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8/13/2019 3as Math Sci Corrections2
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8/13/2019 3as Math Sci Corrections2
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:/
: . : . : :/ : . : . : :/ : . : . : : .
:/ : . : . : :/ : . : . :
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.
(((( ))))
; ;O u v
.
1 .A + =1 2a i z
M A 1 2z i= .M A 1 2z i= M A 1 2z i= +.M A 2z i= .
2 .1 2 4z i= 2 3z i= +3 3 3z i= 4 5
2z i=.
3 .ABCD 2 i +1 4i +3 2i+2 i.
( ) ( )1 4 2 1 3B Az z i i i = + + = +( ) ( )3 2 2 1 3C Dz z i i i = + = + B A C D
z z z z = AB DC= A BCD .4 .
Az,
Bz
Cz
Dz ( )3 ;1A( )3; 1B ( )0;2C
( )3 ;3D =.1(4 2Az = =,2Bz =2Cz .= A,BC O 2.2(2 2D AAD z z i= = =,3 2D CCD z z i= = + =
2A O OC CD DA= = = . AOCD= z r .z r =( )Arg z = +.z r=( )A rg z = .
1 1 1
z z r= =
1A rg
z
=
.
7
6
19
30
37
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33 3z z r= =( )3 3A rg z =.
1 1 1 1nn nnz rz z
= = =
( ) ( )1 nnA rg A rg z nA rg z nz
= = =
5 .1
2 2cos sin
2 2 4 4z i i
= + = +.
2
2 2 3 3cos sin
2 2 4 4z i i
= + = +.
3
2 2 3 3cos sin
2 2 4 4z i i
= = + .
1
3 1cos sin
2 2 6 6z i i
= + = +2
3 1 5 5cos sin
2 2 6 6z i i
= + = +
3
1 3cos sin
2 2 3 3z i i
= = + 4 1 3 2 2cos sin
2 2 3 3z i i
= = + .
( )1 3 3 cos 0 sin 0z i= = +( )2 5 5 cos sinz i = = +3 cos sin
2 2z i i
= = +4
1 1cos sin
2 2 2 2z i i
= = +
5 7 7 cos sin2 2
z i i
= = +
.
z x= 0x < ( )cos0 sin0z x i= + 0x > ( )cos sinz x i = +.z yi= 0y < cos sin
2 2z y i
= +
0y > cos sin
2 2z y i
= +
.
1
2 21 2 2 cos sin
2 2 4 4z i i i
= + = + = +
2
2 23 3 3 2 3 2 cos sin
2 2 4 4z i i i
= = = +
3
1 3 2 25 15 2 5 2 5 cos sin
2 2 3 3
z i i i
= = = +
4
3 1 5 56 2 2 2 2 2 cos sin
2 2 6 6z i i i
= + = + = +
.
4 4
1 3
iZ
i
+=
4 41 3
iZ
i
+=
.
1(( )( )
( )4 4 1 34 4
1 3 1 341 3
i iiZ i
i
+ ++= = = + +
.
41
42
43
44
46
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2(
4 2 cos sin5 54 4
2 2 cos sin12 12
2 cos sin6 6
i
Z i
i
+
= = +
+
.
3(1 1 5 5
cos sin12 122 2
iZ
= +
( )
20092009
2 2 cos sin12 12
Z i
= +
5 5
2 2 cos sin12 12
Z i
= +
.
6 .42
i
Az e
= 2AOA z= =( )4
AArg z
=.
23
2
i
Bz e
= 32
BOB z= =( )
2B
Arg z
=.
iCz e
= 1COC z= =( )CArg z =.5
62i
Dz e
= 2DOD z= =( ) 56
DA rg z
= .
3
4 3 3 2 2
6 6 cos 6 3 2 3 24 4 2 2
i
e isin i i
= + = + = +
.
3
2 3 3
5 5 cos 52 2
i
e isin i
= + =
.
( )1 1 1
cos
2 2 2
ie isin
= + = .
2
3 2 2 1 3
2 3 2 3 cos 2 3 3 33 3 2 2
i
e isin i i
= + = = .
7 .22 6 5 0z z + =2' 1 i = '3=
2
iz
=3"
2
iz
+=.
2 3 0z = + 2 23z i= 3z i=3z i= .1 2
1 2
5
2
z z
z z
=
+ = 1z2z 2 2 5 0z z+ + =.
48
49
56
A
O
B
C
D
0 1
1
x
y
A
O
B
C
D
58
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:/
: . : " ." : :/ : . : " ." : :/ : . : " ." : .
+ ='z az b :/ : . : .
: :/ : . : . :
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8/13/2019 3as Math Sci Corrections2
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:/ : . : . :
.
1 .2(
" " " " " "3
A B B C C A
AB BC CA= = =.
2(" " " " " "
2A B B C C A
AB BC CA= = =.
1(' ' 2M N MN=.
)2 ( )1 4z i z= + + 4z i= .A3(
'2
AM
A M=( ) ( ); ' arg 1
4AM A M i
= + =
.
)4 ' ' ' ' ' ' 2B C B D C DBC BD CD
= = = BCD' ' 'B C D .2 .
)1 22
A K
A B=
4
.
)2 2 4
.
)3 2 4
.
1(3
2
6
.2(
3
3
2
.3(
2
3
6
.4(
1
2
3
.
3 .2(
9 9 3 6 3 5 3 3 10'
2 2 2 2z i z i
+ = + +
.
T ( )' 1z i z i= + ( )1 4
2.
10
11
17
16
1
2
5
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5 . 1S
2
3
.( )S A C= . OCD OA B S . 1S O 2
3
.
.1 .
1(( )0A( )8 8C i+( )4 4P i+( )4 8Q i+.
2(( )1
1
4
= +0 =.
3(2
4 =
4
=.
4(( )16
1 25
i = +
3 .)1 3 9
4z iz i= +.
.=( )A O=( )O B)21. 0O A = C ( )1 2f C = C( )f = 2 C ) ) ( )O A ( ) ( )O B ( )A ( )B ( ) ( )O A B
O .( )AB ( )f A O=( )f O B=( )f = O A
B O
=
2A B O = .
35
26
50
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:
. :
. :
. :
. .
: .
: .
)1())2 2( 1) 0k k kk G A k G B k G C + + = )2 1) ( )kk G A k BA A C + =
(2( 1 kk G A k BC + = 2 1k kG A BC k= +
3(1 1MG MG =
)E( [ ]1 1G G 4(12 2 2MG MA MB MC MC= +
12 2MG MC=
1G C=
)F( 1G =1R G C5((1 2 1(0;0;8) ; (0;0;2) ; (0;0;0)G G G
1 1 3 302
G GR = = ))F))E
) 2 2 2
( ) : 4
( ) : 30
E z
F x y z
=
+ + =)2 2 ) : 26C x y+ =
)2(:I(1( G { }( ;1);( ;1);( ;3)O A C 3 0GO GA GC + + =
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5GC AC OC = + 5 2GC A j OJ JC = + + 0A j OJ+ = 5 2GC JC = 5 ;.. .2.GC JC .
2(1 3
; ;05 5
G
II(1(3 5MO MA MC MG+ + =
2(( )3 . 0MO MA MC MB+ + =
MG MB= )E( ]]GB
4
1
2
A B
13
4
AC
. 4 ( 1) ( 1)(3) (2)( 4) 15A B A C = + + = 2 2 24 ( 1) 2 21AB = + + =26A C =
. 15cos 0,6419546.
A B A CBA C
A B A C
= =
0130BA C .
) ; ; )M x y z )P ) ( )M P . 0AM u = = +:2 7 0x z))P . ( 1;2;1)n))P
)P(:2 2 0x y z + + =)1 A )p()q( 5
6
)2 M )p()q: ( :2 2
6 6
x y z x y z+ += 2 2x y z x y z+ = +
3 0x z+ = = +2 3 0x y z
13
14
30
2
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:
.
: d .
: . . )1(I( A( 1 ; 2 ; 2 )B( 3 ; 2 ; 1 )C( 1 ; 3 ; 3 )
)1 ABAC ABA C
2( 1p2p (2;0; 1)u . 0AM u = k 0k. 0M
( ) A ( ) 0A M A ( ).
)2: (1(( .........3( ) : 0
15 5
x t
AB y t
z t
=
=
= +
(y = z = 0 t = - 3 ( E( 9 ; 0 ; 0(30
5
A B
320
10
AC
A BA C
)1 1 1
2
n
8
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2(.........
4
( ) : 2
1 2
x t
D y t t
z t
= +
=
= +
3(H ))D))P ( 4 + t ) ( -2 t ) + 2 ( 1 + 2t ) = 0 43t =
8 2 5; ;
3 3 3H
=( ; )d A u '(2;1;1)u(3;3;0)A ( ; )d B v= (3;1;3)B(1;2; 1)u A B u v = +
2 02 23
+ =
+ =
=
1 23 2
=
=
=
.
: 3 2
3
3 '
1 2 '
3 '
x t
y t
z t
x t
y t
z t
= +
= + =
= + = +
=
3 2
3
3 '
' 6
' 5
x t
y t
z t
t t
t
t
= +
= + =
= =
=
1( an bc
. 0
. 0
n A B
n A C
=
=
1
2
2
A B
23
1
AC
2 2 02 3 0
a b c
a b c
+ + =
+ + =
4
3
1
3
a b
c b
=
=
431
n
( M( x ; y ; z )2 . 0A M n = )4 1) 3( 1) ( 1) 0x y z + + + + = ( ) : 4 3 0P x y z+ + =1( A . 0AB A C =
2( ))P1
1
1
n
3AB n=
)p( ))AB3(( ') : 0A P ax by cz d + + + =3 2 2d a b c= +
3
57
13
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( ')P A C ( ')P 'a
n b
c
A C
) ') : 0P x z =4(
3 0
0
x y z
x z
+ + =
=
2 3
x z
y z
=
= + .........3 2x ty t t
z t
==
=
II(1(D( 0 ; 4 ; - 1 )
3
6
3
AD
ABC1 2
1
n
3AD n= ( ) ( ')A D P2(
127
3
AB CV S A D= =
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:
. :
:
. .
: .
: :
)1( /
)2: ( .
.
)1 2 2 2y z R+ = A 2 2 13y z+ =)2 )C(1( ) : 2p x = 13. )C(1( ) : 3p y = ( );o i.
)C(1( ) : 4p z = .
1( ( ) ........2
0 .
x t
y tz t
=
= =
)C( 2 2 2 2x y k z+ = 2k )C( )D( )C( )D(
1t = ((2;0; 1 2 4k = 2 2 24x y z+ =
2(( 2 2 24x y zz a
+ =
= 2 2 24x y a+ =24 4a = 1a =
5
7
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