17M,x4
!VM = 3
4
!VG = 1
4 ;
1
4 ;
1
4
;
x43x52,
!VA+ 3
4
!AG
1 =
1
4 ;
1
4 ;
1
4
=!VM:
x23x24Mx6bVGx42AG1x79,x43x5,x3ex53x50:x1ex18G2;G3x11x12x134VABx3a4VACx15x24x7f,x4aCG2x42BG3
x67x40x3x3cx17M0,x3f!VM0 =
1
4 ;
1
4 ;
1
4
·
,x21x4fM0 = M,x3fx3x17x11x73x48x16x74x2ax1dx653, 1.
x1ax28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x73x73
x73x73
x73x60
xcx1dx1dx1d
x1dx1dx1e
x1dx1dx1d
x1dx1dx1d
x1dx1dx1d
x3fx4
xax58x51x51x52x51
x51x51x51x51x51
x51
x1x11x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5
x5
x5x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f x4dx1e x4dx1f x4dx1e x4d
x1e x4cx1f x4dx1e x4dx1f
A
C
B
V
G
G1
x012x1x2
x1ax28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x73x73
x73x73
x73x60
xcx1dx1dx1d
x1dx1dx1e
x1dx1dx1d
x1dx1dx1d
x1dx1dx1d
x3fx4
xax58x51x51x52x51
x51x51x51x51x51
x51
x1x11x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5
x5
x5
x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f x4dx1e x4dx1f x4dx1e
x4dx1e x4cx1f x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f x4dx1e
x4dx1f
A
C
B
D
E
V
x013x1x2
13,x6cxfx10x15x55x65x3x15x37x31x14x75x22x2x14,x73x48x2x2x14x15x16x17x15x4cx74x75x22x6cxfx10x15x76x16x74,x53x54:x6c
xfx10x15x34x31x76x16x74x3x3cx38x15x24x7f,x3fx4fx24x7fx4ex73x48x31x76x16x74x11x2ax3bx77x65x56x15x37x7cx11.
x3x4,x43x79x61x5bx2bx5cx5dx57x55x6a[V;?!VA;?!VB;?!VC],x18Mx13x6cxfx10x15x24x7f,x4a!VM =
1
4 ;
1
4 ;
1
4
·
,x18
D;Ex11x12x22AB;VCx15x16x17,x4a
!AD = 1
2
!AB = 1
2
!VB? 1
2
!VA;
!VD = 1
2
!VA+ 1
2
!VB = 1
2 ;
1
2 ;0
:
!VE = 1
2
!VC = 0;0; 1
2
;
x23x24
!ED =?!VD!VE = 1
2 ;
1
2 ;?
1
2
:
x4ex3c
!VE + 1
2
!ED = 1
4 ;
1
4 ;
1
4
=!VM;
x58x54x24x7fMx6bEDx79x3fx56x11ED,x43x5x3ex53x3cx38.
x1 x2 1–4
￠ 13 ￠
1,x78x67x1dx1e2x79x423x79xdx1ex29:
x5,(1)
flfl
flfl?1 1
2 3
flfl
flfl =?5,(2)
flfl
flflcos?sin
sin cos
flfl
flfl = 1.
(3)
flfl
flfl
flfl
1 2 2
0?1 0
0 0?1
flfl
flfl
flfl = 1,(4)
flfl
flfl
flfl
1 1 1
2 3 4
4 9 16
flfl
flfl
flfl = 2.
(5)
flfl
flfl
flfl
2 1 1
1 1 5
2 3?3
flfl
flfl
flfl =?22,(6)
flfl
flfl
flfl
x y z
z x y
y z x
flfl
flfl
flfl = x3 +y3 +z3?3xyz.
2,x33x1a2x79x443x79xdx1ex29x2dx74x26x40x41x42:
(1)
( 2x?3y = 5;
3x+2y = 1; (2)
8>
<
>:
2x?y +3z = 9;
3x?5y +z =?3;
x+3y?2z =?6:
x5,(1) x =
flfl
flfl5?3
1 2
flfl
flfl
flfl
flfl2?3
3 2
flfl
flfl
= 1313 = 1,y =
flfl
flfl2 5
3 1
flfl
flfl
flfl
flfl2?3
3 2
flfl
flfl
=?1313 =?1.
(2) x =
flfl
flfl
flfl
9?1 3
3?5 1
6 3?2
flfl
flfl
flfl
flfl
flfl
flfl
2?1 3
3?5 1
1 3?2
flfl
flfl
flfl
=?4249 =? 67,y =
flfl
flfl
flfl
2 9 3
3?3 1
1?6?2
flfl
flfl
flfl
flfl
flfl
flfl
2?1 3
3?5 1
1 3?2
flfl
flfl
flfl
= 4249 = 67,z =
flfl
flfl
flfl
2?1 9
3?5?3
1 3?6
flfl
flfl
flfl
flfl
flfl
flfl
2?1 3
3?5 1
1 3?2
flfl
flfl
flfl
=
189
49 =
27
7,
3,x18?!e1,?!e2,?!e3 x22x7a.
(1)x53x54:x1fx20?!a =?!e1 +3?!e2!e3,?!b = 2?!e1?3?!e2?10?!e3,?!c =!e1 +2?!e2 +6?!e3 x74x26x2cx2a;
(2)x73x1fx20?!d = 3?!a?2?!b +?!c x6bx7a?!e1,?!e2,?!e3 x1dx15x57x55;
(3)x73x1fx20?!f,x27!a +2?!b?3?!c +3?!f = 0.
x5,(1)x18x47x32xex1;x2;x3x2dx2ex74x26x2ax6ax29x1?!a +x2?!b +x3?!c = 0,x1bx50x2ax57x55x36x29x6fx13
8>
<
>:
x1 +2x2?x3 = 0
3x1?3x2 +2x3 = 0
x1?10x2 +6x3 = 0;
x38x15x6axexdx1ex29x13 fl
flfl
flfl
fl
1 2?1
3?3 2
1?10 6
flfl
flfl
flfl =?5 6= 0;
x21x4fx77x66x40x41x42x7bx47x6fx2d,x4?!a ;?!b ;?!c x74x26x2cx2a.
(2)?!d = 3?!a? 2?!b +?!c = 3(?!e1 + 3?!e2!e3)? 2(2?!e1? 3?!e2? 10?!e3) + (!e1 + 2?!e2 + 6?!e3) =
2?!e1 +17?!e2 +23?!e3,x21?!d x15x57x55x13(?2;17;23).
(3)?!f = 13 (?!a?2?!b +3?!c ) = 13 (?6?!e1 +15?!e2 +37?!e3) =?2?!e1 +5?!e2 + 373?!e3.
4,x7cx7dx1dx1ex73x42x15x34x66x1fx20?!a ;?!b ;?!c x13x29x28xf? x63x29x76?!c x1bx1cx2ax3cx38x37x66x1fx20x15x74x26x42x54?x1e
x63,x7ex25x7fx10x15x1bx1cx29x0.
(1)?!a (5;2;1),?!b (?1;4;2),?!c (?1;?1;5);
(2)?!a (3;3;2),?!b (6;6;4),?!c (1;?1;0);
￠ 14 ￠
(3)?!a (1;2;?3),?!b (?2;?4;6),?!c (1;0;5).
x5,x1x61x2x22x22x73x2dx1?!a +x2?!b +x3?!c = 0.
(1)x3x48x74x26x40x41x42 8
><
>:
5x1?x2?x3 = 0
2x1 +4x2?x3 = 0
x1 +2x2 +5x3 = 0;
x3cx6axexdx1ex29 fl
flfl
flfl
fl
5?1?1
2 4?1
1 2 5
flfl
flfl
flfl = 121 6= 0;
x40x41x7bx47x6fx2d,x21x4bx1fx20x42x55x28xf.
(2)x3x48x74x26x40x41x42 8
><
>:
3x1 +6x2 +x3 = 0
3x1 +6x2?x3 = 0
2x1 +4x2 = 0;
x3cx6axexdx1ex29 fl
flfl
flfl
fl
3 6 1
3 6?1
2 4 0
flfl
flfl
flfl = 0;
x40x41x42x47x6ex6fx2d,x21x4bx1fx20x42x28xf,x22x76?!c x1bx1cx2a?!a ;?!b x15x74x26x42x54,x3ex7ax3 =?1x51x52,x50x8x40x41x428
><
>:
3x1 +6x2 = 1
3x1 +6x2 = 1
2x1 +4x2 = 0;
x77x13x34x35x40x41x42,x21x4f?!c x55x63x1bx1cx2a?!a ;?!b x15x74x26x42x54.
(3)x3x48x74x26x40x41x42 8
><
>:
x1?2x2 +x3 = 0
2x1?4x2 = 0
3x1 +6x2 +5x3 = 0;
x3cx6axexdx1ex29 fl
flfl
flfl
fl
1?2 1
2?4 0
3 6 5
flfl
flfl
flfl = 0;
x40x41x42x47x6ex6fx2d,x21x4bx1fx20x42x28xf,x22x76?!c x1bx1cx2a?!a ;?!b x15x74x26x42x54,x3ex7ax3 =?1x51x52,x50x8x40x41x428
><
>:
x1?2x2 =?1
2x1?4x2 = 0
3x1 +6x2 =?5;
x77x13x34x35x40x41x42,x21x4f?!c x55x63x1bx1cx2a?!a ;?!b x15x74x26x42x54.
5,x18x1fx20?!a ;?!b ;?!c x15x57x55x11x12x13(1;?1;2),(2;k;1),(1;1?k;k),x1,x62kx7ax2fx30x4x52,?!a ;?!b ;?!c
x28xf? x69x12x6d,kx7ax2fx30x4x52,?!a ;?!c x28x74?
x5,x773x66x1fx20x28xfx15x30x11x40x26x31x32x13x3cx57x55x15xdx1ex29x56x3c0,x4fl
flfl
flfl
fl
1?1 2
2 k 1
1 1?k k
flfl
flfl
flfl = k2?3k +2 = 0:
￠ 15 ￠
x21x4fx62k = 1x442x52x773x66x1fx20x28xf,x26x27?!a ;?!c x28x74x40x41x27x38x15x15x65x2cx57x55x2ax65x6a,x4 11 = 1?k?1 = k2,
x2dx50k = 2,x21x4fx62k = 2x52?!a ;?!c x28x74.
6,x18?!e1,?!e2,?!e3 x22x7a,x1x1fx20?!v x63x29x1bx22x1fx20?!a ;?!b ;?!c x15x74x26x42x54?x8x63,x4ax7ex25x1bx50x29.
(1)?!a =?!e1 +2?!e2 +4?!e3,?!b =?!e1!e2 +?!e3,?!c =?!e1 +?!e2 +3?!e3,?!v = 6?!e1 +3?!e2 +15?!e3;
(2)?!a =?!e1?2?!e2 +3?!e3,?!b =?!e1 +2?!e2 +2?!e3,?!c =!e1 +6?!e2 +5?!e3,?!v = 2?!e1 +?!e2:
x5,x18?!v = x1?!a +x2?!b +x3?!c,x1x61x2x22x22x2dx74x26x40x41x42.
(1)x40x41x42x22 8
><
>:
x1 +x2 +x3 = 6
2x1?x2 +x3 = 3
4x1 +x2 +3x3 = 15;
x6axexdx1ex29
flfl
flfl
flfl
1 1 1
2?1 1
4 1 3
flfl
flfl
flfl = 0,x55x63x7dx1dx40x41x42x13x29x47x2d,x1ax17x5x6x7x44x2dx50
( x
1 = 3? 23 x3
x2 = 3? 13 x3;
x49
x3 = 3kx3ex24x50x8x74x26x1bx1cx29?!v = (3?2k)?!a +(3?k)?!b +3k?!c,x3cx16kx22x0x1xe.
(2)x40x41x42x22
8>
<
>:
x1 +x2?x3 = 2
2x1 +2x2 +6x3 = 1
3x1 +2x2 +5x3 = 0;
x6axexdx1ex29
flfl
flfl
flfl
1 1?1
2 2 6
3 2 5
flfl
flfl
flfl = 36;
x40x41x42x47x2d:
x1 =
flfl
flfl
flfl
2 1?1
1 2 6
0 2 5
flfl
flfl
flfl
36 =?
11
36 ; x2 =
flfl
flfl
flfl
1 2?1
2 1 6
3 0 5
flfl
flfl
flfl
36 =
16
9 ; x3 =
flfl
flfl
flfl
1 1 2
2 2 1
3 2 0
flfl
flfl
flfl
36 =?
19
36,
x74x26x1bx1cx29x22?!v =? 1136?!a + 169?!b? 1936?!c,
7,x62ax22x8x4x52,x1dx1ex6cx17x28xf:
M1(1;a;a2); M2(1;?1;1); M3(2;1;?2); M4(?1;2;2):
x5,x3dx3ex5ex234.5,x4f4x17x28xfx15x30x11x40x26x31x32x13fl
flfl
flfl
fl
1?1 2?1?1?1
1?a 1?a 2?a
1?a2?2?a2 2?a2
flfl
flfl
flfl =?7a2?5a+2 = 0;
x2dx50a =?1x44 27,
x1 x2 1–5
1,x3dx3enx46x1fx20x15x1dx4dx53x54:x2x0x1nx46x1fx20fi,x47
(1) 0fi = 0;
(2) (?1)fi =?fi;
(3) k0 = 0 (x0x1xek);
(4)x43kfi = 0x5ex25k = 0x44fi = 0.
x3x4,x2x0x1x15fi = (a1;￠￠￠ ;an),x4a:
￠ 16 ￠
(1) 0fi = (0a1;￠￠￠ ;0an) = (0;￠￠￠ ;0) = 0.
(2) (?1)fi = ((?1)a1;￠￠￠ ;(?1)an) = (?a1;￠￠￠ ;?an) =?fi.
(3) k0 = (k0;￠￠￠ ;k0) = (0;￠￠￠ ;0) = 0.
(4) kfi = (ka1;￠￠￠ ;kan) = (0;￠￠￠ ;0),x1efi 6= 0,x4ax31x6bai 6= 0,x4ekai = 0x3ex50k = 0.
2,x53x54:x0x48xex9x6dx36xax47x5xex9.
x3x4,x18Kx22x48x66xex9,x4a1 2 K,x23x24x2x0x1x15x72x2bxenx47n = 1+￠￠￠+1| {z }
n
2 K,x57x3fnx15x39xb
n 2 K,x21x4fKxax47x33x7cx2bxe,x51x21x2x0x1x15x2bxen 6= 0,n 2 K,1n x22nx15xcxb,x4a 1n 2 K,x23x24x2x0
x1x15x47x5xe mn (x3cx16m;nx13x2bxe),x47 mn = m￡n 2 K,x21x47x5xex9Q K.
3,x53x54:x33x10x36x8
a+bp2; a;b 2Q
x15xex42x2ax15xdx54x75x2ax48x66xex9.
x3x4,x4ex77x66xdx54x1fx22K,x18a1 +b1p2;a2 +b2p2 2 K,x4a
(a1 +b1p2)§(a2 +b2p2) = (a1 §a2)+(b1 §b2)p2 2 K
(x21x22x47x5xex15x3ax42x5bxex13x47x5xe);
(a1 +b1p2)(a2 +b2p2) = (a1a2 +2b1b2)+(a1b2 +a2b1)p2 2 K
(x21x22x47x5xex15x3ax0x5bx42xfx10xex13x47x5xe);x62a2 +b2p2 6= 0x52,
a1 +b1p2
a2 +b2p2 =
(a1 +b1p2)(a2?b2p2)
a22?2b22 =
a1a2?2b1b2
a22?2b22 +
a2b1?a1b2
a22?2b22 2 K
(x47x5xex2ax3cx11x44x67x13x12x13x15),x21x4fxdx54Kx2ax3cx17x5xfx11x44x6dx12x13,x2ax22x48x66xex9.
4,x18Kx22xex9,V x22Kx79x15nx46x1fx20x70x71,x53x54:x2x23x47x15k 2 K,fi;fl 2 V,x47
(1) k(fi?fl) = kfi?kfl;
(2) fi+fi+￠￠￠+fi| {z }
nx0
= nfi;
(3)x1efi+fl = fi+,x4afl =,
x3x4,x18fi = (a1;￠￠￠ ;an),fl = (b1;￠￠￠ ;bn) (ai;bi 2 K),x4ax2x0x1x15k 2 K,
(1) k(fi?fl) = k(a1?b1;￠￠￠ ;an?bn) = (k(a1?b1);￠￠￠ ;k(an?bn)) = (ka1?kb1;￠￠￠ ;kan?kbn) =
k(a1;￠￠￠ ;an)?k(b1;￠￠￠ ;bn) = kfi?kfl.
(2) fi+￠￠￠+fi| {z }
n
= (a1 +￠￠￠a1| {z }
n;a2 +￠￠￠+a2| {z }
n;￠￠￠ ;an +￠￠￠+an| {z }
n
) = (na1;￠￠￠ ;nan) = nfi.
(3)x1ex18 = (c1;￠￠￠ ;cn),x3ffi+fl = fi+,x4,(a1 +b1;￠￠￠ ;an +bn) = (a1 +c1;￠￠￠ ;an +cn),x4a
x47ai +bi = ai +ci,x4bi = ci,x23x24fl =,
x1 x2 1–6
1,x76x1dx1ex1fx20x14x2fx4c:
(1)?!a = 5?!i?6?!j +3?!k ;
(2)?!b = 12?!i? 13?!k,
x5,(1)?!a0 =?!aj?!a j =
p70
70 (5
!i?6?!j +3?!k ).
￠ 17 ￠
(2)?!b0 =
!b
j?!b j =
p13
13 (3
!i?2?!k ).
2,x78x67x1dx1ex1fx20x15x15x35:
(1)?!a = (1;?2;3),?!b = (2;1;?2);
(2)?!a = (?2;1;?1),?!b = (1;?1;4).
x5,(1) j?!a j = p14,j?!b j = 3,?!a ￠?!b =?6,x23x24
cosh?!a ;?!b i =?6p14￠3 =?
p14
7 ; h
!a ;?!b i = …?arccos
p14
7,
(2) j?!a j = p6,j?!b j = 3p2,?!a ￠?!b =?7,x23x24
cosh?!a ;?!b i =?76p3 =? 7
p3
18 ; h
!a ;?!b i = …?arccos 7
p3
18,
3,x73x1fx20?!a x6b?!e0 x79x15x16x17:
(1)?!a = (1;?1;2),?!e = (1;1;1);
(2)?!a = (?2;1;3),?!e = (1;2;0).
x5,(1)?!e0 =?!ej?!e j =
p3
3 (1;1;1),x4a
pr?!e0?!a = (Π?!e0?!a )?!e0 = (?!a ￠?!e0)?!e0 = 2
p3
3
!e0 = 2p3
3 ￠
p3
3 (1;1;1) =
2
3 (1;1;1):
(2)x21?!a ￠?!e = 0,x23x24pr?!e0?!a = 0.
4,x53x54:x24A(3;?1;2),B(0;?4;2),C(?3;2;1)x22x7ex17x15x34x35x36x13x56x18x34x35x36.
x3x4,?!AB = (?3;?3;0),?!AC = (?6;3;?1),?!BC = (?3;6;?1),j?!ABj = 3p2,j?!ACj = j?!BCj = p46 6=
j?!ABj,x23x244ABCx13x56x18x34x35x36.
5,x53x54:x24A(3;?2;1),B(7;6;9),C(9;1;?5)x22x7ex17x15x34x35x36x13x2ex35x34x35x36.
x3x4,?!AB = (4;8;8),?!AC = (6;3;?6),?!AB ￠?!AC = 12(2+2?4) = 0,x23x244ABCx13x2ex35x34x35x36.
6,x19x1(?!a?!b )?!c =?!a (?!b?!c )x48x1dx2ax2bx19? x1ax1bx25x1cx1fx20x56x29x2ax2bx15x31x32.
x5,x56x29x1dx61x13x42?!c x28x74x15x1fx20,x1ex61x13x42?!a x28x74x15x1fx20,x8x1fx37x61x6dx55x56x3c0,x4a?!a x42?!c x28
x74,x4x31x6bk 6= 0,x27?!a = k?!c, x4dx39x1e?!a = k?!c,x1dx38= k(?!c ￠?!b )?!c = (k?!c )(?!b ￠?!c ) =x1ex38.
x1ex56x29x37x38x6dx56x3c0,x4ax44x20?!a ;?!b ;?!c x16x45x46x47x48x66x6fx1fx20;x44x20x34x66x1fx20x6dx55x56x3c0,x71?!a ￠?!b =
!b ￠?!c = 0,x4?!b x42?!a ;?!c x4bx72x3.
7,x73x2dx1fx20x40x41?!a?!x =?!b?!x,
x5,x21x22(?!a!b )?!x = 0,x11x37x62xfx21,(a)x1e?!a!b 6= 0,x4ax2d?!x x22x0x1x42?!a!b x22x2ex15x1fx20;
(b)x1e?!a!b = 0,x4ax0x1x1fx20?!x x6dx13x2dx1fx20.
8,x18x47x34x66x1fx20?!a ;?!b ;?!c x37x37x75x2a60–x35,x3fxbj?!a j = 4,j?!b j = 2,j?!c j = 6,x73?!a +?!b +?!c x15
x3bx77.
x5,j?!a +?!b +?!c j2 = (?!a +?!b +?!c )￠(?!a +?!b +?!c ) =?!a 2+?!b 2+?!c 2+2?!a ￠?!b +2?!a ￠?!c +2?!b ￠?!c =
16+4+36+2(8+24+12)cos60– = 100,x23x24j?!a +?!b +?!c j = 10.
9,xaxbj?!a j = 3,j?!b j = 2,h?!a ;?!b i = …6, x19x733?!a +2?!b x422?!a?5?!b x15x7bx10.
x5,(3?!a +2?!b )￠(2?!a?5?!b ) = 6?!a 2?10?!b 2?11?!a ￠?!b = 54?40?11￡3￡2￡
p3
2 = 14?33
p3.
10,x18x48x66x6cx38x36x28x38x39x3bx11x12x13a;b;c;d,x3fx3cx2x35x74x64x65x22x2e,x73x53x28x38x39x3bx67x13a;b;c;dx15
x0x48x6cx38x36x15x37x31x2x35x74x67x65x64x22x2e.
￠ 18 ￠
x3x4,x8x9,x47
!AC =?!AB +?!BC =?!AD +?!DC;
!BD =?!BC +?!CD =?!BA+?!AD:
xbxcx24x1dx7bx10:
!AC ￠?!BD = (?!AB +?!BC)￠(?!BC +?!CD) =?!BC2 +?!AB ￠?!BC +?!AB ￠?!CD +?!BC ￠?!CD;
!AC ￠?!BD = (?!AB +?!BC)￠(?!BA+?!AD) =!AB2 +?!AB ￠?!AD!AB ￠?!BC +?!AD￠?!BC;
!AC ￠?!BD = (?!AD +?!DC)￠(?!BC +?!CD) =!CD2 +?!AD￠?!BC +?!AD￠?!CD!BC ￠?!CD;
!AC ￠?!BD = (?!AD +?!DC)￠(?!BA+?!AD) =?!AD2!AB ￠?!AD +?!AB ￠?!CD!AD￠?!CD:
x76x79x534x29x65x17,x3ex50:
4?!AC ￠?!BD =?!AD2 +?!BC2!AB2!CD2 +2?!AB ￠?!CD +2?!AD￠?!BC:
x4e
!AB +?!BC +?!CD +?!DA = 0
x3ex50
!AB +?!CD =?!AD!BC:
x4e
(?!AB +?!CD)2 = (?!AD!BC)2
x2bx5x50
!AD2 +?!BC2!AB2!CD2 = 2(?!AB ￠?!CD +?!AD￠?!BC):
x51x52x79x29x50
4?!AC ￠?!BD = 2(?!AD2 +?!BC2!AB2!CD2) = 2(b2 +d2?a2?c2):
x77x58x54x2x35x74x22x2ex15x30x11x40x26x31x32x13a2 +c2 = b2 +d2,x7bx42x6cx38x36x15x38x3bx47x2a.
x3x17x1bx1a
x1bx1bx1a
x1bx1bx1a
x1bx1bx1a
x1bx1bx1a
x1bx1bx1a
x1bx1bx1a
x1b
xax9xa
xaxax9
xaxaxa
x9xaxa
xax9xa
xax12x1x34x33x34
x33x34x33x34x33x34
x34x33x34x49x1dx75x76x75x75
x76x75x75
x76x75x75
x76x75x75
x76x75x75
x76x75x75
x76x75x2d
xfx8x20x50x75x75x75x75x75x75
x5ax5bx5a
x5ax5ax5b
x5ax5ax5b
x5ax5ax5a
x5bx5ax5a
x5bx5ax5a
x5ax5b
A B
D
C
O
x010x1x2
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6exex75x75x75x75x75x75x75x75x28x28x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28
x28x44x1a
x0x0x74
x74
x74
x74
B H C
A
x011x1x2
11,x34x35x36ABCx16,xaxbBCx38x79x15x23x22AH,x19x1a?!AB,?!ACx1bx1c?!AH.
x5,x76?!AH =?!AB +k?!BCx51x52?!AH ￠?!BC = 0,x50?!AB ￠?!BC +k?!BC2 = 0,x21x4f
k =?
!AB ￠?!BC
!BC2 ;
x50
!AH =?!AB? (?!AB ￠?!BC)?!BC
!BC2,
x12x1a?!BC =?!AC!ABx51x52,x2bx5x7x50
!AH = 1
(?!AC!AB)2
[(?!AC ￠(?!AC!AB))?!AB?(?!AB ￠(?!AC!AB))?!AC]:
￠ 19 ￠
12,x6bx2ex35x57x55x6ax16,?!a ;?!b ;?!c x15x57x55x11x12x13(3;5;7),(0;4;3),(?1;2;?4),x733?!a +4?!b?5?!c x42
2?!b +?!c x15x15x35.
x5,x1f
!p = 3?!a +4?!b?5?!c = (14;21;53);?!q = 2?!b +?!c = (?1;10;2);
x4a
!p 2 = 3446;?!q 2 = 105;?!p ￠?!q = 302:
x23x24
cosh?!p ;?!q i = 151
p361830
180915,
13,x73x1dx1ex1fx20x15x40x1fx24x25:
(1)?!a = (2;?3;?6); (2)?!b = (2;3;?10).
x5,(1) cosfi = 27,cosfl =?37,cos =?67,
(2) cosfi = 2
p113
113,cosfl =
3p113
113,cos =?
10p113
113,
14,x78x67x72x40x10x15x2x35x74x42x38x15x0x48x66xfx15x2x35x74x39x71x15x15x35.
x5,x5bx2bx2ex35x57x55x6a[A;?!AB;?!AD;!AA1],x24x2x35x74AC1x19x78x67x4fx61.
!AC
1 =
!AB +?!BC +!CC
1 =
!AB +?!AD +!AA
1 = (1;1;1):
(a)!AB1 =?!AB +!BB1 =?!AB +!AA1 = (1;0;1),x23x24cosh!AC1;!AB1i = 2p3￠p2 =
p6
3, x4ex2x75x26,
!AC
1x42
!A
1C1;
!AD
1;
!BC
1;
!DC
1;
!ACx15x15x35x24x25x67x22 p6
3,
(b)?!BD =?!AD!AB = (?1;0;1),x23x24cosh!AC1;?!BDi = 0p3￠p2 = 0,x4h!AC1;?!BDi = …2, x43x5.
!AC
1x42
!B
1D,
!DA
1;
!CB
1;
!BA
1;
!CD
1x15x15x35x67x22 …2,
x20x72x75x75x75
x10x70
x74
x74
x74
x75x75x75x72x20x74
x74
x74
x70x10x71
x71x20
x20x72x75x75x75
x20x71
x71
x71x71
x20x74
x74
x74
x70x10
x40x18 x40x18
x40x18 x40x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20x30
x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
A B
D C
A1 B1
C1D1
x014x1x2
x30x0x70x75x75x75x75x75x40x20x8x71x71
x71x71
x71x70
x0x30x8x20x40x75x75x75x75x75
x71x71
x71x71
x71
x75x75x75x75x75x40x20x8x71x71
x71x71
x71
x30x0x70
x1dx4ax1dx4ax1dx4ax1dx4a
x1dx4ax1dx4ax1dx4ax1dx4ax1dx4a
x1dx4ax1dx4ax7x4a
x45x1a x45x1a x45
x1a x45x1a x45x1a
x45x1a x45x1a x45
x1a x6
x40x18 x40x18
x40x18 x40x18
x40x18 x40x18
x40x18 x30x10
x30x10x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x20x8O
A B
C
C1
A1
O1
B1
P
M
N
x015x1x2
15,x8x9,xaxbx3bx40x10OABC–O1A1B1C1x16,jOAj = 8,jOCj = 6,jOO1j = 1,Px13x14OCx79x15x17,
x3fjPCj = 2jOPj,Mx13x14ABx79x15x17,x3fjAMj = 2jMBj,Nx13x14B1C1x15x16x17,x73x2ex74A1Px42x2ex74MN
x23x2ax15x35.
x5,x4ex1fx20?!OA;?!OC;!OO1x15x14x2fx1fx20x1fx22?!i ;?!j ;?!k,x5bx2bx2ex35x57x55x6a[O;?!i ;?!j ;?!k ],x4a
!OA
1 = 8
!i +?!k = (8;0;1);
!OP = 1
3
!OC = 2?!j = (0;2;0);
!OM = 8?!i + 2
3
!OC = 8?!i +4?!j = (8;4;0);
!ON = 1
2
!OA+6?!j +?!k = (4;6;1);
x21x4f
!A
1P =
!OP!OA
1 = (?8;2;?1);
￠ 20 ￠
!MN =!ON!OM = (?4;2;1):
x23x24
cosh!A1P;!MNi = 35p69p21 = 5
p161
69,
16,x1ex1fx20?!a ;?!b ;?!c x55x28xf,x25x3f?!a?!x = 0,?!b?!x = 0,?!c?!x = 0,x4a?!x = 0,x19x53x39.
x3x4,x21x22?!a ;?!b ;?!c x55x28xf,x23x24x38x15x74x26x2cx2a,x3f?!x x3ex4ex38x15x74x26x1bx1c,x4,?!x = k1?!a +k2?!b +
k3?!c, x3cx13?!x 2 = k1(?!a ￠?!x )+k2(?!b ￠?!x )+k3(?!c ￠?!x ) = 0,x4,?!x = 0.
17,x53x54x34x66x1fx20?!a ;?!b ;?!c x28xfx15x30x11x40x26x31x32x13fl
flfl
flfl
fl
!a?!a?!a?!b?!a?!c
!b?!a?!b?!b?!b?!c
!c?!a?!c?!b?!c?!c
flfl
flfl
flfl = 0:
x3x4,?!a ;?!b ;?!c x28xfx62x3fx63x62x47x55x33x22x6fx15x32xek1;k2;k3x27,k1?!a +k2?!b +k3?!c = 0,x43x258
><
>:
k1?!a 2 +k2(?!a ￠?!b )+k3(?!a ￠?!c ) = 0
k1(?!b ￠?!a )+k2?!b 2 +k3(?!b ￠?!c ) = 0
k1(?!c ￠?!a )+k2(?!c ￠?!b )+k3?!c 2 = 0;
x67x4x3x48x74x26x40x41x42 8
><
>:
x?!a 2 +y(?!a ￠?!b )+z(?!a ￠?!c ) = 0
x(?!b ￠?!a )+y?!b 2 +z(?!b ￠?!c ) = 0
x(?!c ￠?!a )+y(?!c ￠?!b )+z?!c 2 = 0
(?)
x47x6ex6fx2dx = k1,y = k2,z = k3,x3dx3ex26x54.1,x6axexdx1ex29fl
flfl
flfl
fl
!a 2?!a ￠?!b?!a ￠?!c
!b ￠?!a?!b 2?!b ￠?!c
!c ￠?!a?!c ￠?!b?!c 2
flfl
flfl
flfl = 0:
x4dx39,x1e fl
flfl
flfl
fl
!a 2?!a ￠?!b?!a ￠?!c
!b ￠?!a?!b 2?!b ￠?!c
!c ￠?!a?!c ￠?!b?!c 2
flfl
flfl
flfl = 0;
x4a(?)x40x47x6ex6fx2d,x18x22x = k1,y = k2,z = k3,x49?!p = k1?!a +k2?!b +k3?!c,x20x30,x43(?)xb:
!p 2 = k1?!p ￠?!a +k2?!p ￠?!b +k3?!p ￠?!c = 0;
x4?!p = 0,?!a ;?!b ;?!c x74x26x2cx2a,x40x1dx28xf.
x1 x2 1–7
1,x6bx2ex35x57x55x6ax16,xaxb?!a ;?!b ;?!cx15x57x55x11x12x13(1;0;1),(1;?2;0),(?1;2;1),x73(3?!a +?!b )￡(?!b!c )
x15x57x55.
x5,3?!a +?!b = (4;?2;3),?!b!c = (2;?4;?1),x23x24
(3?!a +?!b )￡(?!b!c ) =
flfl
flfl?2 3
4?1
flfl
flfl;?
flfl
flfl4 3
2?1
flfl
flfl;
flfl
flfl4?2
2?4
flfl
flfl
= (14;10;?12):
2,x53x54(?!a ￡?!b )2 6?!a 2?!b 2,x57x58x54x56x29x8x52x2ax2b.
x3x4,(?!a ￡?!b )2 = j?!a ￡?!b j2 = j?!a j2j?!b j2 sin2h?!a ;?!b i6j?!a j2j?!b j2,x56x59x2ax2bx62x3fx63x62sinh?!a ;?!b i
= 0,x4,?!a ==?!b,
3,xaxb?!a ;?!b x13x37x66x64x55xcxdx15x1fx20,x73x53(?!a!b )￡(?!a +?!b ) = 2(?!a ￡?!b ),x57x58x54x38x15x27x8
x1x4d.
￠ 21 ￠
x3x4,(?!a!b )￡(?!a +?!b ) =?!a ￡?!a +?!a ￡?!b!b ￡?!a!b ￡?!b = 2(?!a ￡?!b ),x27x8x1x4d:x1e
x24?!a ;?!b x75x2ax48x66xcxdx6cx38x36x15x65x28x37x38,x4a?!a!b,?!a +?!b x22x4fxcxdx6cx38x36x15x37x2x35x74,x79x29x58x54:
x24x2x35x74x75x2ax15xcxdx6cx38x36xfx10x22x4bxcxdx6cx38x36xfx102x18.
4,x73x1fx20?!c,x27?!c!a,?!c!b,x3cx16,
(1)?!a =?!i?2?!j +3?!k,?!b = 4?!j?5?!k ;
(2)?!a = 3?!i!j +?!k,?!b =!i +2?!j!k,
x5,x49?!c =?!a ￡?!b, x4a?!c!a,?!c!b, x78x67x50:
(1)?!c =?!a ￡?!b =?2?!i +5?!j +4?!k,
(2)?!c =?!a ￡?!b =!i +2?!j +5?!k, (x29x61x2ax2bx55x2cx48)
5,x18?!u = 2?!i +3?!j!k,?!v =?8?!i?5?!j +3?!k, x73?!v1,?!v2,x27?!v =?!v1 +?!v2;?!v1!u ;?!v2==?!u,
x5,x49?!n0 =?!uj?!u j = 1p14 (2;3;?1),pr?!u0?!v = (?!v ￠?!u0)?!u0 =?177 (2;3;?1),x49
!v2 =?17
7 (2
!i +3?!j!k );?!v
1 =?!v!v2 =
2
7 (?11
!i +8?!j +2?!k );
x4a,?!v2==?!u,x3f?!v1 ￠?!u = 0.
6,x18?!u x22x1bx1dx15x6ex6fx1fx20,?!v x22x0x48x1fx20.
(1)x53x54,?!v x3ex2cx48x11x2dx22?!v =?!v1 +?!v2,x3cx16,?!v1!u,?!v2==?!u ;
(2)x7fx10x7ex25?!v1,?!v2 x15x1bx50x29.
x3x4,(1)x1e?!v x47x37x62x11x2dx44,?!v =?!v1 +?!v2 =?!v01 +?!v02,x3cx16,?!v1!u,?!v01!u,?!v2==?!u,?!v02==?!u,
x4a?!v1!v01 =?!v02!v2,x71(?!v1!v01)￠?!u = 0,(?!v02!v2)￡?!u = 0,x23x24?!v1!v01!u,?!v1!v01==?!u,?!u 6= 0,
x5ex25,?!v1!v01 = 0,?!v02!v2 = 0,x4?!v1 =?!v01,?!v2 =?!v02.
(2) x49?!v2 = pr?!u0?!v = (?!v ￠?!u0)?!u0 = (
!v ￠?!u )
j?!u j2
!u,?!v1 =?!v!v2,x4a?!v2==?!u,?!v1 ￠?!u =?!v ￠?!u?
(?!v ￠?!u )
j?!u j2
!u 2 = 0,x4?!v1!u, x3fxb,?!v =?!v1 +?!v2,x4e(1)xbx77x62x11x2dx13x2cx48x15,x21x1bx50x29x22
(?!v
2 =
!v ￠?!u
j?!u j2
!u
!v1 =?!v!v2:
x75x75x75x75x75x71x20
x40x74
x74
x70x72x75x75x40x20x71
x70x74
x74
x75x75x72x71x20
x40
x74
x74
x70x53x53x54x59
x8x20x72x75
x4x18x40x20x71
x71x71
x75x75x75x75x75x75x8x71x71
x71x20
x40x18x4x20x75
x40x18 x40x18
x40x10x30x10
x30x10x30x10
x30x10x30x10
x30x10
x18x40x18x40
x18x40x18x40
x4x18x20x8 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20x49x1d x4ax1c x49x1d x49x1d
x49x1c x49x1d x49x1c
DA
B C
C1
A1 D1
B1
P
C2
x07x1x2
7,x8x9,xaxbABCD?A1B1C1D1x13x14x2fx72x40x10,P x13x14DD1x79x0x1x48x66x17,x74x78C1C2x15x16x17
x13B1,x1ax2dx25x1dx1ex28x66x1fx20x10x23x64x1dx15x1fx20:
(1)!A1P ￡!A1A; (2)?!PC ￡!A1A; (3)!A1C ￡!A1A.
x5,x5bx2bx2ex35x55x56[A1;!A1B1;!A1D1;!A1A],x18Px22DD1x79x0x48x17,x4a:
!A
1P =
!A
1D1 +
!D
1P =
!A
1D1 +k
!D
1D =
!A
1D1 +k
!A
1A = (0;1;k);
!A
1C = (1;1;1):
(1)!A1P ￡!A1A = (0;1;k)￡(0;0;1) = (1;0;0) =!A1B1.
￠ 22 ￠
(2)?!PC =!A1C!A1P = (1;0;1?k),?!PC ￡!A1A = (1;0;1?k)￡(0;0;1) = (0;?1;0) =!A1D1 =
!D
1A1.
(3)!A1C ￡!A1A = (1;1;1)￡(0;0;1) = (1;?1;0) =!A1B1!A1D1 =!A1C2.
8,x78x67x4ex1fx20?!a,?!b x23x2ex2ax15xcxdx6cx38x36x15xfx10:
(1)?!a = 3?!i +4?!j +2?!k,?!b = 2?!i +?!j +?!k ;
(2)?!a =!i +2?!j +?!k,?!b = 2?!i?2?!j,
x5,(1)?!a ￡?!b = 2?!i +?!j?5?!k,j?!a ￡?!b j = p30,x23x24?!a ;?!b x2ex2ax15xcxdx6cx38x36xfx10x22p30.
(2)?!a ￡?!b = 2?!i +2?!j?2?!k,j?!a ￡?!b j = 2p3,x23x24?!a ;?!b x2ex2ax15xcxdx6cx38x36xfx10x222p3.
9,x18?!a ;?!b ;?!c x22x37x37x55x28x74x15x1fx20,x53x54,?!a +?!b +?!c = 0x62x3fx63x62?!a ￡?!b =?!b ￡?!c =?!c ￡?!a,
x3x4,x1e?!a +?!b +?!c = 0,x4ax4fx29x42?!b x3a?!c x2fx30x10x7x3ex50?!a ￡?!b +?!c ￡?!b = 0 x24x68
!a ￡?!c +?!b ￡?!c = 0,x4?!a ￡?!b =?!b ￡?!c =?!c ￡?!a,
x4dx39,x18?!p =?!a +?!b +?!c, x4ex79x53x56x29x3ex50?!p ￡?!b =?!a ￡?!b +?!c ￡?!b = 0x24x68?!p ￡?!c =
!a ￡?!c +?!b ￡?!c = 0,x8x1f?!p 6= 0,x4ax4e?!p ;?!b x28x74x24x68?!p ;?!b x28x74x3ex50?!b x42?!c x28x74,x42x31x18x34x35.
10,x18?!a ;?!b x22x37x55x28x74x15x1fx20,?!AB =?!a +?!b,?!BC = 2?!a +8?!b,?!CD = 3(?!a!b ),x53x54,A;B;D
x34x17x28x74.
x3x4,x26x53A;B;Dx34x17x28x74,x7bx41x53x54,?!AB ￡?!BD = 0x4x3e,x4e
!BD =?!BC +?!CD = 5?!a +5?!b = 5(?!a +?!b ) = 5?!AB;
x3ex50?!AB ￡?!BD = 0,x4,A;B;Dx34x17x28x74.
11,x34x66x1fx20?!OA;?!OB;?!OCx2dx2e
!OB ￡?!OC +?!OC ￡?!OA+?!OA￡?!OB = 0:
x73x53,(1)?!OA;?!OB;?!OCx28xf;
(2)x34x17A;B;Cx28x74.
x3x4,(1)x26x53,?!OA;?!DB;?!OCx28xfx7bx41x53x54:x31x6bx6ex6fx1fx20?!p,x27,?!p ￠?!OA =?!p ￠?!OB =?!p ￠?!OC = 0
x4x3e,x49?!p =?!OB ￡?!OC,x4a?!p ￠?!OB =?!p ￠?!OC = 0,x51x21
!OB ￡?!OC +?!OC ￡?!OA+?!OA￡?!OB = 0;
x3ex50
!p ￡?!OA =?(?!OC ￡?!OA)￠?!OA?(?!OA￡?!OB)￠?!OA = 0:
x1e?!p 6= 0,x4a?!OA;?!OB;?!OCx28xf;x8x1f?!p = 0,x4a?!OBx42?!OCx28x74,x67x47?!OA;?!OB;?!OCx28xf.
(2)x26x53A;B;Cx28x74x7bx41x53x54,?!AB ￡?!AC = 0,x21
!AB =?!OB!OA;?!AC =?!OC!OA;
x23x24
!AB ￡?!AC = (?!OB!OA)￡(?!OC!OA) =?!OB ￡?!OC +?!OA￡?!OB +?!OC ￡?!OA = 0;
x21A;B;Cx34x17x28x74.
12,x8x1f?!a =?!p ￡?!n,?!b =?!q ￡?!n,?!c =?!r ￡?!n,x4a?!a ;?!b ;?!c x28xf,x19x53x54x39.
x3x4,x4ex3c?!n ￠?!a =?!n ￠?!b =?!n ￠?!c = 0,x1e?!n 6= 0,x4a?!a ;?!b ;?!c x28xf; x29x4a,x4e?!n = 0x3ex50
!a =?!b =?!c = 0,x67x28xf.
x1 x2 1–8
￠ 23 ￠
1,x7cx7dx1dx1ex1fx20x42x13x29x28xf:
(1)?!a = 3?!i?2?!j!k,?!b = 5?!i +4?!j?3?!k,?!c = 11?!i!k ;
(2)?!a =?2?!i +3?!j!k,?!b =?!i?4?!j?2?!k,?!c =?!i?5?!j +2?!k ;
(3)?!a =?!i +2?!j +3?!k,?!b = 4?!i?5?!j +?!k,?!c = 7?!i?5?!j +8?!k ;
(4)?!a =!j +3?!k,?!b = 2?!i?2?!j,?!c =!i +2?!j?3?!k,
x5,(1) (?!a ;?!b ;?!c ) =
flfl
flfl
flfl
3?2?1
5 4?3
11 0?1
flfl
flfl
flfl = 88 6= 0,x23x24?!a ;
!b ;?!c x55x28xf.
(2) (?!a ;?!b ;?!c =
flfl
flfl
flfl
2 3?1
1?4?2
1?5 2
flfl
flfl
flfl = 25 6= 0,x23x24?!a ;
!b ;?!c x55x28xf.
(3) (?!a ;?!b ;?!c ) =
flfl
flfl
flfl
1 2 3
4?51
7?5 8
flfl
flfl
flfl =?40 6= 0,x23x24?!a ;
!b ;?!c x55x28xf.
(4) (?!a ;?!b ;?!c ) =
flfl
flfl
flfl
0?1 3
2?2 0
1 2?3
flfl
flfl
flfl = 0,x23x24?!a ;
!b ;?!c x28xf.
2,x78x67x4ex1fx20?!a,?!b,?!c x23x2ex2ax15xcxdxexfx10x15x10x10:
(1)?!a =!i +?!j +2?!k,?!b =?!i!j +3?!k,?!c = 2?!i +?!j +?!k ;
(2)?!a = 3?!i +?!k,?!b = 2?!i!j +?!k,?!c =?!i +5?!j!k ;
(3)?!a =?!i!j!k,?!b =?3?!i?2?!j +?!k,?!c =?!i +?!j ;
(4)?!a =?!i,?!b =?!i +?!j ;?!c =?!i +?!j +?!k,
x5,(1) V = j(?!a ;?!b ;?!c )j =
flfl
flfl
flfl
flfl
flfl
flfl
1 1 2
1?1 3
2 1 1
flfl
flfl
flfl
flfl
flfl
flfl = 15.
(2) V = j(?!a ;?!b ;?!c )j =
flfl
flfl
flfl
flfl
flfl
flfl
3 0 1
2?1 1
1 5?1
flfl
flfl
flfl
flfl
flfl
flfl = j?1j = 1.
(3) V = j(?!a ;?!b ;?!c )j =
flfl
flfl
flfl
flfl
flfl
flfl
1?1?1
3?2 1
1 1 0
flfl
flfl
flfl
flfl
flfl
flfl = j?1j = 1.
(4) V = j(?!a ;?!b ;?!c )j =
flfl
flfl
flfl
flfl
flfl
flfl
1 0 0
1 1 0
1 1 1
flfl
flfl
flfl
flfl
flfl
flfl = 1.
3,x64x1dx1dx1ex6cx17x13x29x28xf:
(1) A(1;2;?1),B(0;1;5),C(?1;2;1),D(2;1;3);
(2) A(3;?2;1),B(2;0;?1),C(?1;?4;5),D(3;?2;4);
(3) A(1;2;?3),B(3;5;?1),C(0;?2;7),D(2;1;3);
(4) A(1;0;1),B(0;?1;2),C(1;2;?2),D(2;0;?21).
x5,x26x64x1dA;B;C;Dx6cx17x13x29x28xf,x7bx41x64x1d?!AB;?!AC;?!ADx77x34x66x1fx20x13x29x28xf,x23x24x7bx41x68
(?!AB;?!AC;?!AD)x13x29x22x6f.
￠ 24 ￠
(1)?!AB = (?1;?1;6),?!AC = (?2;0;2),?!AD = (1;?1;4),
(?!AB;?!AC;?!AD) =
flfl
flfl
flfl
1?1 6
2 0 2
1?1 4
flfl
flfl
flfl = 0; x28xf.
(2)?!AB = (?1;2;?2),?!AC = (?4;?2;4),?!AD = (0;0;3),
(?!AB;?!AC;?!AD) =
flfl
flfl
flfl
1 2?2
4?2 4
0 0 3
flfl
flfl
flfl = 30 6= 0; x55x28xf.
(3)?!AB = (2;3;2),?!AC = (?1;?4;10),?!AD = (1;?1;6),
(?!AB;?!AC;?!AD) =
flfl
flfl
flfl
2 3 2
1?4 10
1?1 6
flfl
flfl
flfl = 30 6= 0; x55x28xf.
(4)?!AB = (?1;?1;1),?!AC = (0;2;?3),?!AD = (1;0;?22),
(?!AB;?!AC;?!AD) =
flfl
flfl
flfl
1?1 1
0 2?3
1 0?22
flfl
flfl
flfl = 45 6= 0; x55x28xf.
4,x64x1dx24A,B,C,Dx22x7ex17x15x6cxfx10x15x10x10:
(1) A(?1;0;1),B(?2;1;4),C(1;3;?3),D(?2;?1;3);
(2) A(2;?1;1),B(5;4;4),C(2;3;?1),D(4;1;2);
(3) A(1;0;2),B(1;?1;0),C(2;2;?1),D(3;1;0);
(4) A(2;2;?1),B(1;2;?2),C(2;?2;1),D(1;1;1).
x5,(1)?!AB = (?1;1;3),?!AC = (2;3;?4),?!AD = (?1;?1;2),
V = 16 j(?!AB;?!AC;?!AD)j = 16 ￡
flfl
flfl
flfl
flfl
flfl
flfl
1 1 3
2 3?4
1?1 2
flfl
flfl
flfl
flfl
flfl
flfl =
1
6,
(2)?!AB = (3;5;3),?!AC = (0;4;?2),?!AD = (2;2;1),
V = 16 j(?!AB;?!AC;?!AD)j = 16 ￡
flfl
flfl
flfl
flfl
flfl
flfl
3 5 3
0 4?2
2 2 1
flfl
flfl
flfl
flfl
flfl
flfl =
j?20j
6 =
10
3,
(3)?!AB = (0;?1;?2),?!AC = (1;2;?3),?!AD = (2;1;?2),
V = 16 j(?!AB;?!AC;?!AD)j = 16 ￡
flfl
flfl
flfl
flfl
flfl
flfl
0?1?2
1 2?3
2 1?2
flfl
flfl
flfl
flfl
flfl
flfl =
10
6 =
5
3,
(4)?!AB = (?1;0;?1),?!AC = (0;?4;2),?!AD = (?1;?1;2),
V = 16 j(?!AB;?!AC;?!AD)j = 16 ￡
flfl
flfl
flfl
flfl
flfl
flfl
1 0?1
0?4 2
1?1 2
flfl
flfl
flfl
flfl
flfl
flfl =
5
3,
￠ 25 ￠
x70x74
x74
x75x75x75x75x75x71x20
x40x74
x74
x70x75x75x75x75x75x40x20x71x75x75x75x75x75x71x20x40
x34x9x8
x8x9x8
x8x9x8
x9x8x8
x9x8
x74
x74
x70
x40x18 x40x18
x40x10x30x10
x30x10x30x10
x30x10x30x10
x30x10
x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x20x8
x1cx48x1cx48x1cx48x1cx48
x1cx48x1cx48x1cx48x1cx48
x1cx48x1cx48x1cx48
x1cx48x1cx48x6x1cx22x9 x53x22 x53x22 x54
x22 x53x22 x53x22 x54x21 x53x22 x53x22 x54x21 x53
x22 x22x34x12 x34x12
x34x12 x34x12
x34x12 x34x12
x34x12 x34x12
x34x13
D
A B
C
C1
A1
D1
B1
x05x1x2
5,x8x9,xaxbx3bx40x10ABCD–A1B1C1D1,jABj = 4,jADj = jAA1j = 2,x73
(1)x17A1x8xcxfC1BDx15x32x33;
(2)x2ex74AD1x42xcxfC1BDx15x32x33.
x5,x18?!DA;?!DC;!DD1x15x14x2fx1fx20x13?!i ;?!j ;?!k,x5bx2bx2ex35x57x55x6a[D;?!i ;?!j ;?!k ],x4a
!DA
1 = (2;0;2);
!DB = (2;4;0);!DC
1 = (0;4;2):
(1)x17A1x8xcxfC1BDx15x32x33
d = j(
!DA
1;
!DB;!DC
1)j
j?!DB ￡!DC1j
=
flfl
flfl
flfl
flfl
flfl
flfl
2 0 2
2 4 0
0 4 2
flfl
flfl
flfl
flfl
flfl
flfl
j(8;?4;8)j =
32
12 =
8
3,
(2) x21x22AD1==BC1,x23x24AD1==xcxfC1BD,x21AD1x79x0x48x17x8xcxfC1BDx15x32x33x4x22AD1x8
xcxfC1BDx15x32x33.
d = j(
!DA;?!DB;!DC
1)j
j?!DB ￡!DC1j
=
flfl
flfl
flfl
flfl
flfl
flfl
2 0 0
2 4 0
0 4 2
flfl
flfl
flfl
flfl
flfl
flfl
12 =
16
12 =
4
3,
6,x18?!a = a1?!e1 +b1?!e2 +c1?!e3,?!b = a2?!e1 +b2?!e2 +c2?!e3,?!c = a3?!e1 +b3?!e2 +c3?!e3,x73x53
(?!a ;?!b ;?!c ) =
flfl
flfl
flfl
a1 b1 c1
a2 b2 c2
a3 b3 c3
flfl
flfl
flfl(?!e1;?!e2;?!e3):
x3x4,?!a ￡?!b = (a1?!e1+b1?!e2+c1?!e3)￡(a2?!e1+b2?!e2+c2?!e3) = a1b2?!e1￡?!e2+a1c2?!e1￡?!e3+b1a2?!e2￡?!e1+
b1c2?!e2￡?!e3 +c1a2?!e3￡?!e1 +c1b2?!e3￡?!e2 = (a1b2?b1a2)?!e1￡?!e2 +(a1c2?c1a2)?!e1￡?!e3 +(b1c2?c1b2)?!e2￡?!e3,
(?!a ;?!b ;?!c ) = (?!a ￡?!b )￠?!c = [(a1b2?b1a2)?!e1￡?!e2+(a1c2?c1a2)?!e1￡?!e3+(b1c2?c1b2)?!e2￡?!e3]￠(a3?!e1+
b3?!e2 +c3?!e3) = [(a1b2?b1a2)c3](?!e1;?!e2;?!e3)?[(a1c2?c1a2)b3](?!e1;?!e2;?!e3)+[(b1c2?c1b2)a3](?!e1;?!e2;?!e3) =fl
flfl
flfl
fl
a1 b1 c1
a2 b2 c2
a3 b3 c3
flfl
flfl
flfl(?!e1;?!e2;?!e3),(x7bx26x33x1ax34x79xdx1ex29x15x1dx4dx34x3ex78x67x50).
7,x73x53,j(?!a ;?!b ;?!c )j6j?!a jj?!b jj?!c j.
x3x4,j(?!a ;?!b ;?!c )j = j(?!a ￡?!b ) ￠?!c j =
flfl
flj?!a ￡?!b jj?!c jcosh?!a ￡?!b ;?!c i
flfl
fl 6 j?!a ￡?!b jj?!c j =
j?!a jj?!b jj?!c jsinh?!a ;?!b i6j?!a jj?!b jj?!c j.
8,x53x54x35x3ex65x36x56x29:
!a ￡(?!b ￡?!c )+?!b ￡(?!c ￡?!a )+?!c ￡(?!a ￡?!b ) = 0:
x3x4,x4ex37x617.7xb:
!a ￡(?!b ￡?!c ) = (?!a ￠?!c )?!b?(?!a ￠?!b )?!c ;
￠ 26 ￠
!b ￡(?!c ￡?!a ) = (?!a ￠?!b )?!c?(?!b ￠?!c )￠?!a ;
!c ￡(?!a ￡?!b ) = (?!b ￠?!c )?!a?(?!a ￠?!c )?!b ;
x65x17x4x50x22x23.
9,x53x54,x70x71x16x6cx17A,B,C,P x28xfx15x30x11x40x26x31x32x13,x38x15x23x2x2cx15x2fx59x1fx20?!a,?!b,?!c,?!p
x2dx2e
(?!p ;?!b ;?!c )+(?!a ;?!p ;?!c )+(?!a ;?!b ;?!p )?(?!a ;?!b ;?!c ) = 0:
x3x4,A;B;C;Px28xfx62x3fx63x62?!PA;?!PB;?!PCx28xf,xaxb
!a =?!OA;?!b =?!OB;?!c =?!OC;?!p =?!OP;
x21
!PA =?!a!p ;?!PB =?!b!p ;?!PC =?!c!p,
x21x4f?!PA;?!PB;?!PCx28xfx62x3fx63x62(?!PA;?!PB;?!PC) = 0,x62x3fx63x62
(?!a!p ;?!b!p ;?!c!p ) = 0;
x62x3fx63x62
(?!a ;?!b!p ;?!c!p )?(?!p ;?!b!p ;?!c!p ) = 0;
x62x3fx63x62
(?!a ;?!b ;?!c )?(?!a ;?!b ;?!p )?(?!a ;?!p ;?!c )?(?!p ;?!b ;?!c ) = 0;
x62x3fx63x62
(?!p ;?!b ;?!c )+(?!a ;?!p ;?!c )+(?!a ;?!b ;?!p )?(?!a ;?!b ;?!c ) = 0:
10,x53x54
(1) (?!a ￡?!b )￡(?!c ￡?!d ) = (?!a ;?!b ;?!d )?!c?(?!a ;?!b ;?!c )?!d ;
(2) (?!a ￡?!b )￡(?!c ￡?!d ) = (?!a ;?!c ;?!d )?!b?(?!b ;?!c ;?!d )?!a,
x3x4,(1) (?!a ￡?!b )￡(?!c ￡?!d ) =?(?!c ￡?!d )￡(?!a ￡?!b ) =?[?!c ￠(?!a ￡?!b )?!d!d ￠(?!a ￡?!b )?!c ] =
(?!a ;?!b ;?!d )?!c?(?!a ;?!b ;?!c )?!d,
(2) (?!a ￡?!b )￡(?!c ￡?!d ) = [?!a ￠(?!c ￡?!d )]?!b?[?!b ￠(?!c ￡?!d )]?!a = (?!a ;?!c ;?!d )?!b?(?!b ;?!c ;?!d )?!a,
11,x53x54x2x0x1x6cx66x1fx20?!a ;?!b ;?!c ;?!d x38x47
(?!b ;?!c ;?!d )?!a +(?!c ;?!a ;?!d )?!b +(?!a ;?!b ;?!d )?!c +(?!b ;?!a ;?!c )?!d = 0:
x3x4,x4ex3d10x61x22x23x3ex50
(?!a ;?!b ;?!d )?!c?(?!a ;?!b ;?!c )?!d = (?!a ;?!c ;?!d )?!b?(?!b ;?!c ;?!d )?!a ;
x39x3ax7x12x3bx62x3cx3dx3ex54x10x16x1fx20x48x3fx4x3ex53x50.
12,x53x54x1dx1ex1fx20x36x56x29:
(1) (?!a ￡?!b ;?!b ￡?!c ;?!c ￡?!a ) = (?!a ;?!b ;?!c )2;
(2) (?!b ￡?!c )￡(?!a ￡?!d )+(?!c ￡?!a )￡(?!b ￡?!d )+(?!a ￡?!b )￡(?!c ￡?!d ) =?2(?!a ;?!b ;?!c )?!d ;
(3) (?!a!d )￠(?!b!c )+(?!b!d )￠(?!c!a )+(?!c!d )￠(?!a!b ) = 0;
(4) (?!a!d )￡(?!b!c )+(?!b!d )￡(?!c!a )+(?!c!d )￡(?!a!b ) = 2(?!a ￡?!b +?!b ￡?!c +?!c ￡?!a ).
x3x4,(1) (?!a ￡?!b ;?!b ￡?!c ;?!c ￡?!a ) = [(?!a ￡?!b ) ￡ (?!b ￡?!c )] ￠ (?!c ￡?!a ) = [(?!a ;?!b ;?!c )?!b?
(?!b ;?!c ;?!b )?!a ]￠(?!c ￡?!a ) = (?!a ;?!b ;?!c )￠(?!c ;?!a ;?!b ) = (?!a ;?!b ;?!c )2.
￠ 27 ￠
(2) (?!b ￡?!c ) ￡ (?!a ￡?!d ) =?(?!a ￡?!d ) ￡ (?!b ￡?!c ) =?(?!b ;?!c ;?!a )?!d + (?!b ;?!c ;?!d )?!a,(?!c ￡
!a )￡(?!b ￡?!d ) =?(?!b ￡?!d )￡(?!c ￡?!a ) =?(?!c ;?!a ;?!b )?!d +(?!c ;?!a ;?!d )?!b,(?!a ￡?!b )￡(?!c ￡?!d ) =
(?!c ;?!d ;?!a )?!b?(?!c ;?!d ;?!b )?!a,x23x24
(?!b ￡?!c )￡(?!a ￡?!d )+(?!c ￡?!a )￡(?!b ￡?!d )+(?!a ￡?!b )￡(?!c ￡?!d ) =?2(?!a ;?!b ;?!c )?!d,
(3) (?!a!d )￠(?!b!c ) =?!a ￠?!b!a ￠?!c!b ￠?!d +?!c ￠?!d,
(?!b!d )￠(?!c!a ) =?!b ￠?!c!a ￠?!b!c ￠?!d +?!a ￠?!d,
(?!c!d )￠(?!a!b ) =?!a ￠?!c!b ￠?!c!a ￠?!d +?!b ￠?!d,
x76x79x53x56x29x1dx0 x1ex37x61x11x12x65x17x4a:
(?!a!d )￠(?!b!c )+(?!b!d )￠(?!c!a )+(?!c!d )￠(?!a!b ) = 0:
(4) (?!a!d )￡(?!b!c ) =?!a ￡?!b!a ￡?!c +?!b ￡?!d!c ￡?!d,
(?!b!d )￡(?!c!a ) =?!b ￡?!c +?!a ￡?!b +?!c ￡?!d!a ￡?!d,
(?!c!d )￡(?!a!b ) =!a ￡?!c +?!b ￡?!c +?!a ￡?!d!b ￡?!d,
x76x79x53x56x29x1dx0x1ex37x61x11x12x65x17x50:
(?!a!d )￡(?!b!c )+(?!b!d )￡(?!c!a )(?!c!d )￡(?!a!b ) = 2(?!a ￡?!b +?!b ￡?!c +?!c ￡?!a ):
13,x53x54x2x3cx0x1x1fx20ri (i = 1;2;3;4),x1dx29x2ax2b:
(?!r1 ￡?!r2)(?!r3 ￡?!r4)+(?!r1 ￡?!r3)(?!r4 ￡?!r2)+(?!r1 ￡?!r4)(?!r2 ￡?!r3) = 0:
x3x4,x3dx3ex1dx58.7,(?!r1 ￡?!r2)￠(?!r3 ￡?!r4) = (?!r1 ￠?!r3)(?!r2 ￠?!r4)?(?!r1 ￠?!r4)(?!r2 ￠?!r3),
(?!r1 ￡?!r3)￠(?!r4 ￡?!r2) = (?!r1 ￠?!r4)(?!r2 ￠?!r3)?(?!r1 ￠?!r2)(?!r3 ￠?!r4),
(?!r1 ￡?!r4)￠(?!r2 ￡?!r3) = (?!r1 ￠?!r2)(?!r3 ￠?!r4)?(?!r1 ￠?!r3)(?!r2 ￠?!r4),
x76x79x53x56x29x15x1dx0x1ex37x61x11x12x65x17x7x50x8x22x23.
14,x53x54?!a ;?!b ;?!c x28xfx15x30x11x40x26x31x32x13?!b ￡?!c,?!c ￡?!a,?!a ￡?!b x28xf.
x3x4,x21x22?!b ￡?!c,?!c ￡?!a,?!a ￡?!b x28xfx62x3fx63x62(?!a ￡?!b ;?!b ￡?!c ;?!c ￡?!a ) = 0,x71x4e12x61
x15(1)xb,(?!a ￡?!b ;?!b ￡?!c ;?!c ￡?!a ) = (?!a ;?!b ;?!c )2,x23x24?!a,?!b,?!c x28xfx62x3fx63x62?!a ￡?!b,?!b ￡?!c,
!c ￡?!a x28xf.
x1 x2 1–9
1,x34x35x36ABCx40x38x15x37x66x61x17x22B(?3;0),C(3;0),x7ex17Ax6bx2ex747x?5y?35 = 0x79x39x41,x73
x34x35x36x24x7fx15x42x43.
x5,x18x24x7fx15x57x55x22(x;y),x4a
( x = xA +xB +xC
3 =
xA
3
y = yA +yB +yC3 = yA3 ;
x4
( x
A = 3x
yA = 3y:
x25(xA;yA)x2dx2ex40x417x?5y?35 = 0,x51x52x4x5021x?15y?35 = 0.
￠ 28 ￠
-
6
x12x13x12
x13x12x13
x12x13x12
x15x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x2bx12x13x12
x13x12x13
x12x13x12
x13x12x13
x12x13x12
x15x2x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x2bxe
x5
x49x1d x49x1d x4ax1c x49x1c
x49x1d x49x1d x4ax1c x49x1c
x49x1d x49x1d
A BP O
M
N
Q
x
y
x02x1x2
2,x6bx3bx22lx15x74x78ABx79x47x48x41x17P,x6bABx15x43x44,x24AP,PBx22x38x11x12x2fx56x38x34x35x36AMPx3a
BNP,x73MNx15x16x17Qx15x42x43.
x5,x24ABx15x16x17Ox22x4bx17,x24ABx22xx45,x5bx2bx2ex35x57x55x6a[O;?!i,?!j ],x3cx13
A
l2 ;0
; B
l
2 ;0
; P(t;0);
l2 < t < l2
:
x4a
M
0
@ t?
l
2
2 ;
p3
2
t+ l2
1
A; N
0
@ t+
l
2
2 ;
p3
2
l
2?t
1
A:
x23x24MNx15x16x17Qx15x57x55x22:8
<
:
xQ = t2
l2 < t < l2
·
yQ =
p3
4 l:
x4Qx17x15x42x43x40x41x22,y =
p3
4 l
l4 < x < l4
·
.
￠ 29 ￠
x0x1x2 x3x4x5x6
x1 x2 1–1
1,x8x9,xaxbxcxdxexfx10ABCD?A1B1C1D1,Ex0 F x11x12x13x14BCx0 C1D1x15x16x17,x18?!AB =?!a,
!AD =?!b,!AA
1 =?!c, x19x1a?!a,
!b,?!c x1bx1cx1dx1ex1fx20:
(1)!AC1; (2)!BD1; (3)?!AF; (4)?!EF.
x5,(1)x21x22
!BC =?!AD;!CC
1 =
!AA
1;
!AC
1 =
!AB +?!BC +!CC
1;
x23x24
!AC
1 =?!a +
!b +?!c,
(2)x21x22!BD1 =?!BD +!DD1,x25
!BD =?!AD!AB =?!b!a ;!DD
1 =
!AA
1:
x23x24
!BD
1 =
!b!a +?!c,
(3)?!AF =?!AD +!DD1 +!D1F,x25
!DD
1 =
!AA
1;
!D
1F =
1
2
!D
1C1 =
1
2
!AB;
x23x24
!AF = 1
2
!a +?!b +?!c,
(4)?!EF =?!AF!AE =?!AF?(?!AB +?!BE) =?!AF?(?!AB +?!BE) =?!AF?
!
AB + 12?!BC
·
= 12?!a +
!b +?!c!a? 1
2
!b =? 1
2
!a + 1
2
!b +?!c,
x75x75x75x8x1cx1dx1c
x1dx1cx1d
x17x3x8x75x75x75
x3x17x1dx1c
x1dx1cx1d
x1c
x1x13xcxc
xcxcxc
xcxcxc
xcxcxc
x75x75x75x8x1cx1dx1c
x1dx1cx1d
x17x3xcxc
xcxcxc
xcxcxc
xcxcxc
x13x1
xcxcxc
xcxcxc
xcxcxc
xcxcx13x1
x7x1fx3ax9x21x46
x1x12x6x2x3x13
x3x2x14x5x27
x76x2exfx0x10x2 x7dx2fxfx1x12x8x3x2x14x5x27 x3x1cx1cx2e
x3ex17 x3ex17
x3fx17 x3ex17
x17x3
x31x11x31x10
x31x11x31x10
x31x11x31x10
x31x11x31x10
x31x11x31x10x3fx17x17x3fx18x3e
x17x3fx18x3e
x17x3fx18x3e
x3fx13 x36x13
x36x13 x36x13
x36x13 x36x13
x36x13 x36x13
x6ex2fx6ex2f
x6ex2fx6ex2f
x6fx2fx6ex2f
x6ex2fx6ex2f
x6fx2fx6ex2f
x6ex2fx6ex2f
x43x1a x43x1a
x43x19 x43x1a
x43x1a x44x19
x43x19 x43x1a
x43x1a x43x19
x43x1a x43x1a
x44x19 x43x19
x43x1a
x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x8
A B
D C
A1 B1
D1 C
1
F
E
!a
!c?!
b
x01x1x2
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6exex75x75x75x75x75x75x75x75x28x28x28x28x28x28
x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x44x1ax3fx1ex2fx30
x7x3bx3ax9x45x46
x67x63xdx76x75
!b!a?!b
!a
x03(1)x1x2
2,x26x27x1dx1ex28x29x2ax2b,x1fx20?!a ;?!b x2cx2dx2ex2fx30x31x32?
(1) j?!a +?!b j = j?!a j+j?!b j;
(2) j?!a +?!b j = j?!a j?j?!b j;
(3) j?!a!b j = j?!a j?j?!b j;
(4) j?!a!b j = j?!a j+j?!b j.
x5,(1) x33x1a“x34x35x36x37x38x39x3ax3bx3cx3dx34x38”x3exb,?!a ==?!b, x3fx26x27j?!a +?!b j = j?!a j+j?!b jx40x41,?!a
x42?!b x43x1f,x44?!a,?!b x16x45x46x47x48x220.
￠ 1 ￠
(2)x49?!c =?!a +?!b,x4a?!a =?!c!b,x4bx29x4cx22,j?!c!b j = j?!c j+j?!b j,x23x24?!b ==?!c x3fx4dx1f,x4e
x4fx3ex50,?!a ==?!b,x4dx1f,x3fj?!a j>j?!b j,x44?!b = 0.
(3)x49?!c =?!a!b,x4a?!a =?!b +?!c,x4bx29x4cx22,j?!b j+j?!c j = j?!b +?!c j,x4e(1)xb,?!b ==?!c x3fx43x1f.
x23x24?!a ==?!b x3fx43x1f,x51x21j?!a!b j> 0,x23x24j?!a j>j?!b j,x44?!b = 0.
(4)x49?!c =?!a!b,x4a?!a =?!b +?!c,x4bx29x4cx22,j?!b +?!c j = j?!c j?j?!b j,x4e(2)xb,?!c ==?!b x3fx4dx1f,
x44?!b = 0,x43x52,j?!c j>j?!b j,x23x24?!a ==?!b x3fx4dx1f,x44?!b = 0x44?!a = 0.
3,x53x54x1dx1ex55x56x29,x57x58x54x56x59x2fx30x52x5ax2ax2b.
(1) j?!b!a j>j?!a j?j?!b j;
(2) j?!a +?!b +?!c j6j?!a j+j?!b j+j?!c j.
x3x4,(1)x8x9,x33x1a“x34x35x36x37x38x39x5bx5cx3cx3dx34x38”x3ex50x5dx53x15x55x56x29.x56x29x2ax2bx15x31x32x3ex5ex5fx60
x612(3),?!a ==?!b,x43x1f,x3fj?!a j>j?!b j,x44?!b = 0.
(2)x49?!d =?!b +?!c, x4a,j?!a +?!b +?!c j = j?!a +?!d j6j?!a j+j?!d j = j?!a j+j?!b +?!c j6j?!a j+j?!b j+j?!c j.
x56x59x2ax2bx62x3fx63x62(i)?!a,?!b,?!c x64x65xcxdx3fx43x1f,x44(ii)?!a,?!b,?!c x16x45x46x37x66x220(x67x3ex68x2a(i)x15x69
x6a).
4,x6bx6cx38x36ABCDx16,?!AB =?!a + 2?!b,?!BC =?4?!a!b,?!CD =?5?!a?3?!b (?!a,?!b x6dx13x6ex6f
x1fx20),x53x54ABCDx22x70x36.
x3x4,x6?!AD =?!AB +?!BC +?!CD =?8?!a? 2?!b = 2?!BC,x7?!AD==?!BC,x71j?!ADj = 2j?!BCj,x23x24
ABCDx13x70x36.
5,x18ABCDEFx22x72xex38x36,x73?!AB +?!AC +?!AD +?!AE +?!AF.
x5,x6?!AD =?!AC +?!AF =?!AE +?!AB,x7?!AB +?!AC +?!AD +?!AE +?!AF = 3?!AD.
x12x13x12
x13x12x13
x12x13x12
x13x12x13
x12x13x12
x15x2x75x75x75x75x40x50x20x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x2bxe
x2x15x12x13
x12x13x12
x13x12x13
x12x13x12
x13x12x13
x12
x20x50x40x75x75x75x75
xex2bx6ex6d
x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x6dx6ex6d
x6e
x75x75x75x75x75x75x75x75x72x40x50x8x6x46x2ex2dx2ex2d
x2ex2dx2ex2e
x2dx2ex2dx2e
x2dx2ex2dx2e
x2dx2ex2dx2e
x2dx2ex2dx2e
x2dx2e
x52x53x52x53
x52x53x52x53
x52x53x52x53
x52x53x52x53
x52x53x52x52
x53x52x53x52
x53x52x5axa
xbxcx4x3dx1b
xcx63x65x75x74
x4x43x25x35x34
x4bx4cxcx5dx5b
x7x3bx3ax9x45x46A D
F
B
E
F
x05x1x2
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6exex75x75x75x75x75x75x75x75x28x28x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28
x28x44x1a
x2xcxdxd
xcxdxd
xcxdxd
xcxdxd
xcxdxd
xcxd
x36x36x36x36x36x36x36x36
x36x36x36x36x36x36x36x36
x36x36x36x37x36x36x36x36
x36x36x36x1dx7xax56x4dx4ex4dx4dx4dx4ex4dx4d
x4dx4ex4dx4dx4dx4e
x4dx4dx4dx4ex4dx4d
x6x6x1x13x3ax15x2
x2fx30x8x3cx3cx47x48x8x55x5ax25
B L C
N M
A
x06x1x2
6,x18L;M;Nx11x12x13x34x35x36ABCx15x34x38BC;CA;ABx15x16x17,x53x54x34x16x74x1fx20?!AL,!BM,!CNx3e
x24x75x2ax48x66x34x35x36.
x3x4,x21x22?!AL;!BM;!CNx3ex24x75x2ax48x66x34x35x36,x62x3fx63x62x76x77x34x66x1fx20x39x3ax22x6fx1fx20,x4e
!AL = 1
2 (
!AB +?!AC);!BM = 1
2 (
!BA+?!BC);!CN = 1
2 (
!CA+?!CB);
x3ex50,?!AL+!BM +!CN = 0.
7,x6bx34x35x36ABCx16x73x48x17O,x27
!OA+?!OB +?!OC = 0:
￠ 2 ￠
x5,x18Lx13BCx15x16x17,x4a?!AL = 12 (?!AB+?!AC),x6bx74x78ALx79x7aOx17,x27?!AO = 23?!AL = 13 (?!AB+?!AC),
x4aOx17x6b4ABCx15x7bx7c,x7dx53:
!OA+?!OB +?!OC =?!OA+(?!OA+?!AB)+(?!OA+?!AC) = 3?!OA+?!AB +?!AC
=?(?!AB +?!AC)+?!AB +?!AC = 0:
8,x18A;B;C;Dx13x48x66x6cxfx10x15x6cx66x7ex17,M;Nx11x12x13x38AB,CDx15x16x17,x53x54:
!MN = 1
2 (
!AD +?!BC):
x3x4,x8x9,
!CM = 1
2 (
!CA+?!CB);!CN = 1
2
!CD;
x23x24
!MN =!CN!CM = 1
2
!CD? 1
2 (
!CA+?!CB) = 1
2 (
!AD +?!BC):
x1ax28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x73x73
x73x73
x73x60
xcx1dx1dx1d
x1dx1dx1e
x1dx1dx1d
x1dx1dx1d
x1dx1dx1d
x3fx4
xax58x51x51x52x51
x51x51x51x51x51
x51
x1x11x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5
x5
x5x51x50x61x62
x5cx26 x5cx26
x5cx26 x5dx26
x5cx27 x5cx26
x5cx26 xbx4dx1e x4dx1f x4d
x1e x4dx1e x4cx1f x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f
x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f x4dx1e x4dx1f
B
D
C
A
M
N
x08x1x2
x30x31x30x30x31
x30x30x31x30x31
x30x30x31x30x30
x31x30x30x31x30
x30x31x30x31x30
x30x31x30x30x31
x30x30
x2x37xexf
xexfxf
xexfxe
xfxexf
xexfxe
xf
x8x20x50x75x75x75x75x75x75xfxexf
xexfxe
xfxexf
xexfxf
xexfxe
x37x2x75x75x75x75x75x75x50x20x8x27x5ex5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5c
x5
A
C
B
D
M
O
x09x1x2
9,x18Mx13xcxdx6cx38x36ABCDx15x16x7f,Ox13x0x1x48x17,x53x54:
!OA+?!OB +?!OC +?!OD = 4!OM:
x3x4,x8x9,x21x22
!OM = 1
2 (
!OA+?!OC);!OM = 1
2 (
!OB +?!OD);
x23x24
!OA+?!OB +?!OC +?!OD = 4!OM:
10,x18ABCDx13xcxdx6cx38x36,P;Qx11x12x13x38BC;CDx15x16x17,x53x54AP;AQx42x2x35x74BDx65x3x3c
E;F,x25x76BDx34x56x11.
x3x4,x18?!AB =?!a,?!AD =?!b,x4a
!BD =?!AD!AB =?!b!a ;
!AP =?!AB + 1
2
!BC =?!a + 1
2
!b ;
!AQ =?!AD + 1
2
!DC =?!b + 1
2
!a,
x51x18
!AE = k?!AP (k > 0);?!AF = m?!AQ (m > 0);
x4a
!AE = k?!a + k
2
!b ;?!AF = m?!b + m
2
!a,
x71x13
!AE =?!AB +t?!BD =?!a +t(?!b!a ) = (1?t)?!a +t?!b (t > 0):
￠ 3 ￠
x23x24
k?!a + k2?!b = (1?t)?!a +t?!b ;
x4:
(k +t?1)?!a =
t? k2
!
b ;
x4ex3c?!a x42?!b x55xcxd,x23x24 (
k +t?1 = 0
t? k2 = 0 x4
( k = 2
3
t = 13,
x43x5,x4e
!AF =?!AB +s?!BD = (1?s)?!b +s?!b (s > 0);
x3ex50,( m
2 +s?1 = 0
s?m = 0; x4:
( m = 2
3
s = 23,
x6x7x50x8:
!BF = 2
3
!BD;?!BE = 1
3
!BD;
x58x54E;Fx13x74x78BDx15x34x56x11x17.
11,Ox22x72x9x38x36A1A2￠￠￠Anx15x16x7f,x53x54:
!OA
1 +
!OA
2 +￠￠￠+
!OA
n = 0:
x3x4,xaxbxcnx22xdxex15xfx36,x4fx52.x10x11x47,!OA1 +￠￠￠+!OAn = 0,x12x68nx22x13xex15xfx36,x14x15x16
x17x48x18x7ex17B1;￠￠￠ ;Bnx27x4bx19x72nx38x36A1￠￠￠Anx2ax22,A1B1A2B2￠￠￠An?1Bn?1AnBn,x77x13x48x662nx38
x36,x23x24
!OA
1 +
!OB
1 +
!OA
2 +
!OB
2 +￠￠￠+
!OA
n +
!OB
n = 0:
x1ax1x8!OBix13x4e!OAix1bx1cx48x66x1dx35 x25x50x8(0 < < …),x1ex1f:
!p =!OA1 +￠￠￠+!OAn;
!q =!OB1 +￠￠￠+!OBn;
x20x30?!q x13x4e?!p x1bx1c x35x25x50x8,x4ex3c0 < < …,?!q x42?!p x55xcxd,x21?!p +?!q = 0x62x3fx63x62?!p =?!q = 0.
12,Ox22x72x9x38x36A1A2￠￠￠Anx15x16x7f,Px13x0x1x48x17,x53x54:
!PA
1 +
!PA
2 +￠￠￠+
!PA
n = n
!PO:
x3x4,x21x22
!PO =!PA
i +
!A
iO (i = 1;2;￠￠￠ ;n);
x23x24
n?!PO =!PA1 +￠￠￠+!PAn +(!A1O +￠￠￠+!AnO) =!PA1 +￠￠￠+!PAn
(x33x1ax3d11x61x15x22x23).
x1 x2 1–2
1,xaxbxcxdx6cx38x36ABCDx15x2x35x74x22ACx3aBD,x18?!AC =?!a,?!BD =?!b, x73?!AB,?!CD,?!DA.
x5,x8x9,
!AB =?!AO +?!OB = 1
2
!AC + 1
2
!DB = 1
2 (
!a!b );
￠ 4 ￠
!CD =!AB =? 1
2 (
!a!b );
!DA =!AD =?(?!AO +?!OD) =? 1
2 (
!a +?!b ):
x30x31x30x30x31
x30x30x31x30x31
x30x30x31x30x30
x31x30x30x31x30
x30x31x30x31x30
x30x31x30x30x31
x30x30
x2x37xexf
xexfxf
xexfxe
xfxexf
xexfxe
xf
x8x20x50x75x75x75x75x75x75xfxexf
xexfxe
xfxexf
xexfxf
xexfxe
x37x2x75x75x75x75x75x75x50x20x8x27x5ex5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5c
x1ax44x29x37x37x53x5ax25xax66x63xd
A
C
B
D
O
x01x1x2
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6exex75x75x75x75x75x75x75x75x28x28x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28
x28x28x44x1a
x17x16x17
x17x16x17
x17x16x17
x17x16x17
x17x16x17
x17x16x3
x1x2x3x3
x2x3x3
x2x3x3
x2x3x3
x2x3x3
x2x3
A BNM
C
x02x1x2
2,x6bx34x35x36ABCx16,x17M;Nx13ABx38x79x15x34x56x11x17,x18?!CA =?!a,?!CB =?!b, x73!CM,!CN.
x5,x8x9,x21x22
!AM = 1
3
!AB;?!AN = 2
3
!AB;
x23x24
!CM =?!CA+!AM =?!CA+ 1
3
!AB =?!CA+ 1
3 (
!CB!CA) = 1
3
!b + 2
3
!a ;
!CN =?!CA+?!AN =?!CA+ 2
3
!AB =?!CA+ 2
3 (
!CB!CA) = 2
3
!b + 1
3
!a,
3,x6bx6cxfx10O?ABCx16,x18x17Gx13x34x35x36ABCx15x24x7f,x1a?!OA,?!OB,?!OCx19x1bx1cx1fx20?!OG.
x5,x8x9,x4e
!OG =?!OA+?!AG;?!AG = 2
3
!AD;
!AD = 1
2 (
!AB +?!AC);
x3ex50?!AG = 13 (?!AB +?!AC),x25?!AB =?!OB!OA,?!AC =?!OC!OA,x23x24
!AG = 1
3 (
!OB +?!OC?2?!OA);?!OG = 1
3 (
!OA+?!OB +?!OC):
x1x10x2x2
x3x2x2
x2x2x2
x2x3x2
x2x2x2
x2x2x2
x3x2x2
x2
x25x59x51x52x52x52
x52x51x52x52x52
x51x52x52
x71x71
x71x71
x71x71
x71x18x6cx6cx6cx6cx6c
x6cx6cx6c
x6cx6cx6c
x2bxd
x1bx1cx1b
x1bx1cx1b
x1cx1bx1c
x1b
x70x0xcxb
x1fx4ex1fx4dx1fx4ex1fx4ex1fx4ex1fx4ex1fx4dx1fx4ex1fx4ex1fx4ex1fx4ex1fx4d
x1fx4ex1fx4ex8x1fx21x9 x51x21 x53x20 x51x22 x51x22 x51x22 x51x22 x51x22 x51x22 x51x22 x51x22
x51x22
x11x33x11x33
x12x33x12x33
x11x33x11x33
x12x33x11x33
x11x33x12x32
x1x33
B
D
C
G
O
A
x03x1x2
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6exex75x75x75x75x75x75x75x75x28x28x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x44x1a
x35x36x35x35x36x35x35
x36x35x35x36x35x35x36
x35x35x36x35x35x36x35
x35x36x35x35x36x35x7x6x1cx47x2fx7x3cx3b
xfx2ex6fx66x69
A
T
C
B
x04x1x2
4,x18ATx13x34x35x36ABCx16\Ax15xcx11x74(x42BCx3x3cTx17),x76?!ATx1a?!AB;?!ACx19x1bx25.
x5,x18?!BT = k?!BC,x4a?!TC = (1?k)?!BC,x4ex35xcx11x74x15x26x27x3exb,j?!ABj, j?!ACj = k, (1?k),x21x4f
k = j
!ABj
j?!ABj+j?!ACj, x3cx13
!AT =?!AB +?!BT =?!AB +k?!BC = (1?k)?!AB +k?!AC
= 1
j?!ABj+j?!ACj
(j?!ACj?!AB +j?!ABj?!AC):
5,xaxb?!a ;?!b x55x28x74,x4ax1fx20?!c = 3?!a +?!b x42?!d = 2?!a!b x13x29x74x26x65x2a?
￠ 5 ￠
x5,x18x47kx0 mx27,k?!c +m?!d = 0,x4
3k?!a +k?!b +2m?!a?m?!b = 0;
x2bx5x7x22
(3k +2m)?!a +(k?m)?!b = 0:
x4ex3c?!a ;?!b x55x28x74,x21?!a ;?!b x74x26x2cx2a,x23x24(
3k +2m = 0
k?m = 0 x2dx50k = m = 0;
x4?!c ;?!d x74x26x2cx2a.
6,x53x54x34x66x1fx20k1?!a?k2?!b,k2?!b?k3?!c,k3?!c?k1?!a x28xf.
x3x4,x4ex56x29
(k1?!a?k2?!b )+(k2?!b?k3?!c )+(k3?!c?k1?!a ) = 0;
x3exbx773x66x1fx20x74x26x65x2a,x23x24x28xf.
7,Ox13x48x66x1dx17,x53x54:x2x3cx55x6bx48x2ex74x79x153x66x17A;B;C,x17Mx2fx3cxcxfABCx79x15x30x11x40x26
x31x32x13x31x6bx32xek1;k2;k3,x27x50
!OM = k
1
!OA+k
2
!OB +k
3
!OC; x3fk
1 +k2 +k3 = 1.
x3x4,xaxbAx0 Bx0 C x34x17x55x28x74,x21?!AB;?!AC x74x26x2cx2a,x0x1x17M x2fx3cxcxfABC x79x62x3fx63x62
!AM;?!AB;?!ACx28xf,x4,!AM;?!AB;?!ACx74x26x65x2a,x62x3fx63x62x31x6bx55x33x220x15x32xem
1;m2;m3,x27
m1!AM +m2?!AB +m3?!AC = 0;
x62x3fx63x62x2x3cx1dx17Ox47:
m1(!OM!OA)+m2(?!OB!OA)+m3(?!OC!OA) = 0;
x62x3fx63x62
m1!OM = (m1 +m2 +m3)?!OA?m2?!OB?m3?!OC:
x10x11m1 6= 0,x55x11x42?!AB;?!ACx74x26x2cx2ax34x35,x21x4fx1ex1f:
k1 = 1m
1
(m1 +m2 +m3); k2 =? m2m
1;
k3 =? m3m
1;
x4a
!OM = k
1
!OA+k
2
!OB +k
3
!OC;
x3fk1 +k2 +k3 = 1.
8,O x13x48x66x1dx17,x53x54,x17 M x2fx3c4ABC x79 (x36x37x38x15x38) x15x30x11x40x26x31x32x13x31x6bx6ex39x32xe
k1;k2;k3,x27x50
!OM = k
1
!OA+k
2
!OB +k
3
!OC; x3fk
1 +k2 +k3 = 1.
x3x4,x3ax3bAM,x40x3ex3BCx3cDx17,x21x4f!AM = l?!AD,x3cx160 6 l 6 1,x4ex3cDx6bx74x78BCx79,x3dx3e
x6a2.1,x31x6bx32xem1;m2,x27x50
!OD = m
1
!OB +m
2
!OC; m
1 +m2 = 1;m1;m2 > 0:
x3cx13
!OM =?!OA+!AM = (1?l)?!OA+l?!OD = (1?l)?!OA+lm
1
!OB +lm
2
!OC:
￠ 6 ￠
x49k1 = 1?l,k2 = lm1,k3 = lm2,x4x50
!OM = k
1
!OA+k
2
!OB +k
3
!OC; k
1 +k2 +k3 = 1; k1;k2;k3 > 0:
x4dx39,x55x3fx18k1 6= 1,x2dx40x41x42
8>
<
>:
1?l = k1
lm1 = k2
lm2 = k3
x3ex50
8>
><
>>:
l = 1?k1;
m1 = k21?k
1;
m2 = k31?k
1;
x4ax47
m1 +m2 = 1; m1;m2 > 0; 0 < l 6 1:
x49
!OD = m
1
!OB +m
2
!OC;
x4aDx17x6bx74x78BCx79,x4e
!OM = (1?l)?!OA+l?!OD
x3ex24x50x25!AM = l?!AD,x21x4fMx6bx74x78ADx79,x43x25x6b4ABCx79.
9,x53x54:x0x1x55x43x15x34x17A;B;Cx28x74x15x30x11x40x26x31x32x13x31x6bx55x33x22x6fx15x32xek1;k2;k3,x27x50
0 = k1?!OA+k2?!OB +k3?!OC; x3fk1 +k2 +k3 = 0:
x3x4,Ax0 Bx0 Cx28x74,x62x3fx63x62l?!AB +m?!AC = 0 (l;mx6dx55x22x6f),x62x3fx63x62
l(?!OB!OA)+m(?!OC!OA) = 0;
x62x3fx63x62
(l +m)?!OA+l?!OB +m?!OC = 0:
x49k1 =?(l +m),k2 = l,k3 = m,x10x11x38x15x55x33x22x6f,x3f:
k1?!OA+k2?!OB +k3?!OC = 0; k1 +k2 +k3 = 0:
10,x53x54:x0x1x55x43x15x6cx17A;B;C;Dx28xfx15x30x11x40x26x31x32x13x31x6bx6cx66x55x33x22x6fx15x32xe,x27x50
0 = k1?!OA+k2?!OB +k3?!OC +k4?!OD; x3fk1 +k2 +k3 +k4 = 0:
x3x4,Ax0 Bx0 Cx0 Dx28xfx62x3fx63x62?!AB;?!AC;?!ADx74x26x65x2a,x62x3fx63x62x47x55x33x22x6fx15xel;m;nx27:
l?!AB +m?!AC +n?!AD = 0;
x62x3fx63x62
l(?!OB!OA)+m(?!OC!OA)+n(?!OD!OA) = 0;
x62x3fx63x62
(l +m+n)?!OA+l?!OB +m?!OC +n?!OD = 0:
x1fk1 =?(l +m+n),k2 = l,k3 = m,k4 = n,x10x11x38x15x55x33x22x6f,x27x50
k1?!OA+k2?!OB +k3?!OC +k4?!OD = 0; k1 +k2 +k3 +k4 = 0:
￠ 7 ￠
x13x12x13
x13x12x13
x13x12x13
x13x12x13
x13x12x13
x13x12x3x6fx6ex6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
xex75x75x75x75x75x75x75x75x28x28x28x28
x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x44
x1a
x35x34x35x35x35x34x35
x35x35x34x35x35x35x34
x35x35x35x34x35x35x35
x34x35x35x35x34x4bx1dx6xax9x56x4bx4cx4bx4cx4bx4bx4c
x4bx4cx4bx4cx4bx4cx4b
x4bx4cx4bx4cx4bx4cx4b
A F B
C
DE
M
x011x1x2
11,x1ax1fx20x15x40x44x53x54x45x46x1dx5,x1ex34x35x36ABCx15x34x31x38AB,BC,CAx47x48x49x11x4ax2aAF, FB =
k1, k2,BD, DC = k3, k1,CE, EA = k2, k3,x3cx16,k1;k2:k3x4bx22x72xe,x4ax34x35x36ABCx15x7ex17x42x38x2
x38x15x11x17x15x4cx74x3x3cx48x17M,x3fx2x3cx0x1x48x17Ox47
!OM = 1
k1 +k2 +k3(k2
!OA+k
1
!OB +k
3
!OC):
x3x4,x3dx3ex11x17Dx42Ex15x1dx4dx3ex50
!BD = k3
k1 +k3
!BC;?!AE = k3
k2 +k3
!AC:
x3cx13
!AD =?!AB +?!BD =?!AB + k3
k1 +k3
!BC =?!AB + k3
k1 +k3 (
!AC!AB)
= k1k
1 +k3
!AB + k3
k1 +k3
!AC;
!BE =?!AE!AB = k3
k2 +k3
!AC!AB:
x18ADx42BEx3x3cM,x4ax47
!AM = l?!AD;!BM = m?!BE:
x4ex4fxfx50x8x15x1bx50x29x51x52x24x1dx56x29,!AM =?!AB +!BM,x50x8
l
k
1
k1 +k3
!AB + k3
k1 +k3
!AC? =?!AB +m k3
k2 +k3
!AC!AB?:
x4ex3c?!ABx42?!ACx74x26x2cx2a,x4ex79x53x56x29x50x8x40x41x42:8
<
:
lk3
k1 +k3 =
mk3
k2 +k3
lk1
k1 +k3 = 1?m
x2dx50
8<
:
l = k1 +k3k
1 +k2 +k3
m = k2 +k3k
1 +k2 +k3
:
x4
!AM = k1 +k3
k1 +k2 +k3
!AD:
x51x18ADx42CFx65x3x3cM0,x43x5x3ex50
!AM0 = k1 +k3
k1 +k2 +k3
!AD;
x4Mx42M0x24x54,x21x4fAD;BE;CFx3x3cx43x48x17M.
x2x0x1x17O,x47
!OM =?!OA+!AM =?!OA+ k1 +k3
k1 +k2 +k3
k
2
k1 +k3
!AB + k3
k1 +k3
!AC?
=?!OA+ k1k
1 +k2 +k3
(?!OB!OA)+ k3k
1 +k2 +k3
(?!OC!OA)
= 1k
1 +k2 +k3
(k2?!OA+k1?!OB +k3?!OC):
￠ 8 ￠
x1 x2 1–3
1,x18P,Qx37x17x6bx55x56[O;?!e1;?!e2;?!e3]x1dx15x57x55x11x12x13(2;2;1),(?1;?1;3),x19x58x25P;Qx17x15x2fx59.
x5,x5fx5ax9.
x2ex2ex2fx2ex2e
x2ex1cx6x75x40
x2ex6cx78x78
x78x78x78
x5
x5
x26x43x5x35x35
x7x3bx3ax9x45x46
x72x72x7ex7exf
x6x1cx2ex2ex2ex2f
x2ex2e
x75x75x72
x2ex6cx78x78
x78x78x78
x47x1c x47x1c x47x1b
x47x1c x1cx6x8x20x20x50x20x50x20x50
x2ex6cx2ex6c
x2ex6cx2ex6c
x2ex6cx2ex6c
x2ex6cx2ex6c
x2ex6cx2ex6c
x2ex6c
O
!e1
!e2
!e3
P(2;2;1)
Q(?1;?1;3)
x01x1x2
x30x31x30x30x31
x30x30x31x30x31
x30x30x31x30x30
x31x30x30x31x30
x30x31x30x31x30
x30x31x30x30x31
x30x30
x2x37xexf
xexfxf
xexfxe
xfxexf
xexfxe
xf
x8x20x50x75x75x75x75x75x75xfxexf
xexfxe
xfxexf
xexfxf
xexfxe
x37x2x75x75x75x75x75x75x50x20x8x27x5ex5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5cx5c
x5cx5dx5c
x5cx5c
A
C
B
D
O
x02x1x2
2,x2x3cxcxdx6cx38x36ABCD,x73A;D,?!AD;?!DBx6bx55x56[C;?!AC;?!BD]x1dx15x57x55.
x5,?!CA =!AC = (?1)?!AC +0?!BD,x17Ax57x55x22(?1;0);
!CD = 1
2
!CA+ 1
2
!BD =? 1
2
!AC + 1
2
!BD;
x17Dx57x55x22
12 ; 12
·;
!AD = 1
2
!AC + 1
2
!BD;
!ADx57x55x22?1
2 ;
1
2
·;
!DB =!BD = 0?!AC +(?1)?!BD;
!DBx57x55x22(0;?1).
3,x18?!a,?!b,?!c x15x57x55x11x12x13(1;5;2),(0;?3;4),(?2;3;?1),x73x1fx202?!a +?!c,?3?!a +2?!b +4?!c
x15x57x55.
x5,2?!a +?!c = 2(1;5;2)+(?2;3;?1) = (2;10;4)+(?2;3;?1) = (0;13;3).
3?!a +2?!b +4?!c =?3(1;5;2)+2(0;?3;4)+4(?2;3;?1)
= (?3;?15;?6)+(0;?6;8)+(?8;12;?4) = (?11;?9;?2):
4,x53x54x34x35x36x15x34x31x16x74x3x3cx48x17(x24x7f).
x3x4,x18D;E;Fx11x12x13x38BC;CA;ABx79x15x16x17,ADx42BEx3x3cG,ADx42CFx3x3cG0,x4a
!AG = k?!AD = k 1
2
!AB + 1
2
!AC? = k
2
!AB + k
2
!AC:
x1ex5bx2bx5cx5dx55x56[A;?!AB;?!AC],x4ax17Gx57x55x22
k
2 ;
k
2
·
,x51
!BG = m?!BE = m 1
2
!BA+ 1
2
!BC? = m 1
2
!AB + 1
2 (
!AC!AB)?
= m2?!AC?m?!AB;
x23x24?!BGx57x55x22
m; m2
·
,x71?!AG =?!AB +?!BG,x23x24,
k
2 ;
k
2
= (1;0)+
m; m2
·;
￠ 9 ￠
x2dx40x41x42 ( k
2 = 1?m
k
2 =
m
2
x50k = m = 23, x23x24Gx15x57x55x22
1
3 ;
1
3
·
,x43x5,x3ex24x5ex50G0x15x57x55x22
1
3 ;
1
3
·
,x53x50G = G0.
xdxcxd
xdxcxd
xdxcxd
xdxcxd
xdxcxd
xdxcx2x6fx6ex6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
xex75x75x75x75x75x75x75x75x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28
x44x1a
x36x36x36x36x36x36x36x36
x36x36x36x36x36x36x36x36
x36x36x36x37x36x36x36x36
x36x36x36x1dx7xax56x4dx4ex4dx4dx4dx4ex4dx4d
x4dx4ex4dx4dx4dx4e
x4dx4dx4dx4ex4dx4d
A F B
C
DE
M
x04x1x2
x7x7x7
x7x7x7
x7x8x7
x7x7x7
x7x7x7
x7x7x1x6fx6ex6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
x6fx6fx6e
xex75x75x75x75x75x75x75x75x28x28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28
x44x1a
x35x36x35x35x36x35x35
x36x35x35x36x35x35x36
x35x35x36x35x35x36x35
x35x36x35x35x36x35x7
xbx25x59x53x53x53
x53x53x53x53
x53x53x53x53
x53x53x53x53
x53x53
A F B
C
DE
M
x05x1x2
5,x53x54x34x35x36x15x34x31x35xcx11x74x3x3cx48x17.
x3x4,x184ABCx15x34x31x35xcx11x74x11x12x22AD;BEx3aCF,x3fx18ADx42BEx3x3cTx17,x49
!AT = k?!AD = k
j?!ABj+j?!ACj
(j?!ACj?!AB +j?!ABj?!AC):
x5bx2bx5cx5dx55x56[A;?!AB;?!AC],x3fx49?!AB =?!a,?!AC =?!b, x4aTx17x57x55
kj?!b j
j?!a j+j?!b j ;
kj?!a j
j?!a j+j?!b j
!
,x14x15x5f
xbx60
!BE = 1
j?!BAj+j?!BCj
(j?!BCj?!BA+j?!BAj?!BC);
x23x24
!BT = m?!BE = m
j?!a j+j?!b!a j
(?j?!b!a j?!a +j?!a j(?!b!a ))
=?m?!a + mj
!a j
j?!a j+j?!b!a j
!b,
x4ex3c?!AT =?!AB +?!BT,x23x24?
kj?!b j
j?!a j+j?!b j; kj
!a j
j?!a j+j?!b j
!
= (1;0)+
m; mj
!a j
j?!a j+j?!b!a j
!;
x4,8
>><
>>:
kj?!b j
j?!a j+j?!b j = 1?m
kj?!a j
j?!a j+j?!b j =
mj?!a j
j?!a j+j?!b!a jx2dx50:
k = j
!a j+j?!b j
j?!a j+j?!b j+j?!b!a j
:
x51x18ADx42CFx3x3cT0x17,
!AT0 = s?!AD = sj?!b j
j?!a j+j?!b j
!a + sj?!a j
j?!a j+j?!b j
!b,
x50T0x17x15x57x55x22
sj?!b j
j?!a j+j?!b j ;
sj?!a j
j?!a j+j?!b j
!
.
!CT0 = t?!CF = t
j?!CAj+j?!CBj
(j?!CBj?!CA+j?!CAj?!CB) = tj
!b j
j?!b j+j?!a!b j
!a?t?!b ;
￠ 10 ￠
x4e!AT0 =?!AC +!CT0,x50:?
sj?!b j
j?!a j+j?!b j; sj
!a j
j?!a j+j?!b j
!
= (0;1)+
tj?!b j
j?!b j+j?!a!b j;?t
!;
x4,8
>><
>>:
sj?!b j
j?!a j+j?!b j =
tj?!b j
j?!b j+j?!a!b j
sj?!a j
j?!a j+j?!b j = 1?tx2dx50:
s = j
!a j+j?!b j
j?!a j+j?!b j+j?!a!b j
:
x4ex4fx3ex5fs = k,x4T = T0.
6,xaxbx74x78ABx49x17C(2;0;2)x3aD(5;?2;0)x34x56x11,x19x73x25x77x74x78x15x37x66x61x17A;Bx15x57x55.
x5,x55x3fx18A;B;C;Dx6cx17x8x9x23x1c,x18A;Bx37x17x15x57x55x11x12x22(xA;yA;zA)x42(xB;yB;zB),x4a
!AC =?!CD,?!CD =?!DB,x23x24
(xC?xA;yC?yA;zC?zA) = (xD?xC;yD?yC;zD?zC);
x4,8
><
>:
xA = 2xC?xD =?1
yA = 2yC?yD = 2
zA = 2zC?zD = 4:
x43x5,8
><
>:
xB = 2xD?xC = 8
yB = 2yD?yC =?4
zB = 2zD?zC =?2:
x21x4fA;Bx37x17x15x57x55x11x12x22(?1;2;4)x42(8;?4;?2)(x37x62x3ex63).
7,xaxbAx0 Bx37x17x15x57x55x11x12x22(1;?2;3),(4;1;2).
(1)x19x64x1dx17Px15x57x55,x27x17Px11x74x78ABx2ax1dx653, 2;
(2)x19x64x1dx17Px15x57x55,x27x17Px11x74x78BAx2ax1dx65?2, 3.
x5,(1)x4ej?!APj, j?!PBj = 3, 2x3ex50?!AP = 32?!PB,x33x1ax6a3.1x15x1dx65x11x17x66x29,x7ak = 32,x3ex50Px17
x57x55
14
5 ;?
1
5 ;
12
5
·
.
(2)x4exaxbx31x32x3ex50?!BP =? 23?!PA,x1ax1dx65x11x17x66x29x67x50Px17x57x55(10;7;0).
x75x75x75x75x75x75x75x75x5 x5 x5 x5
A C BD
x06x1x2
x75x75x75x75x75x75x75x75x72x40x50x8x5 x5 x5 x5 x5 x5
A C BD E F
x09x1x2
8,ABCDx22xcxdx6cx38x36,xaxbA,Bx68x2x35x74x3x17x15x57x55x11x12x22(?3;1;5),(2;?3;4),(1;?1;2).
x19x64x1dx17C,Dx15x57x55.
x5,x18x2x35x74x3x17x22M,C;Dx15x57x55x11x12x22(xC;yC;zC);(xD;yD;zD),x4ex3cMx13A;Cx15x16x17,x21
x4f 8
>><
>>:
1
2 (?3+xC) = 1
1
2 (1+yC) =?1
1
2 (5+zC) = 2;x2dx50Cx17x57x55x22(5;?3;?1),x4ex3cMx67x13B;Dx15x16x17,x43x5x3ex50Dx17x57x55x22(0;1;0).
￠ 11 ￠
9,x18A;Bx37x17x15x57x55x11x12x22(?6;5;?8),(4;0;7),x19x64x1dx17C;D;E;F,x27C;D;E;F x76x74x78AB
x69x56x11.
x5,x55x3fx18A;B;C;D;E;Fx8x9,x23x24
!AC = 1
4
!CB;?!AD = 2
3
!DB;
!AE = 3
2
!EB;?!AF = 4?!FB:
x33x1ax1dx65x11x17x66x29x67x50Cx17x57x55x22(?4;4;?5),Dx17x57x55x22(?2;3;?2),Ex17x57x55x22(0;2;1),F x17x57x55
x22(2;1;4).
10,x8x9,xaxbxcxdxexfx10OABC?O1A1B1C1x16,x17Px6bx14AA1x79,x3fjAPj = 2jPA1j,x17Sx6bx14
CC1x79,x3fjCSj = 12 jSC1j,x17Q;Rx11x12x13x14O1C1;ABx15x16x17,x73x53:x2ex74PQx42x2ex74RSxcxd.
x3x4,x5bx2bx57x55x6a[O;?!OA;?!OC;!OO1],x21x22j?!APj = 2j!PA1j,x23x24
!AP = 2!PA
1 =
2
3
!AA
1;
!OP =?!OA+?!AP =?!OA+ 2
3
!AA
1 =
!OA+ 2
3
!OO
1;
!OQ =!OO
1 +
!O
1Q =
!OO
1 +
1
2
!OC;
x43x25
!PQ =?!OQ!OP =!OA+ 2
3
!OO
1 +
1
2
!OC:
x6bx6cx6d,
!OR =?!OA+?!AR =?!AB + 1
2
!OC;
!OS =?!OC +?!CS =?!OC + 2
3
!OO
1;
x23x24
!RS =?!OS!OR =!OA+ 2
3
!OO
1 +
1
2
!OC:
x77x6ex6fx47?!PQ =?!RS,x3cx13?!PQ==?!RS.
x10x70
x74
x74
x74
x75x75x75x75x75x75x75x50x71x71
x71x20
x40x74
x74
x74
x70x10x50x75x75x75x75x75x75x75
x40x20x71
x71x71
x75x75x75x75x75x75x75x50x71x71
x71x20
x40
x74
x74
x74
x70x10
x40x18 x40x18
x40x18 x40x18
x40x18 x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10
x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20x47x1b
x47x1b x47x1b x47x1b
x47x1b x47x1b x47x1c
x47x1c x47x1c x47x1c
x47x1c x47x1c x47x1b
x47x1b x47x1b
x47x1b x47x1b x47x1b
x47x1b x47x1b x47x1b
x47x1c x47x1c x47x1c
x47x1c x47x1c x47x1c
x47x1b x47x1b x47x1b
A B
CO
A1 B1
O1 C
1
P
Q
R
S
x010x1x2
x29x29x29x2a
x29x29x29x29
x29x29x2ax29
x29x29x29x6dx6dx6cx6dx6d
x6dx6dx6c
x6dx6dx66
xe
x38x10x10
x10x10x10
x10x10x10
xax50x50x50x50x50
x50x50x50x50x50
x50x50x50x50x50
x50x50
x5 x5
x5 x5
A
DB
C
E F
H G
x011x1x2
11,xaxbx70x71x6cx38x36ABCD,x76AB;AD;CDx68CBx24x65x43x65x11x39,x53x54x77x6cx66x11x17x75x2ax48x66xc
xdx6cx38x36.
x3x4,x8x9,x18x11x17x22E;F;G;H,x18
!AE = k?!EB = 1
1+k
!AB;?!AF = k?!FD = 1
1+k
!AD;
x4a
!EF =?!AF!AE = 1
1+k
!BD:
x6bx6cx6d,x4e!CH = k?!CBx24x68?!CG = k?!CDx3ex24x50x8!HG = 12?!BD,x21x4f?!EF =!HG,x53x54x72EFGHx13xc
xdx6cx38x36.
￠ 12 ￠
12,x53x54:x6cxfx10x15x6cx31x16x74x3x3cx48x17(x4x6cxfx10x15x24x7f),x3fx4fx3x17x76x73x48x31x16x74x11x2ax1dx65x223, 1
(x4ex7ex17x67x74)x15x37x7cx11,(x1a,x6cxfx10x15x16x74x4x6cxfx10x15x7ex17x8x3cx2xfx15x24x7fx15x4cx74)
x3x4,x5bx2bx5cx5dx57x55x6a[V;?!VA;?!VB;?!VC],Gx17x224ABCx15x24x7f,G1x224VBCx15x24x7f,x4ex60x611–2
x15x3d3x61xb:
!VG = 1
3 (
!VA+?!VB +?!VC) = 1
3 ;
1
3 ;
1
3
;
!AG
1 =
1
3 (
!AB +?!AC +?!AV) = 1
3 (
!VB!VA+?!VC!VA!VA)
= 13 (?3?!VA+?!VB +?!VC) =
1; 13 ; 13
:
x7ax4ex16x74VGx11x2a3, 1x15x11x17M,x4
!VM = 3
4
!VG = 1
4 ;
1
4 ;
1
4
;
x43x52,
!VA+ 3
4
!AG
1 =
1
4 ;
1
4 ;
1
4
=!VM:
x23x24Mx6bVGx42AG1x79,x43x5,x3ex53x50:x1ex18G2;G3x11x12x134VABx3a4VACx15x24x7f,x4aCG2x42BG3
x67x40x3x3cx17M0,x3f!VM0 =
1
4 ;
1
4 ;
1
4
·
,x21x4fM0 = M,x3fx3x17x11x73x48x16x74x2ax1dx653, 1.
x1ax28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x73x73
x73x73
x73x60
xcx1dx1dx1d
x1dx1dx1e
x1dx1dx1d
x1dx1dx1d
x1dx1dx1d
x3fx4
xax58x51x51x52x51
x51x51x51x51x51
x51
x1x11x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5
x5
x5x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f x4dx1e x4dx1f x4dx1e x4d
x1e x4cx1f x4dx1e x4dx1f
A
C
B
V
G
G1
x012x1x2
x1ax28x28x28
x28x28x28
x28x28x28x28
x28x28x28x28
x73x73
x73x73
x73x60
xcx1dx1dx1d
x1dx1dx1e
x1dx1dx1d
x1dx1dx1d
x1dx1dx1d
x3fx4
xax58x51x51x52x51
x51x51x51x51x51
x51
x1x11x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5x5x6
x5
x5
x5
x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f x4dx1e x4dx1f x4dx1e
x4dx1e x4cx1f x4dx1e x4dx1f x4dx1e x4dx1e x4cx1f x4dx1e
x4dx1f
A
C
B
D
E
V
x013x1x2
13,x6cxfx10x15x55x65x3x15x37x31x14x75x22x2x14,x73x48x2x2x14x15x16x17x15x4cx74x75x22x6cxfx10x15x76x16x74,x53x54:x6c
xfx10x15x34x31x76x16x74x3x3cx38x15x24x7f,x3fx4fx24x7fx4ex73x48x31x76x16x74x11x2ax3bx77x65x56x15x37x7cx11.
x3x4,x43x79x61x5bx2bx5cx5dx57x55x6a[V;?!VA;?!VB;?!VC],x18Mx13x6cxfx10x15x24x7f,x4a!VM =
1
4 ;
1
4 ;
1
4
·
,x18
D;Ex11x12x22AB;VCx15x16x17,x4a
!AD = 1
2
!AB = 1
2
!VB? 1
2
!VA;
!VD = 1
2
!VA+ 1
2
!VB = 1
2 ;
1
2 ;0
:
!VE = 1
2
!VC = 0;0; 1
2
;
x23x24
!ED =?!VD!VE = 1
2 ;
1
2 ;?
1
2
:
x4ex3c
!VE + 1
2
!ED = 1
4 ;
1
4 ;
1
4
=!VM;
x58x54x24x7fMx6bEDx79x3fx56x11ED,x43x5x3ex53x3cx38.
x1 x2 1–4
￠ 13 ￠
1,x78x67x1dx1e2x79x423x79xdx1ex29:
x5,(1)
flfl
flfl?1 1
2 3
flfl
flfl =?5,(2)
flfl
flflcos?sin
sin cos
flfl
flfl = 1.
(3)
flfl
flfl
flfl
1 2 2
0?1 0
0 0?1
flfl
flfl
flfl = 1,(4)
flfl
flfl
flfl
1 1 1
2 3 4
4 9 16
flfl
flfl
flfl = 2.
(5)
flfl
flfl
flfl
2 1 1
1 1 5
2 3?3
flfl
flfl
flfl =?22,(6)
flfl
flfl
flfl
x y z
z x y
y z x
flfl
flfl
flfl = x3 +y3 +z3?3xyz.
2,x33x1a2x79x443x79xdx1ex29x2dx74x26x40x41x42:
(1)
( 2x?3y = 5;
3x+2y = 1; (2)
8>
<
>:
2x?y +3z = 9;
3x?5y +z =?3;
x+3y?2z =?6:
x5,(1) x =
flfl
flfl5?3
1 2
flfl
flfl
flfl
flfl2?3
3 2
flfl
flfl
= 1313 = 1,y =
flfl
flfl2 5
3 1
flfl
flfl
flfl
flfl2?3
3 2
flfl
flfl
=?1313 =?1.
(2) x =
flfl
flfl
flfl
9?1 3
3?5 1
6 3?2
flfl
flfl
flfl
flfl
flfl
flfl
2?1 3
3?5 1
1 3?2
flfl
flfl
flfl
=?4249 =? 67,y =
flfl
flfl
flfl
2 9 3
3?3 1
1?6?2
flfl
flfl
flfl
flfl
flfl
flfl
2?1 3
3?5 1
1 3?2
flfl
flfl
flfl
= 4249 = 67,z =
flfl
flfl
flfl
2?1 9
3?5?3
1 3?6
flfl
flfl
flfl
flfl
flfl
flfl
2?1 3
3?5 1
1 3?2
flfl
flfl
flfl
=
189
49 =
27
7,
3,x18?!e1,?!e2,?!e3 x22x7a.
(1)x53x54:x1fx20?!a =?!e1 +3?!e2!e3,?!b = 2?!e1?3?!e2?10?!e3,?!c =!e1 +2?!e2 +6?!e3 x74x26x2cx2a;
(2)x73x1fx20?!d = 3?!a?2?!b +?!c x6bx7a?!e1,?!e2,?!e3 x1dx15x57x55;
(3)x73x1fx20?!f,x27!a +2?!b?3?!c +3?!f = 0.
x5,(1)x18x47x32xex1;x2;x3x2dx2ex74x26x2ax6ax29x1?!a +x2?!b +x3?!c = 0,x1bx50x2ax57x55x36x29x6fx13
8>
<
>:
x1 +2x2?x3 = 0
3x1?3x2 +2x3 = 0
x1?10x2 +6x3 = 0;
x38x15x6axexdx1ex29x13 fl
flfl
flfl
fl
1 2?1
3?3 2
1?10 6
flfl
flfl
flfl =?5 6= 0;
x21x4fx77x66x40x41x42x7bx47x6fx2d,x4?!a ;?!b ;?!c x74x26x2cx2a.
(2)?!d = 3?!a? 2?!b +?!c = 3(?!e1 + 3?!e2!e3)? 2(2?!e1? 3?!e2? 10?!e3) + (!e1 + 2?!e2 + 6?!e3) =
2?!e1 +17?!e2 +23?!e3,x21?!d x15x57x55x13(?2;17;23).
(3)?!f = 13 (?!a?2?!b +3?!c ) = 13 (?6?!e1 +15?!e2 +37?!e3) =?2?!e1 +5?!e2 + 373?!e3.
4,x7cx7dx1dx1ex73x42x15x34x66x1fx20?!a ;?!b ;?!c x13x29x28xf? x63x29x76?!c x1bx1cx2ax3cx38x37x66x1fx20x15x74x26x42x54?x1e
x63,x7ex25x7fx10x15x1bx1cx29x0.
(1)?!a (5;2;1),?!b (?1;4;2),?!c (?1;?1;5);
(2)?!a (3;3;2),?!b (6;6;4),?!c (1;?1;0);
￠ 14 ￠
(3)?!a (1;2;?3),?!b (?2;?4;6),?!c (1;0;5).
x5,x1x61x2x22x22x73x2dx1?!a +x2?!b +x3?!c = 0.
(1)x3x48x74x26x40x41x42 8
><
>:
5x1?x2?x3 = 0
2x1 +4x2?x3 = 0
x1 +2x2 +5x3 = 0;
x3cx6axexdx1ex29 fl
flfl
flfl
fl
5?1?1
2 4?1
1 2 5
flfl
flfl
flfl = 121 6= 0;
x40x41x7bx47x6fx2d,x21x4bx1fx20x42x55x28xf.
(2)x3x48x74x26x40x41x42 8
><
>:
3x1 +6x2 +x3 = 0
3x1 +6x2?x3 = 0
2x1 +4x2 = 0;
x3cx6axexdx1ex29 fl
flfl
flfl
fl
3 6 1
3 6?1
2 4 0
flfl
flfl
flfl = 0;
x40x41x42x47x6ex6fx2d,x21x4bx1fx20x42x28xf,x22x76?!c x1bx1cx2a?!a ;?!b x15x74x26x42x54,x3ex7ax3 =?1x51x52,x50x8x40x41x428
><
>:
3x1 +6x2 = 1
3x1 +6x2 = 1
2x1 +4x2 = 0;
x77x13x34x35x40x41x42,x21x4f?!c x55x63x1bx1cx2a?!a ;?!b x15x74x26x42x54.
(3)x3x48x74x26x40x41x42 8
><
>:
x1?2x2 +x3 = 0
2x1?4x2 = 0
3x1 +6x2 +5x3 = 0;
x3cx6axexdx1ex29 fl
flfl
flfl
fl
1?2 1
2?4 0
3 6 5
flfl
flfl
flfl = 0;
x40x41x42x47x6ex6fx2d,x21x4bx1fx20x42x28xf,x22x76?!c x1bx1cx2a?!a ;?!b x15x74x26x42x54,x3ex7ax3 =?1x51x52,x50x8x40x41x428
><
>:
x1?2x2 =?1
2x1?4x2 = 0
3x1 +6x2 =?5;
x77x13x34x35x40x41x42,x21x4f?!c x55x63x1bx1cx2a?!a ;?!b x15x74x26x42x54.
5,x18x1fx20?!a ;?!b ;?!c x15x57x55x11x12x13(1;?1;2),(2;k;1),(1;1?k;k),x1,x62kx7ax2fx30x4x52,?!a ;?!b ;?!c
x28xf? x69x12x6d,kx7ax2fx30x4x52,?!a ;?!c x28x74?
x5,x773x66x1fx20x28xfx15x30x11x40x26x31x32x13x3cx57x55x15xdx1ex29x56x3c0,x4fl
flfl
flfl
fl
1?1 2
2 k 1
1 1?k k
flfl
flfl
flfl = k2?3k +2 = 0:
￠ 15 ￠
x21x4fx62k = 1x442x52x773x66x1fx20x28xf,x26x27?!a ;?!c x28x74x40x41x27x38x15x15x65x2cx57x55x2ax65x6a,x4 11 = 1?k?1 = k2,
x2dx50k = 2,x21x4fx62k = 2x52?!a ;?!c x28x74.
6,x18?!e1,?!e2,?!e3 x22x7a,x1x1fx20?!v x63x29x1bx22x1fx20?!a ;?!b ;?!c x15x74x26x42x54?x8x63,x4ax7ex25x1bx50x29.
(1)?!a =?!e1 +2?!e2 +4?!e3,?!b =?!e1!e2 +?!e3,?!c =?!e1 +?!e2 +3?!e3,?!v = 6?!e1 +3?!e2 +15?!e3;
(2)?!a =?!e1?2?!e2 +3?!e3,?!b =?!e1 +2?!e2 +2?!e3,?!c =!e1 +6?!e2 +5?!e3,?!v = 2?!e1 +?!e2:
x5,x18?!v = x1?!a +x2?!b +x3?!c,x1x61x2x22x22x2dx74x26x40x41x42.
(1)x40x41x42x22 8
><
>:
x1 +x2 +x3 = 6
2x1?x2 +x3 = 3
4x1 +x2 +3x3 = 15;
x6axexdx1ex29
flfl
flfl
flfl
1 1 1
2?1 1
4 1 3
flfl
flfl
flfl = 0,x55x63x7dx1dx40x41x42x13x29x47x2d,x1ax17x5x6x7x44x2dx50
( x
1 = 3? 23 x3
x2 = 3? 13 x3;
x49
x3 = 3kx3ex24x50x8x74x26x1bx1cx29?!v = (3?2k)?!a +(3?k)?!b +3k?!c,x3cx16kx22x0x1xe.
(2)x40x41x42x22
8>
<
>:
x1 +x2?x3 = 2
2x1 +2x2 +6x3 = 1
3x1 +2x2 +5x3 = 0;
x6axexdx1ex29
flfl
flfl
flfl
1 1?1
2 2 6
3 2 5
flfl
flfl
flfl = 36;
x40x41x42x47x2d:
x1 =
flfl
flfl
flfl
2 1?1
1 2 6
0 2 5
flfl
flfl
flfl
36 =?
11
36 ; x2 =
flfl
flfl
flfl
1 2?1
2 1 6
3 0 5
flfl
flfl
flfl
36 =
16
9 ; x3 =
flfl
flfl
flfl
1 1 2
2 2 1
3 2 0
flfl
flfl
flfl
36 =?
19
36,
x74x26x1bx1cx29x22?!v =? 1136?!a + 169?!b? 1936?!c,
7,x62ax22x8x4x52,x1dx1ex6cx17x28xf:
M1(1;a;a2); M2(1;?1;1); M3(2;1;?2); M4(?1;2;2):
x5,x3dx3ex5ex234.5,x4f4x17x28xfx15x30x11x40x26x31x32x13fl
flfl
flfl
fl
1?1 2?1?1?1
1?a 1?a 2?a
1?a2?2?a2 2?a2
flfl
flfl
flfl =?7a2?5a+2 = 0;
x2dx50a =?1x44 27,
x1 x2 1–5
1,x3dx3enx46x1fx20x15x1dx4dx53x54:x2x0x1nx46x1fx20fi,x47
(1) 0fi = 0;
(2) (?1)fi =?fi;
(3) k0 = 0 (x0x1xek);
(4)x43kfi = 0x5ex25k = 0x44fi = 0.
x3x4,x2x0x1x15fi = (a1;￠￠￠ ;an),x4a:
￠ 16 ￠
(1) 0fi = (0a1;￠￠￠ ;0an) = (0;￠￠￠ ;0) = 0.
(2) (?1)fi = ((?1)a1;￠￠￠ ;(?1)an) = (?a1;￠￠￠ ;?an) =?fi.
(3) k0 = (k0;￠￠￠ ;k0) = (0;￠￠￠ ;0) = 0.
(4) kfi = (ka1;￠￠￠ ;kan) = (0;￠￠￠ ;0),x1efi 6= 0,x4ax31x6bai 6= 0,x4ekai = 0x3ex50k = 0.
2,x53x54:x0x48xex9x6dx36xax47x5xex9.
x3x4,x18Kx22x48x66xex9,x4a1 2 K,x23x24x2x0x1x15x72x2bxenx47n = 1+￠￠￠+1| {z }
n
2 K,x57x3fnx15x39xb
n 2 K,x21x4fKxax47x33x7cx2bxe,x51x21x2x0x1x15x2bxen 6= 0,n 2 K,1n x22nx15xcxb,x4a 1n 2 K,x23x24x2x0
x1x15x47x5xe mn (x3cx16m;nx13x2bxe),x47 mn = m￡n 2 K,x21x47x5xex9Q K.
3,x53x54:x33x10x36x8
a+bp2; a;b 2Q
x15xex42x2ax15xdx54x75x2ax48x66xex9.
x3x4,x4ex77x66xdx54x1fx22K,x18a1 +b1p2;a2 +b2p2 2 K,x4a
(a1 +b1p2)§(a2 +b2p2) = (a1 §a2)+(b1 §b2)p2 2 K
(x21x22x47x5xex15x3ax42x5bxex13x47x5xe);
(a1 +b1p2)(a2 +b2p2) = (a1a2 +2b1b2)+(a1b2 +a2b1)p2 2 K
(x21x22x47x5xex15x3ax0x5bx42xfx10xex13x47x5xe);x62a2 +b2p2 6= 0x52,
a1 +b1p2
a2 +b2p2 =
(a1 +b1p2)(a2?b2p2)
a22?2b22 =
a1a2?2b1b2
a22?2b22 +
a2b1?a1b2
a22?2b22 2 K
(x47x5xex2ax3cx11x44x67x13x12x13x15),x21x4fxdx54Kx2ax3cx17x5xfx11x44x6dx12x13,x2ax22x48x66xex9.
4,x18Kx22xex9,V x22Kx79x15nx46x1fx20x70x71,x53x54:x2x23x47x15k 2 K,fi;fl 2 V,x47
(1) k(fi?fl) = kfi?kfl;
(2) fi+fi+￠￠￠+fi| {z }
nx0
= nfi;
(3)x1efi+fl = fi+,x4afl =,
x3x4,x18fi = (a1;￠￠￠ ;an),fl = (b1;￠￠￠ ;bn) (ai;bi 2 K),x4ax2x0x1x15k 2 K,
(1) k(fi?fl) = k(a1?b1;￠￠￠ ;an?bn) = (k(a1?b1);￠￠￠ ;k(an?bn)) = (ka1?kb1;￠￠￠ ;kan?kbn) =
k(a1;￠￠￠ ;an)?k(b1;￠￠￠ ;bn) = kfi?kfl.
(2) fi+￠￠￠+fi| {z }
n
= (a1 +￠￠￠a1| {z }
n;a2 +￠￠￠+a2| {z }
n;￠￠￠ ;an +￠￠￠+an| {z }
n
) = (na1;￠￠￠ ;nan) = nfi.
(3)x1ex18 = (c1;￠￠￠ ;cn),x3ffi+fl = fi+,x4,(a1 +b1;￠￠￠ ;an +bn) = (a1 +c1;￠￠￠ ;an +cn),x4a
x47ai +bi = ai +ci,x4bi = ci,x23x24fl =,
x1 x2 1–6
1,x76x1dx1ex1fx20x14x2fx4c:
(1)?!a = 5?!i?6?!j +3?!k ;
(2)?!b = 12?!i? 13?!k,
x5,(1)?!a0 =?!aj?!a j =
p70
70 (5
!i?6?!j +3?!k ).
￠ 17 ￠
(2)?!b0 =
!b
j?!b j =
p13
13 (3
!i?2?!k ).
2,x78x67x1dx1ex1fx20x15x15x35:
(1)?!a = (1;?2;3),?!b = (2;1;?2);
(2)?!a = (?2;1;?1),?!b = (1;?1;4).
x5,(1) j?!a j = p14,j?!b j = 3,?!a ￠?!b =?6,x23x24
cosh?!a ;?!b i =?6p14￠3 =?
p14
7 ; h
!a ;?!b i = …?arccos
p14
7,
(2) j?!a j = p6,j?!b j = 3p2,?!a ￠?!b =?7,x23x24
cosh?!a ;?!b i =?76p3 =? 7
p3
18 ; h
!a ;?!b i = …?arccos 7
p3
18,
3,x73x1fx20?!a x6b?!e0 x79x15x16x17:
(1)?!a = (1;?1;2),?!e = (1;1;1);
(2)?!a = (?2;1;3),?!e = (1;2;0).
x5,(1)?!e0 =?!ej?!e j =
p3
3 (1;1;1),x4a
pr?!e0?!a = (Π?!e0?!a )?!e0 = (?!a ￠?!e0)?!e0 = 2
p3
3
!e0 = 2p3
3 ￠
p3
3 (1;1;1) =
2
3 (1;1;1):
(2)x21?!a ￠?!e = 0,x23x24pr?!e0?!a = 0.
4,x53x54:x24A(3;?1;2),B(0;?4;2),C(?3;2;1)x22x7ex17x15x34x35x36x13x56x18x34x35x36.
x3x4,?!AB = (?3;?3;0),?!AC = (?6;3;?1),?!BC = (?3;6;?1),j?!ABj = 3p2,j?!ACj = j?!BCj = p46 6=
j?!ABj,x23x244ABCx13x56x18x34x35x36.
5,x53x54:x24A(3;?2;1),B(7;6;9),C(9;1;?5)x22x7ex17x15x34x35x36x13x2ex35x34x35x36.
x3x4,?!AB = (4;8;8),?!AC = (6;3;?6),?!AB ￠?!AC = 12(2+2?4) = 0,x23x244ABCx13x2ex35x34x35x36.
6,x19x1(?!a?!b )?!c =?!a (?!b?!c )x48x1dx2ax2bx19? x1ax1bx25x1cx1fx20x56x29x2ax2bx15x31x32.
x5,x56x29x1dx61x13x42?!c x28x74x15x1fx20,x1ex61x13x42?!a x28x74x15x1fx20,x8x1fx37x61x6dx55x56x3c0,x4a?!a x42?!c x28
x74,x4x31x6bk 6= 0,x27?!a = k?!c, x4dx39x1e?!a = k?!c,x1dx38= k(?!c ￠?!b )?!c = (k?!c )(?!b ￠?!c ) =x1ex38.
x1ex56x29x37x38x6dx56x3c0,x4ax44x20?!a ;?!b ;?!c x16x45x46x47x48x66x6fx1fx20;x44x20x34x66x1fx20x6dx55x56x3c0,x71?!a ￠?!b =
!b ￠?!c = 0,x4?!b x42?!a ;?!c x4bx72x3.
7,x73x2dx1fx20x40x41?!a?!x =?!b?!x,
x5,x21x22(?!a!b )?!x = 0,x11x37x62xfx21,(a)x1e?!a!b 6= 0,x4ax2d?!x x22x0x1x42?!a!b x22x2ex15x1fx20;
(b)x1e?!a!b = 0,x4ax0x1x1fx20?!x x6dx13x2dx1fx20.
8,x18x47x34x66x1fx20?!a ;?!b ;?!c x37x37x75x2a60–x35,x3fxbj?!a j = 4,j?!b j = 2,j?!c j = 6,x73?!a +?!b +?!c x15
x3bx77.
x5,j?!a +?!b +?!c j2 = (?!a +?!b +?!c )￠(?!a +?!b +?!c ) =?!a 2+?!b 2+?!c 2+2?!a ￠?!b +2?!a ￠?!c +2?!b ￠?!c =
16+4+36+2(8+24+12)cos60– = 100,x23x24j?!a +?!b +?!c j = 10.
9,xaxbj?!a j = 3,j?!b j = 2,h?!a ;?!b i = …6, x19x733?!a +2?!b x422?!a?5?!b x15x7bx10.
x5,(3?!a +2?!b )￠(2?!a?5?!b ) = 6?!a 2?10?!b 2?11?!a ￠?!b = 54?40?11￡3￡2￡
p3
2 = 14?33
p3.
10,x18x48x66x6cx38x36x28x38x39x3bx11x12x13a;b;c;d,x3fx3cx2x35x74x64x65x22x2e,x73x53x28x38x39x3bx67x13a;b;c;dx15
x0x48x6cx38x36x15x37x31x2x35x74x67x65x64x22x2e.
￠ 18 ￠
x3x4,x8x9,x47
!AC =?!AB +?!BC =?!AD +?!DC;
!BD =?!BC +?!CD =?!BA+?!AD:
xbxcx24x1dx7bx10:
!AC ￠?!BD = (?!AB +?!BC)￠(?!BC +?!CD) =?!BC2 +?!AB ￠?!BC +?!AB ￠?!CD +?!BC ￠?!CD;
!AC ￠?!BD = (?!AB +?!BC)￠(?!BA+?!AD) =!AB2 +?!AB ￠?!AD!AB ￠?!BC +?!AD￠?!BC;
!AC ￠?!BD = (?!AD +?!DC)￠(?!BC +?!CD) =!CD2 +?!AD￠?!BC +?!AD￠?!CD!BC ￠?!CD;
!AC ￠?!BD = (?!AD +?!DC)￠(?!BA+?!AD) =?!AD2!AB ￠?!AD +?!AB ￠?!CD!AD￠?!CD:
x76x79x534x29x65x17,x3ex50:
4?!AC ￠?!BD =?!AD2 +?!BC2!AB2!CD2 +2?!AB ￠?!CD +2?!AD￠?!BC:
x4e
!AB +?!BC +?!CD +?!DA = 0
x3ex50
!AB +?!CD =?!AD!BC:
x4e
(?!AB +?!CD)2 = (?!AD!BC)2
x2bx5x50
!AD2 +?!BC2!AB2!CD2 = 2(?!AB ￠?!CD +?!AD￠?!BC):
x51x52x79x29x50
4?!AC ￠?!BD = 2(?!AD2 +?!BC2!AB2!CD2) = 2(b2 +d2?a2?c2):
x77x58x54x2x35x74x22x2ex15x30x11x40x26x31x32x13a2 +c2 = b2 +d2,x7bx42x6cx38x36x15x38x3bx47x2a.
x3x17x1bx1a
x1bx1bx1a
x1bx1bx1a
x1bx1bx1a
x1bx1bx1a
x1bx1bx1a
x1bx1bx1a
x1b
xax9xa
xaxax9
xaxaxa
x9xaxa
xax9xa
xax12x1x34x33x34
x33x34x33x34x33x34
x34x33x34x49x1dx75x76x75x75
x76x75x75
x76x75x75
x76x75x75
x76x75x75
x76x75x75
x76x75x2d
xfx8x20x50x75x75x75x75x75x75
x5ax5bx5a
x5ax5ax5b
x5ax5ax5b
x5ax5ax5a
x5bx5ax5a
x5bx5ax5a
x5ax5b
A B
D
C
O
x010x1x2
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6ex6f
x6fx6exex75x75x75x75x75x75x75x75x28x28x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28x28
x28x28x28
x28x44x1a
x0x0x74
x74
x74
x74
B H C
A
x011x1x2
11,x34x35x36ABCx16,xaxbBCx38x79x15x23x22AH,x19x1a?!AB,?!ACx1bx1c?!AH.
x5,x76?!AH =?!AB +k?!BCx51x52?!AH ￠?!BC = 0,x50?!AB ￠?!BC +k?!BC2 = 0,x21x4f
k =?
!AB ￠?!BC
!BC2 ;
x50
!AH =?!AB? (?!AB ￠?!BC)?!BC
!BC2,
x12x1a?!BC =?!AC!ABx51x52,x2bx5x7x50
!AH = 1
(?!AC!AB)2
[(?!AC ￠(?!AC!AB))?!AB?(?!AB ￠(?!AC!AB))?!AC]:
￠ 19 ￠
12,x6bx2ex35x57x55x6ax16,?!a ;?!b ;?!c x15x57x55x11x12x13(3;5;7),(0;4;3),(?1;2;?4),x733?!a +4?!b?5?!c x42
2?!b +?!c x15x15x35.
x5,x1f
!p = 3?!a +4?!b?5?!c = (14;21;53);?!q = 2?!b +?!c = (?1;10;2);
x4a
!p 2 = 3446;?!q 2 = 105;?!p ￠?!q = 302:
x23x24
cosh?!p ;?!q i = 151
p361830
180915,
13,x73x1dx1ex1fx20x15x40x1fx24x25:
(1)?!a = (2;?3;?6); (2)?!b = (2;3;?10).
x5,(1) cosfi = 27,cosfl =?37,cos =?67,
(2) cosfi = 2
p113
113,cosfl =
3p113
113,cos =?
10p113
113,
14,x78x67x72x40x10x15x2x35x74x42x38x15x0x48x66xfx15x2x35x74x39x71x15x15x35.
x5,x5bx2bx2ex35x57x55x6a[A;?!AB;?!AD;!AA1],x24x2x35x74AC1x19x78x67x4fx61.
!AC
1 =
!AB +?!BC +!CC
1 =
!AB +?!AD +!AA
1 = (1;1;1):
(a)!AB1 =?!AB +!BB1 =?!AB +!AA1 = (1;0;1),x23x24cosh!AC1;!AB1i = 2p3￠p2 =
p6
3, x4ex2x75x26,
!AC
1x42
!A
1C1;
!AD
1;
!BC
1;
!DC
1;
!ACx15x15x35x24x25x67x22 p6
3,
(b)?!BD =?!AD!AB = (?1;0;1),x23x24cosh!AC1;?!BDi = 0p3￠p2 = 0,x4h!AC1;?!BDi = …2, x43x5.
!AC
1x42
!B
1D,
!DA
1;
!CB
1;
!BA
1;
!CD
1x15x15x35x67x22 …2,
x20x72x75x75x75
x10x70
x74
x74
x74
x75x75x75x72x20x74
x74
x74
x70x10x71
x71x20
x20x72x75x75x75
x20x71
x71
x71x71
x20x74
x74
x74
x70x10
x40x18 x40x18
x40x18 x40x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20x30
x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
x30x10x30x10
A B
D C
A1 B1
C1D1
x014x1x2
x30x0x70x75x75x75x75x75x40x20x8x71x71
x71x71
x71x70
x0x30x8x20x40x75x75x75x75x75
x71x71
x71x71
x71
x75x75x75x75x75x40x20x8x71x71
x71x71
x71
x30x0x70
x1dx4ax1dx4ax1dx4ax1dx4a
x1dx4ax1dx4ax1dx4ax1dx4ax1dx4a
x1dx4ax1dx4ax7x4a
x45x1a x45x1a x45
x1a x45x1a x45x1a
x45x1a x45x1a x45
x1a x6
x40x18 x40x18
x40x18 x40x18
x40x18 x40x18
x40x18 x30x10
x30x10x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x20x8O
A B
C
C1
A1
O1
B1
P
M
N
x015x1x2
15,x8x9,xaxbx3bx40x10OABC–O1A1B1C1x16,jOAj = 8,jOCj = 6,jOO1j = 1,Px13x14OCx79x15x17,
x3fjPCj = 2jOPj,Mx13x14ABx79x15x17,x3fjAMj = 2jMBj,Nx13x14B1C1x15x16x17,x73x2ex74A1Px42x2ex74MN
x23x2ax15x35.
x5,x4ex1fx20?!OA;?!OC;!OO1x15x14x2fx1fx20x1fx22?!i ;?!j ;?!k,x5bx2bx2ex35x57x55x6a[O;?!i ;?!j ;?!k ],x4a
!OA
1 = 8
!i +?!k = (8;0;1);
!OP = 1
3
!OC = 2?!j = (0;2;0);
!OM = 8?!i + 2
3
!OC = 8?!i +4?!j = (8;4;0);
!ON = 1
2
!OA+6?!j +?!k = (4;6;1);
x21x4f
!A
1P =
!OP!OA
1 = (?8;2;?1);
￠ 20 ￠
!MN =!ON!OM = (?4;2;1):
x23x24
cosh!A1P;!MNi = 35p69p21 = 5
p161
69,
16,x1ex1fx20?!a ;?!b ;?!c x55x28xf,x25x3f?!a?!x = 0,?!b?!x = 0,?!c?!x = 0,x4a?!x = 0,x19x53x39.
x3x4,x21x22?!a ;?!b ;?!c x55x28xf,x23x24x38x15x74x26x2cx2a,x3f?!x x3ex4ex38x15x74x26x1bx1c,x4,?!x = k1?!a +k2?!b +
k3?!c, x3cx13?!x 2 = k1(?!a ￠?!x )+k2(?!b ￠?!x )+k3(?!c ￠?!x ) = 0,x4,?!x = 0.
17,x53x54x34x66x1fx20?!a ;?!b ;?!c x28xfx15x30x11x40x26x31x32x13fl
flfl
flfl
fl
!a?!a?!a?!b?!a?!c
!b?!a?!b?!b?!b?!c
!c?!a?!c?!b?!c?!c
flfl
flfl
flfl = 0:
x3x4,?!a ;?!b ;?!c x28xfx62x3fx63x62x47x55x33x22x6fx15x32xek1;k2;k3x27,k1?!a +k2?!b +k3?!c = 0,x43x258
><
>:
k1?!a 2 +k2(?!a ￠?!b )+k3(?!a ￠?!c ) = 0
k1(?!b ￠?!a )+k2?!b 2 +k3(?!b ￠?!c ) = 0
k1(?!c ￠?!a )+k2(?!c ￠?!b )+k3?!c 2 = 0;
x67x4x3x48x74x26x40x41x42 8
><
>:
x?!a 2 +y(?!a ￠?!b )+z(?!a ￠?!c ) = 0
x(?!b ￠?!a )+y?!b 2 +z(?!b ￠?!c ) = 0
x(?!c ￠?!a )+y(?!c ￠?!b )+z?!c 2 = 0
(?)
x47x6ex6fx2dx = k1,y = k2,z = k3,x3dx3ex26x54.1,x6axexdx1ex29fl
flfl
flfl
fl
!a 2?!a ￠?!b?!a ￠?!c
!b ￠?!a?!b 2?!b ￠?!c
!c ￠?!a?!c ￠?!b?!c 2
flfl
flfl
flfl = 0:
x4dx39,x1e fl
flfl
flfl
fl
!a 2?!a ￠?!b?!a ￠?!c
!b ￠?!a?!b 2?!b ￠?!c
!c ￠?!a?!c ￠?!b?!c 2
flfl
flfl
flfl = 0;
x4a(?)x40x47x6ex6fx2d,x18x22x = k1,y = k2,z = k3,x49?!p = k1?!a +k2?!b +k3?!c,x20x30,x43(?)xb:
!p 2 = k1?!p ￠?!a +k2?!p ￠?!b +k3?!p ￠?!c = 0;
x4?!p = 0,?!a ;?!b ;?!c x74x26x2cx2a,x40x1dx28xf.
x1 x2 1–7
1,x6bx2ex35x57x55x6ax16,xaxb?!a ;?!b ;?!cx15x57x55x11x12x13(1;0;1),(1;?2;0),(?1;2;1),x73(3?!a +?!b )￡(?!b!c )
x15x57x55.
x5,3?!a +?!b = (4;?2;3),?!b!c = (2;?4;?1),x23x24
(3?!a +?!b )￡(?!b!c ) =
flfl
flfl?2 3
4?1
flfl
flfl;?
flfl
flfl4 3
2?1
flfl
flfl;
flfl
flfl4?2
2?4
flfl
flfl
= (14;10;?12):
2,x53x54(?!a ￡?!b )2 6?!a 2?!b 2,x57x58x54x56x29x8x52x2ax2b.
x3x4,(?!a ￡?!b )2 = j?!a ￡?!b j2 = j?!a j2j?!b j2 sin2h?!a ;?!b i6j?!a j2j?!b j2,x56x59x2ax2bx62x3fx63x62sinh?!a ;?!b i
= 0,x4,?!a ==?!b,
3,xaxb?!a ;?!b x13x37x66x64x55xcxdx15x1fx20,x73x53(?!a!b )￡(?!a +?!b ) = 2(?!a ￡?!b ),x57x58x54x38x15x27x8
x1x4d.
￠ 21 ￠
x3x4,(?!a!b )￡(?!a +?!b ) =?!a ￡?!a +?!a ￡?!b!b ￡?!a!b ￡?!b = 2(?!a ￡?!b ),x27x8x1x4d:x1e
x24?!a ;?!b x75x2ax48x66xcxdx6cx38x36x15x65x28x37x38,x4a?!a!b,?!a +?!b x22x4fxcxdx6cx38x36x15x37x2x35x74,x79x29x58x54:
x24x2x35x74x75x2ax15xcxdx6cx38x36xfx10x22x4bxcxdx6cx38x36xfx102x18.
4,x73x1fx20?!c,x27?!c!a,?!c!b,x3cx16,
(1)?!a =?!i?2?!j +3?!k,?!b = 4?!j?5?!k ;
(2)?!a = 3?!i!j +?!k,?!b =!i +2?!j!k,
x5,x49?!c =?!a ￡?!b, x4a?!c!a,?!c!b, x78x67x50:
(1)?!c =?!a ￡?!b =?2?!i +5?!j +4?!k,
(2)?!c =?!a ￡?!b =!i +2?!j +5?!k, (x29x61x2ax2bx55x2cx48)
5,x18?!u = 2?!i +3?!j!k,?!v =?8?!i?5?!j +3?!k, x73?!v1,?!v2,x27?!v =?!v1 +?!v2;?!v1!u ;?!v2==?!u,
x5,x49?!n0 =?!uj?!u j = 1p14 (2;3;?1),pr?!u0?!v = (?!v ￠?!u0)?!u0 =?177 (2;3;?1),x49
!v2 =?17
7 (2
!i +3?!j!k );?!v
1 =?!v!v2 =
2
7 (?11
!i +8?!j +2?!k );
x4a,?!v2==?!u,x3f?!v1 ￠?!u = 0.
6,x18?!u x22x1bx1dx15x6ex6fx1fx20,?!v x22x0x48x1fx20.
(1)x53x54,?!v x3ex2cx48x11x2dx22?!v =?!v1 +?!v2,x3cx16,?!v1!u,?!v2==?!u ;
(2)x7fx10x7ex25?!v1,?!v2 x15x1bx50x29.
x3x4,(1)x1e?!v x47x37x62x11x2dx44,?!v =?!v1 +?!v2 =?!v01 +?!v02,x3cx16,?!v1!u,?!v01!u,?!v2==?!u,?!v02==?!u,
x4a?!v1!v01 =?!v02!v2,x71(?!v1!v01)￠?!u = 0,(?!v02!v2)￡?!u = 0,x23x24?!v1!v01!u,?!v1!v01==?!u,?!u 6= 0,
x5ex25,?!v1!v01 = 0,?!v02!v2 = 0,x4?!v1 =?!v01,?!v2 =?!v02.
(2) x49?!v2 = pr?!u0?!v = (?!v ￠?!u0)?!u0 = (
!v ￠?!u )
j?!u j2
!u,?!v1 =?!v!v2,x4a?!v2==?!u,?!v1 ￠?!u =?!v ￠?!u?
(?!v ￠?!u )
j?!u j2
!u 2 = 0,x4?!v1!u, x3fxb,?!v =?!v1 +?!v2,x4e(1)xbx77x62x11x2dx13x2cx48x15,x21x1bx50x29x22
(?!v
2 =
!v ￠?!u
j?!u j2
!u
!v1 =?!v!v2:
x75x75x75x75x75x71x20
x40x74
x74
x70x72x75x75x40x20x71
x70x74
x74
x75x75x72x71x20
x40
x74
x74
x70x53x53x54x59
x8x20x72x75
x4x18x40x20x71
x71x71
x75x75x75x75x75x75x8x71x71
x71x20
x40x18x4x20x75
x40x18 x40x18
x40x10x30x10
x30x10x30x10
x30x10x30x10
x30x10
x18x40x18x40
x18x40x18x40
x4x18x20x8 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20x49x1d x4ax1c x49x1d x49x1d
x49x1c x49x1d x49x1c
DA
B C
C1
A1 D1
B1
P
C2
x07x1x2
7,x8x9,xaxbABCD?A1B1C1D1x13x14x2fx72x40x10,P x13x14DD1x79x0x1x48x66x17,x74x78C1C2x15x16x17
x13B1,x1ax2dx25x1dx1ex28x66x1fx20x10x23x64x1dx15x1fx20:
(1)!A1P ￡!A1A; (2)?!PC ￡!A1A; (3)!A1C ￡!A1A.
x5,x5bx2bx2ex35x55x56[A1;!A1B1;!A1D1;!A1A],x18Px22DD1x79x0x48x17,x4a:
!A
1P =
!A
1D1 +
!D
1P =
!A
1D1 +k
!D
1D =
!A
1D1 +k
!A
1A = (0;1;k);
!A
1C = (1;1;1):
(1)!A1P ￡!A1A = (0;1;k)￡(0;0;1) = (1;0;0) =!A1B1.
￠ 22 ￠
(2)?!PC =!A1C!A1P = (1;0;1?k),?!PC ￡!A1A = (1;0;1?k)￡(0;0;1) = (0;?1;0) =!A1D1 =
!D
1A1.
(3)!A1C ￡!A1A = (1;1;1)￡(0;0;1) = (1;?1;0) =!A1B1!A1D1 =!A1C2.
8,x78x67x4ex1fx20?!a,?!b x23x2ex2ax15xcxdx6cx38x36x15xfx10:
(1)?!a = 3?!i +4?!j +2?!k,?!b = 2?!i +?!j +?!k ;
(2)?!a =!i +2?!j +?!k,?!b = 2?!i?2?!j,
x5,(1)?!a ￡?!b = 2?!i +?!j?5?!k,j?!a ￡?!b j = p30,x23x24?!a ;?!b x2ex2ax15xcxdx6cx38x36xfx10x22p30.
(2)?!a ￡?!b = 2?!i +2?!j?2?!k,j?!a ￡?!b j = 2p3,x23x24?!a ;?!b x2ex2ax15xcxdx6cx38x36xfx10x222p3.
9,x18?!a ;?!b ;?!c x22x37x37x55x28x74x15x1fx20,x53x54,?!a +?!b +?!c = 0x62x3fx63x62?!a ￡?!b =?!b ￡?!c =?!c ￡?!a,
x3x4,x1e?!a +?!b +?!c = 0,x4ax4fx29x42?!b x3a?!c x2fx30x10x7x3ex50?!a ￡?!b +?!c ￡?!b = 0 x24x68
!a ￡?!c +?!b ￡?!c = 0,x4?!a ￡?!b =?!b ￡?!c =?!c ￡?!a,
x4dx39,x18?!p =?!a +?!b +?!c, x4ex79x53x56x29x3ex50?!p ￡?!b =?!a ￡?!b +?!c ￡?!b = 0x24x68?!p ￡?!c =
!a ￡?!c +?!b ￡?!c = 0,x8x1f?!p 6= 0,x4ax4e?!p ;?!b x28x74x24x68?!p ;?!b x28x74x3ex50?!b x42?!c x28x74,x42x31x18x34x35.
10,x18?!a ;?!b x22x37x55x28x74x15x1fx20,?!AB =?!a +?!b,?!BC = 2?!a +8?!b,?!CD = 3(?!a!b ),x53x54,A;B;D
x34x17x28x74.
x3x4,x26x53A;B;Dx34x17x28x74,x7bx41x53x54,?!AB ￡?!BD = 0x4x3e,x4e
!BD =?!BC +?!CD = 5?!a +5?!b = 5(?!a +?!b ) = 5?!AB;
x3ex50?!AB ￡?!BD = 0,x4,A;B;Dx34x17x28x74.
11,x34x66x1fx20?!OA;?!OB;?!OCx2dx2e
!OB ￡?!OC +?!OC ￡?!OA+?!OA￡?!OB = 0:
x73x53,(1)?!OA;?!OB;?!OCx28xf;
(2)x34x17A;B;Cx28x74.
x3x4,(1)x26x53,?!OA;?!DB;?!OCx28xfx7bx41x53x54:x31x6bx6ex6fx1fx20?!p,x27,?!p ￠?!OA =?!p ￠?!OB =?!p ￠?!OC = 0
x4x3e,x49?!p =?!OB ￡?!OC,x4a?!p ￠?!OB =?!p ￠?!OC = 0,x51x21
!OB ￡?!OC +?!OC ￡?!OA+?!OA￡?!OB = 0;
x3ex50
!p ￡?!OA =?(?!OC ￡?!OA)￠?!OA?(?!OA￡?!OB)￠?!OA = 0:
x1e?!p 6= 0,x4a?!OA;?!OB;?!OCx28xf;x8x1f?!p = 0,x4a?!OBx42?!OCx28x74,x67x47?!OA;?!OB;?!OCx28xf.
(2)x26x53A;B;Cx28x74x7bx41x53x54,?!AB ￡?!AC = 0,x21
!AB =?!OB!OA;?!AC =?!OC!OA;
x23x24
!AB ￡?!AC = (?!OB!OA)￡(?!OC!OA) =?!OB ￡?!OC +?!OA￡?!OB +?!OC ￡?!OA = 0;
x21A;B;Cx34x17x28x74.
12,x8x1f?!a =?!p ￡?!n,?!b =?!q ￡?!n,?!c =?!r ￡?!n,x4a?!a ;?!b ;?!c x28xf,x19x53x54x39.
x3x4,x4ex3c?!n ￠?!a =?!n ￠?!b =?!n ￠?!c = 0,x1e?!n 6= 0,x4a?!a ;?!b ;?!c x28xf; x29x4a,x4e?!n = 0x3ex50
!a =?!b =?!c = 0,x67x28xf.
x1 x2 1–8
￠ 23 ￠
1,x7cx7dx1dx1ex1fx20x42x13x29x28xf:
(1)?!a = 3?!i?2?!j!k,?!b = 5?!i +4?!j?3?!k,?!c = 11?!i!k ;
(2)?!a =?2?!i +3?!j!k,?!b =?!i?4?!j?2?!k,?!c =?!i?5?!j +2?!k ;
(3)?!a =?!i +2?!j +3?!k,?!b = 4?!i?5?!j +?!k,?!c = 7?!i?5?!j +8?!k ;
(4)?!a =!j +3?!k,?!b = 2?!i?2?!j,?!c =!i +2?!j?3?!k,
x5,(1) (?!a ;?!b ;?!c ) =
flfl
flfl
flfl
3?2?1
5 4?3
11 0?1
flfl
flfl
flfl = 88 6= 0,x23x24?!a ;
!b ;?!c x55x28xf.
(2) (?!a ;?!b ;?!c =
flfl
flfl
flfl
2 3?1
1?4?2
1?5 2
flfl
flfl
flfl = 25 6= 0,x23x24?!a ;
!b ;?!c x55x28xf.
(3) (?!a ;?!b ;?!c ) =
flfl
flfl
flfl
1 2 3
4?51
7?5 8
flfl
flfl
flfl =?40 6= 0,x23x24?!a ;
!b ;?!c x55x28xf.
(4) (?!a ;?!b ;?!c ) =
flfl
flfl
flfl
0?1 3
2?2 0
1 2?3
flfl
flfl
flfl = 0,x23x24?!a ;
!b ;?!c x28xf.
2,x78x67x4ex1fx20?!a,?!b,?!c x23x2ex2ax15xcxdxexfx10x15x10x10:
(1)?!a =!i +?!j +2?!k,?!b =?!i!j +3?!k,?!c = 2?!i +?!j +?!k ;
(2)?!a = 3?!i +?!k,?!b = 2?!i!j +?!k,?!c =?!i +5?!j!k ;
(3)?!a =?!i!j!k,?!b =?3?!i?2?!j +?!k,?!c =?!i +?!j ;
(4)?!a =?!i,?!b =?!i +?!j ;?!c =?!i +?!j +?!k,
x5,(1) V = j(?!a ;?!b ;?!c )j =
flfl
flfl
flfl
flfl
flfl
flfl
1 1 2
1?1 3
2 1 1
flfl
flfl
flfl
flfl
flfl
flfl = 15.
(2) V = j(?!a ;?!b ;?!c )j =
flfl
flfl
flfl
flfl
flfl
flfl
3 0 1
2?1 1
1 5?1
flfl
flfl
flfl
flfl
flfl
flfl = j?1j = 1.
(3) V = j(?!a ;?!b ;?!c )j =
flfl
flfl
flfl
flfl
flfl
flfl
1?1?1
3?2 1
1 1 0
flfl
flfl
flfl
flfl
flfl
flfl = j?1j = 1.
(4) V = j(?!a ;?!b ;?!c )j =
flfl
flfl
flfl
flfl
flfl
flfl
1 0 0
1 1 0
1 1 1
flfl
flfl
flfl
flfl
flfl
flfl = 1.
3,x64x1dx1dx1ex6cx17x13x29x28xf:
(1) A(1;2;?1),B(0;1;5),C(?1;2;1),D(2;1;3);
(2) A(3;?2;1),B(2;0;?1),C(?1;?4;5),D(3;?2;4);
(3) A(1;2;?3),B(3;5;?1),C(0;?2;7),D(2;1;3);
(4) A(1;0;1),B(0;?1;2),C(1;2;?2),D(2;0;?21).
x5,x26x64x1dA;B;C;Dx6cx17x13x29x28xf,x7bx41x64x1d?!AB;?!AC;?!ADx77x34x66x1fx20x13x29x28xf,x23x24x7bx41x68
(?!AB;?!AC;?!AD)x13x29x22x6f.
￠ 24 ￠
(1)?!AB = (?1;?1;6),?!AC = (?2;0;2),?!AD = (1;?1;4),
(?!AB;?!AC;?!AD) =
flfl
flfl
flfl
1?1 6
2 0 2
1?1 4
flfl
flfl
flfl = 0; x28xf.
(2)?!AB = (?1;2;?2),?!AC = (?4;?2;4),?!AD = (0;0;3),
(?!AB;?!AC;?!AD) =
flfl
flfl
flfl
1 2?2
4?2 4
0 0 3
flfl
flfl
flfl = 30 6= 0; x55x28xf.
(3)?!AB = (2;3;2),?!AC = (?1;?4;10),?!AD = (1;?1;6),
(?!AB;?!AC;?!AD) =
flfl
flfl
flfl
2 3 2
1?4 10
1?1 6
flfl
flfl
flfl = 30 6= 0; x55x28xf.
(4)?!AB = (?1;?1;1),?!AC = (0;2;?3),?!AD = (1;0;?22),
(?!AB;?!AC;?!AD) =
flfl
flfl
flfl
1?1 1
0 2?3
1 0?22
flfl
flfl
flfl = 45 6= 0; x55x28xf.
4,x64x1dx24A,B,C,Dx22x7ex17x15x6cxfx10x15x10x10:
(1) A(?1;0;1),B(?2;1;4),C(1;3;?3),D(?2;?1;3);
(2) A(2;?1;1),B(5;4;4),C(2;3;?1),D(4;1;2);
(3) A(1;0;2),B(1;?1;0),C(2;2;?1),D(3;1;0);
(4) A(2;2;?1),B(1;2;?2),C(2;?2;1),D(1;1;1).
x5,(1)?!AB = (?1;1;3),?!AC = (2;3;?4),?!AD = (?1;?1;2),
V = 16 j(?!AB;?!AC;?!AD)j = 16 ￡
flfl
flfl
flfl
flfl
flfl
flfl
1 1 3
2 3?4
1?1 2
flfl
flfl
flfl
flfl
flfl
flfl =
1
6,
(2)?!AB = (3;5;3),?!AC = (0;4;?2),?!AD = (2;2;1),
V = 16 j(?!AB;?!AC;?!AD)j = 16 ￡
flfl
flfl
flfl
flfl
flfl
flfl
3 5 3
0 4?2
2 2 1
flfl
flfl
flfl
flfl
flfl
flfl =
j?20j
6 =
10
3,
(3)?!AB = (0;?1;?2),?!AC = (1;2;?3),?!AD = (2;1;?2),
V = 16 j(?!AB;?!AC;?!AD)j = 16 ￡
flfl
flfl
flfl
flfl
flfl
flfl
0?1?2
1 2?3
2 1?2
flfl
flfl
flfl
flfl
flfl
flfl =
10
6 =
5
3,
(4)?!AB = (?1;0;?1),?!AC = (0;?4;2),?!AD = (?1;?1;2),
V = 16 j(?!AB;?!AC;?!AD)j = 16 ￡
flfl
flfl
flfl
flfl
flfl
flfl
1 0?1
0?4 2
1?1 2
flfl
flfl
flfl
flfl
flfl
flfl =
5
3,
￠ 25 ￠
x70x74
x74
x75x75x75x75x75x71x20
x40x74
x74
x70x75x75x75x75x75x40x20x71x75x75x75x75x75x71x20x40
x34x9x8
x8x9x8
x8x9x8
x9x8x8
x9x8
x74
x74
x70
x40x18 x40x18
x40x10x30x10
x30x10x30x10
x30x10x30x10
x30x10
x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x50x20 x20x8
x1cx48x1cx48x1cx48x1cx48
x1cx48x1cx48x1cx48x1cx48
x1cx48x1cx48x1cx48
x1cx48x1cx48x6x1cx22x9 x53x22 x53x22 x54
x22 x53x22 x53x22 x54x21 x53x22 x53x22 x54x21 x53
x22 x22x34x12 x34x12
x34x12 x34x12
x34x12 x34x12
x34x12 x34x12
x34x13
D
A B
C
C1
A1
D1
B1
x05x1x2
5,x8x9,xaxbx3bx40x10ABCD–A1B1C1D1,jABj = 4,jADj = jAA1j = 2,x73
(1)x17A1x8xcxfC1BDx15x32x33;
(2)x2ex74AD1x42xcxfC1BDx15x32x33.
x5,x18?!DA;?!DC;!DD1x15x14x2fx1fx20x13?!i ;?!j ;?!k,x5bx2bx2ex35x57x55x6a[D;?!i ;?!j ;?!k ],x4a
!DA
1 = (2;0;2);
!DB = (2;4;0);!DC
1 = (0;4;2):
(1)x17A1x8xcxfC1BDx15x32x33
d = j(
!DA
1;
!DB;!DC
1)j
j?!DB ￡!DC1j
=
flfl
flfl
flfl
flfl
flfl
flfl
2 0 2
2 4 0
0 4 2
flfl
flfl
flfl
flfl
flfl
flfl
j(8;?4;8)j =
32
12 =
8
3,
(2) x21x22AD1==BC1,x23x24AD1==xcxfC1BD,x21AD1x79x0x48x17x8xcxfC1BDx15x32x33x4x22AD1x8
xcxfC1BDx15x32x33.
d = j(
!DA;?!DB;!DC
1)j
j?!DB ￡!DC1j
=
flfl
flfl
flfl
flfl
flfl
flfl
2 0 0
2 4 0
0 4 2
flfl
flfl
flfl
flfl
flfl
flfl
12 =
16
12 =
4
3,
6,x18?!a = a1?!e1 +b1?!e2 +c1?!e3,?!b = a2?!e1 +b2?!e2 +c2?!e3,?!c = a3?!e1 +b3?!e2 +c3?!e3,x73x53
(?!a ;?!b ;?!c ) =
flfl
flfl
flfl
a1 b1 c1
a2 b2 c2
a3 b3 c3
flfl
flfl
flfl(?!e1;?!e2;?!e3):
x3x4,?!a ￡?!b = (a1?!e1+b1?!e2+c1?!e3)￡(a2?!e1+b2?!e2+c2?!e3) = a1b2?!e1￡?!e2+a1c2?!e1￡?!e3+b1a2?!e2￡?!e1+
b1c2?!e2￡?!e3 +c1a2?!e3￡?!e1 +c1b2?!e3￡?!e2 = (a1b2?b1a2)?!e1￡?!e2 +(a1c2?c1a2)?!e1￡?!e3 +(b1c2?c1b2)?!e2￡?!e3,
(?!a ;?!b ;?!c ) = (?!a ￡?!b )￠?!c = [(a1b2?b1a2)?!e1￡?!e2+(a1c2?c1a2)?!e1￡?!e3+(b1c2?c1b2)?!e2￡?!e3]￠(a3?!e1+
b3?!e2 +c3?!e3) = [(a1b2?b1a2)c3](?!e1;?!e2;?!e3)?[(a1c2?c1a2)b3](?!e1;?!e2;?!e3)+[(b1c2?c1b2)a3](?!e1;?!e2;?!e3) =fl
flfl
flfl
fl
a1 b1 c1
a2 b2 c2
a3 b3 c3
flfl
flfl
flfl(?!e1;?!e2;?!e3),(x7bx26x33x1ax34x79xdx1ex29x15x1dx4dx34x3ex78x67x50).
7,x73x53,j(?!a ;?!b ;?!c )j6j?!a jj?!b jj?!c j.
x3x4,j(?!a ;?!b ;?!c )j = j(?!a ￡?!b ) ￠?!c j =
flfl
flj?!a ￡?!b jj?!c jcosh?!a ￡?!b ;?!c i
flfl
fl 6 j?!a ￡?!b jj?!c j =
j?!a jj?!b jj?!c jsinh?!a ;?!b i6j?!a jj?!b jj?!c j.
8,x53x54x35x3ex65x36x56x29:
!a ￡(?!b ￡?!c )+?!b ￡(?!c ￡?!a )+?!c ￡(?!a ￡?!b ) = 0:
x3x4,x4ex37x617.7xb:
!a ￡(?!b ￡?!c ) = (?!a ￠?!c )?!b?(?!a ￠?!b )?!c ;
￠ 26 ￠
!b ￡(?!c ￡?!a ) = (?!a ￠?!b )?!c?(?!b ￠?!c )￠?!a ;
!c ￡(?!a ￡?!b ) = (?!b ￠?!c )?!a?(?!a ￠?!c )?!b ;
x65x17x4x50x22x23.
9,x53x54,x70x71x16x6cx17A,B,C,P x28xfx15x30x11x40x26x31x32x13,x38x15x23x2x2cx15x2fx59x1fx20?!a,?!b,?!c,?!p
x2dx2e
(?!p ;?!b ;?!c )+(?!a ;?!p ;?!c )+(?!a ;?!b ;?!p )?(?!a ;?!b ;?!c ) = 0:
x3x4,A;B;C;Px28xfx62x3fx63x62?!PA;?!PB;?!PCx28xf,xaxb
!a =?!OA;?!b =?!OB;?!c =?!OC;?!p =?!OP;
x21
!PA =?!a!p ;?!PB =?!b!p ;?!PC =?!c!p,
x21x4f?!PA;?!PB;?!PCx28xfx62x3fx63x62(?!PA;?!PB;?!PC) = 0,x62x3fx63x62
(?!a!p ;?!b!p ;?!c!p ) = 0;
x62x3fx63x62
(?!a ;?!b!p ;?!c!p )?(?!p ;?!b!p ;?!c!p ) = 0;
x62x3fx63x62
(?!a ;?!b ;?!c )?(?!a ;?!b ;?!p )?(?!a ;?!p ;?!c )?(?!p ;?!b ;?!c ) = 0;
x62x3fx63x62
(?!p ;?!b ;?!c )+(?!a ;?!p ;?!c )+(?!a ;?!b ;?!p )?(?!a ;?!b ;?!c ) = 0:
10,x53x54
(1) (?!a ￡?!b )￡(?!c ￡?!d ) = (?!a ;?!b ;?!d )?!c?(?!a ;?!b ;?!c )?!d ;
(2) (?!a ￡?!b )￡(?!c ￡?!d ) = (?!a ;?!c ;?!d )?!b?(?!b ;?!c ;?!d )?!a,
x3x4,(1) (?!a ￡?!b )￡(?!c ￡?!d ) =?(?!c ￡?!d )￡(?!a ￡?!b ) =?[?!c ￠(?!a ￡?!b )?!d!d ￠(?!a ￡?!b )?!c ] =
(?!a ;?!b ;?!d )?!c?(?!a ;?!b ;?!c )?!d,
(2) (?!a ￡?!b )￡(?!c ￡?!d ) = [?!a ￠(?!c ￡?!d )]?!b?[?!b ￠(?!c ￡?!d )]?!a = (?!a ;?!c ;?!d )?!b?(?!b ;?!c ;?!d )?!a,
11,x53x54x2x0x1x6cx66x1fx20?!a ;?!b ;?!c ;?!d x38x47
(?!b ;?!c ;?!d )?!a +(?!c ;?!a ;?!d )?!b +(?!a ;?!b ;?!d )?!c +(?!b ;?!a ;?!c )?!d = 0:
x3x4,x4ex3d10x61x22x23x3ex50
(?!a ;?!b ;?!d )?!c?(?!a ;?!b ;?!c )?!d = (?!a ;?!c ;?!d )?!b?(?!b ;?!c ;?!d )?!a ;
x39x3ax7x12x3bx62x3cx3dx3ex54x10x16x1fx20x48x3fx4x3ex53x50.
12,x53x54x1dx1ex1fx20x36x56x29:
(1) (?!a ￡?!b ;?!b ￡?!c ;?!c ￡?!a ) = (?!a ;?!b ;?!c )2;
(2) (?!b ￡?!c )￡(?!a ￡?!d )+(?!c ￡?!a )￡(?!b ￡?!d )+(?!a ￡?!b )￡(?!c ￡?!d ) =?2(?!a ;?!b ;?!c )?!d ;
(3) (?!a!d )￠(?!b!c )+(?!b!d )￠(?!c!a )+(?!c!d )￠(?!a!b ) = 0;
(4) (?!a!d )￡(?!b!c )+(?!b!d )￡(?!c!a )+(?!c!d )￡(?!a!b ) = 2(?!a ￡?!b +?!b ￡?!c +?!c ￡?!a ).
x3x4,(1) (?!a ￡?!b ;?!b ￡?!c ;?!c ￡?!a ) = [(?!a ￡?!b ) ￡ (?!b ￡?!c )] ￠ (?!c ￡?!a ) = [(?!a ;?!b ;?!c )?!b?
(?!b ;?!c ;?!b )?!a ]￠(?!c ￡?!a ) = (?!a ;?!b ;?!c )￠(?!c ;?!a ;?!b ) = (?!a ;?!b ;?!c )2.
￠ 27 ￠
(2) (?!b ￡?!c ) ￡ (?!a ￡?!d ) =?(?!a ￡?!d ) ￡ (?!b ￡?!c ) =?(?!b ;?!c ;?!a )?!d + (?!b ;?!c ;?!d )?!a,(?!c ￡
!a )￡(?!b ￡?!d ) =?(?!b ￡?!d )￡(?!c ￡?!a ) =?(?!c ;?!a ;?!b )?!d +(?!c ;?!a ;?!d )?!b,(?!a ￡?!b )￡(?!c ￡?!d ) =
(?!c ;?!d ;?!a )?!b?(?!c ;?!d ;?!b )?!a,x23x24
(?!b ￡?!c )￡(?!a ￡?!d )+(?!c ￡?!a )￡(?!b ￡?!d )+(?!a ￡?!b )￡(?!c ￡?!d ) =?2(?!a ;?!b ;?!c )?!d,
(3) (?!a!d )￠(?!b!c ) =?!a ￠?!b!a ￠?!c!b ￠?!d +?!c ￠?!d,
(?!b!d )￠(?!c!a ) =?!b ￠?!c!a ￠?!b!c ￠?!d +?!a ￠?!d,
(?!c!d )￠(?!a!b ) =?!a ￠?!c!b ￠?!c!a ￠?!d +?!b ￠?!d,
x76x79x53x56x29x1dx0 x1ex37x61x11x12x65x17x4a:
(?!a!d )￠(?!b!c )+(?!b!d )￠(?!c!a )+(?!c!d )￠(?!a!b ) = 0:
(4) (?!a!d )￡(?!b!c ) =?!a ￡?!b!a ￡?!c +?!b ￡?!d!c ￡?!d,
(?!b!d )￡(?!c!a ) =?!b ￡?!c +?!a ￡?!b +?!c ￡?!d!a ￡?!d,
(?!c!d )￡(?!a!b ) =!a ￡?!c +?!b ￡?!c +?!a ￡?!d!b ￡?!d,
x76x79x53x56x29x1dx0x1ex37x61x11x12x65x17x50:
(?!a!d )￡(?!b!c )+(?!b!d )￡(?!c!a )(?!c!d )￡(?!a!b ) = 2(?!a ￡?!b +?!b ￡?!c +?!c ￡?!a ):
13,x53x54x2x3cx0x1x1fx20ri (i = 1;2;3;4),x1dx29x2ax2b:
(?!r1 ￡?!r2)(?!r3 ￡?!r4)+(?!r1 ￡?!r3)(?!r4 ￡?!r2)+(?!r1 ￡?!r4)(?!r2 ￡?!r3) = 0:
x3x4,x3dx3ex1dx58.7,(?!r1 ￡?!r2)￠(?!r3 ￡?!r4) = (?!r1 ￠?!r3)(?!r2 ￠?!r4)?(?!r1 ￠?!r4)(?!r2 ￠?!r3),
(?!r1 ￡?!r3)￠(?!r4 ￡?!r2) = (?!r1 ￠?!r4)(?!r2 ￠?!r3)?(?!r1 ￠?!r2)(?!r3 ￠?!r4),
(?!r1 ￡?!r4)￠(?!r2 ￡?!r3) = (?!r1 ￠?!r2)(?!r3 ￠?!r4)?(?!r1 ￠?!r3)(?!r2 ￠?!r4),
x76x79x53x56x29x15x1dx0x1ex37x61x11x12x65x17x7x50x8x22x23.
14,x53x54?!a ;?!b ;?!c x28xfx15x30x11x40x26x31x32x13?!b ￡?!c,?!c ￡?!a,?!a ￡?!b x28xf.
x3x4,x21x22?!b ￡?!c,?!c ￡?!a,?!a ￡?!b x28xfx62x3fx63x62(?!a ￡?!b ;?!b ￡?!c ;?!c ￡?!a ) = 0,x71x4e12x61
x15(1)xb,(?!a ￡?!b ;?!b ￡?!c ;?!c ￡?!a ) = (?!a ;?!b ;?!c )2,x23x24?!a,?!b,?!c x28xfx62x3fx63x62?!a ￡?!b,?!b ￡?!c,
!c ￡?!a x28xf.
x1 x2 1–9
1,x34x35x36ABCx40x38x15x37x66x61x17x22B(?3;0),C(3;0),x7ex17Ax6bx2ex747x?5y?35 = 0x79x39x41,x73
x34x35x36x24x7fx15x42x43.
x5,x18x24x7fx15x57x55x22(x;y),x4a
( x = xA +xB +xC
3 =
xA
3
y = yA +yB +yC3 = yA3 ;
x4
( x
A = 3x
yA = 3y:
x25(xA;yA)x2dx2ex40x417x?5y?35 = 0,x51x52x4x5021x?15y?35 = 0.
￠ 28 ￠
-
6
x12x13x12
x13x12x13
x12x13x12
x15x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x2bx12x13x12
x13x12x13
x12x13x12
x13x12x13
x12x13x12
x15x2x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x6dx6ex6d
x6ex6dx6e
x2bxe
x5
x49x1d x49x1d x4ax1c x49x1c
x49x1d x49x1d x4ax1c x49x1c
x49x1d x49x1d
A BP O
M
N
Q
x
y
x02x1x2
2,x6bx3bx22lx15x74x78ABx79x47x48x41x17P,x6bABx15x43x44,x24AP,PBx22x38x11x12x2fx56x38x34x35x36AMPx3a
BNP,x73MNx15x16x17Qx15x42x43.
x5,x24ABx15x16x17Ox22x4bx17,x24ABx22xx45,x5bx2bx2ex35x57x55x6a[O;?!i,?!j ],x3cx13
A
l2 ;0
; B
l
2 ;0
; P(t;0);
l2 < t < l2
:
x4a
M
0
@ t?
l
2
2 ;
p3
2
t+ l2
1
A; N
0
@ t+
l
2
2 ;
p3
2
l
2?t
1
A:
x23x24MNx15x16x17Qx15x57x55x22:8
<
:
xQ = t2
l2 < t < l2
·
yQ =
p3
4 l:
x4Qx17x15x42x43x40x41x22,y =
p3
4 l
l4 < x < l4
·
.
￠ 29 ￠
