Аналитическая геометрия. Комплексные числа: Методические указания и контрольные работы по курсу ''Высшая математика

  • Published on
    08-Dec-2016

  • View
    213

  • Download
    1

Embed Size (px)

Transcript

  • .

    : .

    1

    : . .

    2002

  • 2

    1.

    z x iy= + , x y - , i - ( , : 2 1i = ).

    x y z . : Re ; Imx z y z= = . z x iy= + .

    1 1 1z x iy= + 2 2 2z x iy= + , , .. 1 2x x= 1 2y y= . 0 0 0 i= + . 1 1 0 i= + . 0z i y= + . z x i y= z x i y= + .

    , , :

    1) . 1 1 1z x iy= + 2 2 2z x iy= + , ( ) ( )1 2 1 2 1 2z z x x i y y+ = + + + .

    2) . 1 1 1z x iy= + 2 2 2z x iy= + , ( ) ( )1 2 1 2 1 2z z x x i y y = + .

    3) . 1 1 1z x iy= + 2 2 2z x iy= + , ( ) ( )1 2 1 2 1 2 1 2 2 1z z x x y y i x y x y = + + .

    , , 2i 1.

    4) . 1 1 1z x iy= + 2 2 2z x iy= + , 1 1 2 1 2 2 1 1 2

    2 2 2 22 2 2 2 2

    z x x y y x y x yiz x y x y

    + = +

    + +.

    , , .

    z x iy= + ( , )M x y .

    , .

    M , z z . :

    y ( , )M x y x 0

  • 3

    2 2 ; argz x y z = = + = . 0z

    : - z , 2 k + , 0, 1, 2,...k = .

    arg z = , 2 2 2 2

    cos , sinx yx y x y

    = =+ +

    tg yx

    = .

    x y , cos ; sinx y = = .

    : ( )cos sinz i = + . .

    :

    10. . ( )1 1 1 1cos sinz i = + ( )2 2 2 2cos sinz i = + , ( ) ( )( )1 2 1 2 1 2 1 2cos sinz z i = + + + , ..

    .

    20. . ( )1 1 1 1cos sinz i = + ( )2 2 2 2cos sinz i = + ,

    ( ) ( )( )1 1 1 2 1 22 2

    cos sinz iz

    = + , ..

    .

    n : ( )cos sinn nz n i n = + , ( )cos sinz i = + .

    n - ( )cos sinz i = + n , :

    2 2cos sinnkk kz i

    n n

    + + = +

    , 0, 1, ..., 1 ,k n n N= .

    1.

    25 22 1iz

    i +

    = + .

    , ( )25 4 2i i i i i i= = = . , ( ) ( )( ) ( )

    ( )2 22 22 1 2 4 32 4 3 9 24 9 242 1 1 2 1 2 5 25 25 25 24

    i i ii i iz ii i i

    + + = = = = = = + + .

    , 9 24Re ; Im25 25

    z z= = .

    2.

  • 4

    (3 ) (4 2 ) 2 6.

    (4 2 ) (2 3 ) 5 4i x i y ii x i y i

    + + = + + + = +

    . 23 4 2 (3 )(2 3 ) (4 2 ) ( 9 7 ) (12 16 )

    4 2 (2 3 )21 23 ,

    2 6 4 2(2 6 )(2 3 ) (4 2 )(5 4 )

    5 4 (2 3 )(14 18 ) (12 6 ) 2 44 ,

    3 2 6(3 )(5 4 ) (4 2 )(2 6 ) (19 7 ) ( 4 28

    4 2 5 4

    x

    y

    i ii i i i i

    i ii

    i ii i i i

    i ii i i

    i ii i i i i

    i i

    + = = + + = + =

    + +

    =

    + + = = + + + + =

    + +

    = + =

    + = = + + + = + +

    + +)

    23 21 .

    i

    i

    =

    =

    , 2 44 2(1 22 )(21 23 ) 2( 485 485 ) 970 970 1 ,21 23 (21 23 )(21 23 ) 970 97023 21 (23 21 )(21 23 ) 0 970 .21 23 (21 23 )(21 23 ) 970

    x

    y

    i i i i ix ii i ii i i iy ii i i

    += = = = = = +

    +

    = = = = = +

    , 1 ,x i y i= + = . 3. 1z ,

    ( )(1 2 ) (1 )(3 4 ) 1 7i z i iz i i + + = + .

    , ( 5 5 ) 10i z i = ,

    210 2 2 (1 ) 2 2 1 .5(1 ) 1 (1 )(1 ) 1 1

    i i i i i iz ii i i i

    += = = = =

    + + + +

    1z i= - .

    32 ; arg ctg14

    z z ar = = = = = ,

    z : 3 32 cos sin4 4

    z i = + .

    1.

    :

  • 5

    1) ( )61 2 )i+ ; 2) ( ) ( )7 72 2i i+ + ; 3) ( )5

    3

    1(1 )

    ii

    +

    .

    2.

    1) 11

    ii

    +

    ; 2) ( )31 3 3 5i i+ + + ; 3) ( )2 3

    3 2

    1 2 (1 )(3 2 ) (2 )

    i ii i

    + + +

    ;

    4) 12

    32

    i +

    ; 5) ( )52 2i+ ; 6) 3

    1 22 2

    i +

    .

    3.

    1) 1 32 2

    i + ; 2) 11

    ii

    +

    ; 3) ( )34 3i + ;

    4) 5( 1 )i + ; 5) 1 3i ; 6) 2

    1 32 2

    i +

    .

    4.

    1) (2 ) (2 ) 6

    (3 2 ) (3 2 ) 8i x i yi x i y

    + + = + + =

    ;

    2) 2 10

    2 203 (1 ) 30

    x yi zx y zi

    xi yi i z

    + = + = + + =

    .

    5. 1) 2 (2 ) ( 1 7 ) 0x i x i+ + + + = ; 2) 2 (3 2 ) (5 5 ) 0x i x i + = ; 3) 2(2 ) (5 ) (2 2 ) 0i x i x i+ + = .

    2.

    0Ax By Cz D+ + + = ( 1)

    , , , . .

    (1) , . , ,

  • 6

    , . , , . , . , , , , .

    a, b , , :

    1x y za b c

    + + = . ( 2 )

    , , (cos ,cos , cos ), :

    cos cos cos 0.x y z p + + = (3)

    (1) , :

    2 2 2

    1 .MA B C

    = + +

    (4)

    '

    (1). (1)

    :

    2 2 2

    2 2 2 2 2 2 2 2 2cos ; cos ; cos

    DpA B C

    A B CA B C A B C A B C

    = + + = = = + + + + + +

    m

    . ( 5 )

    (', ',z') (1) / / /

    2 2 2

    Ax By Cz DA B C

    + + +

    =+ +

    (6)

    , , / / /cos cos cosx y z p = + + , ( 6/ )

  • 7

    . . , , ,

    1 1 1 1

    00

    Ax By Cz DA x B y C z D

    + + + = + + + =

    ( 7 )

    :

    1 1 12 2 2 2 2 2

    1 1 1

    cos AA BB CCA B C A B C

    + +

    = + + + +

    . ( 8 )

    (7):

    1 1 1

    A B CA B C

    = = . ( 9 )

    :

    1 1 1 0AA BB CC+ + = . ( 10 ) ( ) ( )1 1 1 2 2 2; ; , ; ;x y z x y z ( )3 3 3; ;x y z , :

    1 1 1

    2 2 2

    3 2 3

    11

    011

    x y zx y zx y zx y x

    = . ( 11)

    ( ) ( )1 1 1 2 2 2; ; , ; ;x y z x y z , ( )3 3 3; ;x y z ( )4 4 4; ;x y z ,

    1 1 1

    2 2 2

    3 3 3

    4 4 4

    11

    011

    x y zx y zx y zx y x

    = . ( 12)

    , , , :

    1 1 1

    2 2 2

    3 3 3

    4 4 4

    111 0161

    x y zx y z

    Vx y zx y x

    = = . ( 13 )

    , (V >0).

  • 8

    1. 4 3 0x y z + = : (-1; 6; 3), B(3; -2; -5), (0; 4; 1), D(2; 0; 5), E(2; 7; 0), F (0; 1; 0)?

    2. , Ax + By +Cz + D == 0 , , - , .

    3. : 1. 3 5 1 0x y + = ; 2. 2 3 7 0x y z+ = ; 3. 9 2 0y = ; 4. 8 3 0y z = ; 5. 5 0x y+ = .

    4. : 1. ( )xz (2; 5;3) ; 2. z ( 3;1; 2) ; 3. x (4;0; 2)

    (5;1;7) . 5. ,

    : 1. 2 3 12 0x y z + = ; 2. 5 3 15 0x y z+ = ; 3. 1 0x y z + = ; 4. 4 6 0x z + = ; 5. 5 2 0x y z + = ; 6. 7 0x = . 6.

    5 2 10 0x y+ = . 7. 3 2 18 0x y z+ =

    . , , , , , .

    8. (7; 5;1)P , .

    9. , , . , , : 6; 29 ; 5AB BC CA= = = .

    10. : 1) 2 9 6 22 0x y z + = 2) 10 2 11 60 0x y z+ + = 3) 6 6 7 33 0x y z + = 11. 15 10 6 190 0x y z + =

    .

  • 9

    12. , 6 , : : : 1:3: 2a b c= .

    13. , 2 2 9 0x y z + = .

    14. : 11 , 55 , 10a b c= = = . ,

    . 15. 2 5 0x y z + =

    ( )yz . 16. ,

    6 2 9 121 0x y z+ + = . 17. , , (3: 6;2)P

    , . 18. :

    1) (3;1; 1) 22 4 20 45 0x y z+ = . 2) ( )4;3; 2 3 5 1 0x y z + + = .

    3) 12;0;2

    4 4 2 17 0x y z + + = .

    19. ( ) ( ) ( ) ( )0;6;4 , 3;5;3 , 2;11; 5 , 1; 1;4A B C D .

    20. ( )1;3; 2A ( )7; 4;4B . B , AB .

    21. : 1) 4 5 3 1 0x y z + = 4 9 0x y z + = ; 2) 3 2 15 0x y z + + = 5 9 3 1 0x y z+ = ; 3) 6 2 4 17 0x y z+ + = 9 3 6 4 0x y z+ = . 22. : 1) ( )2;7;3

    4 5 1 0x y z + = ; 2)

    : 2 5 3 0x y z + + = 3 7 0x y z+ = ; 3) ( )0;0;1L ( )3;0;0N

    3 ( )xy .

    23. : 11 2 10 15 0x y z = 11 2 10 45 0x y z = .

    24. 3 6 2 14 0x y z + = .

  • 10

    25. : ( ) ( ) ( )0;0;2 , 3;0;5 , 1;1;0A B C ( )4;1;2D . .

    26. , . 27. ,

    : 1) ( ) ( ) ( )3; 1; 0 , 0; 7; 2 , 1; 0; 5 ( )4; 1; 5 ; 2) ( ) ( ) ( )1; 1; 1 , 0; 2; 4 , 1; 3; 3 ( )4; 0; 3 . 28. : 1) 4 2 3 0 , 3 5 0 , 3 12 62 7 0x y z x y z x y z + = + + = + + = ; 2) 5 8 0 , 2 3 1 0 , 2 3 2 9 0x y z x y z x y z+ = + + = + = ; 3) 2 5 4 0 , 5 2 13 23 0 , 3 5 0x y z x y z x z + = + + = + = . 29. 4 3 1 0x y z + =

    5 2 0x y z+ + = : 1) ; 2) ( )1;1;1 ; 3) y ; 4) 2 5 3 0x y z + = .

    3.

    ; :

    1 1 1 1

    00

    Ax By Cz DA x B y C z D

    + + + = + + + =

    . ( 14 )

    , (14). , ; , . , (14) :

    x mz ay nz b

    = + = +

    . ( 15 )

    . ( )1 1 1; ;x y z ( )2 2 2; ;x y z , , :

  • 11

    1 1 1

    2 1 2 1 2 1

    x x y y z zx x y y z z

    = =

    . ( 16 )

    : 2 1 2 1 2 1; ;x x m y y n z z p = = = , :

    1 1 1x x y y z zm n p

    = = ; ( 17 )

    ;m n p

    : : cos : cos : cosm n p = , ( 18 ) :

    2 2 2

    2 2 2

    2 2 2

    cos

    cos

    cos

    mm n p

    nm n p

    pm n p

    =

    + + =

    + + = + +

    . ( 19 )

    (19) , . .

    x a y b z c

    m n p

    = = 1 1 11 1 1

    x a y b z zm n p

    = = ( 20 )

    :

    1 1 12 2 2 2 2 2

    1 1 1

    cos mm nn ppm n p m n p

    + +

    = + + + +

    . ( 21 )

    :

    1 1 1

    m n pm n p

    = = . ( 22 )

    :

    1 1 1 0mm nn pp+ + = . ( 23 ) ,

    (17), (14) . : (14) , , (15), z (15) :

  • 12

    1x a y b z

    m n

    = = ;

    0c = 1p = . : (14)

    ,a b c - , ,m n p ,

    1 1 1 1 1 1

    : : : :B C C A A B

    m n pB C C A A B

    = . ( 24 )

    ,m n p , , , .

    0x a y b z c

    n p

    = =

    0x a = y b z cn p

    = ;

    x . 0 0

    x a y b z cp

    = = 0x a =

    0y b = ; z . ( ), ,a b c

    cos , cos , cos , :

    cos cos cosx a y b z c

    = = . ( 25 )

    (17)

    2 2 2

    1m n p

    + +

    .

    (25) ( ); ;x y z ( ), ,a b c . (17) .

    (20)

    1 1 1

    1 1 1

    0a a b b c c

    m n pm n p

    = . ( 26 )

  • 13

    1. :

    1) 1 1 1

    00

    Ax By CzA x B y C z

    + + = + + =

    ; 2) 1 1

    00

    Ax CzA x C z

    + = + =

    ;

    3) 1 1

    00

    Ax DA x D

    + = + =

    ; 4) 1 1

    00

    Ax By Cz DB y D

    + + + = + =

    .

    2. D

    3 2 6 0

    4 0x y z

    x y z D + =

    + + =

    z ?

    3. B D

    2 9 0

    3 0x y zx By z D

    + = + + + =

    ( )xy ?

    4.

    1 1 1 1

    00

    Ax By Cz DA x B y C z D

    + + + = + + + =

    :

    1) x ;

    2) y ;

    3) z

    4) ( )yz ;

    5) ( )xz ;

    6) ?

    5. ,

    5 8 3 9 02 4 1 0x y zx y z+ + =

    + =

  • 14

    .

    6.

    3 2 5 0

    1 0x y zx y z

    + + = + =

    ?

    7.

    4 2 5 0

    3 2 0x y z

    x y z + =

    + + =

    2 3 0x y z+ + = .

    8. , ( , , )a b c .

    9. , : ( ) ( ) ( ) ( )0; 0;2 , 4; 0; 5 ; 5; 3; 0 ; 1; 4; 2A B C D .

    10. , : ( )3;0;1 ,

    ( ) 40;2;4 ; 1; ; 33

    .

    11. ( )1 1, ,0x y ( )2 2;0;x z . .

    12.

    1) 1 5 24 3 12

    x y z += =

    ; 2) 7 3

    12 9 20x y z +

    = = .

    13. , ( )1; 5;3A , 600, 450 1200.

    14. , : 1 2 53 6 2

    x y z + = =

    3 12 9 6x y z +

    = = .

    15. , : ( ) ( ) ( ) ( )3; 1;0 , 0; 7;3 , 2;1; 1 , 3;2;6A B C D .

  • 15

    16.

    2 3 3 9 0

    2 3 0x y zx y z

    = + + =

    .

    17.

    5 6 2 21 0

    3 0x y z

    x z + + =

    + = .

    18.

    3 4 2 02 2 0x y zx y z =

    + =

    4 6 2 03 2 0

    x y zy z+ =

    + = .

    19. ( )2; 5;3 :

    1) z ;

    2) 1 2 34 6 9

    x y z += =

    ;

    3) 2 3 1 05 4 7 0

    x y zx y z

    + = =

    .

    20....

Recommended

View more >