МІНІСТЕРСТВО ОСВІТИ І НАУКИ УКРАЇНИ ХАРКІВСЬКА НАЦІОНАЛЬНА АКАДЕМІЯ МІСЬКОГО ГОСПОДАРСТВА

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  • .. , .. , ..

    I:

    2010

  • ________________..

    .. , .. , ..

    :

    ( 1- 2 6.050701 , )

    2010

  • 514.123 . I: : ( 1-2 6.050701 , ) / . . . . -; .: .. , .. , .. . .: , 2010. 79 .

    : , , ..-.., . ..

    , 10 26.05.2010 .

  • i Ii ...1. 2. ..2.1. .2.1.1. , 2.1.2. ...2.1.3. , .2.1.4. 2.2. . ..2.3. 3. ...3.1. ..............7.............10............13.............14............15............18.............20.............224.............8............24.............25.............26.............14

  • 3.1.1. 3.1.2. ..3.1.3. , ...............3.1.4. ..3.2. . 3.3. . ...4. , 4.1. . .4.2. ..4.3. 4.4. 5. ...............26............27.............28.............29.............31.............32.............34.............35.............36.............38.............39............425

  • 5.1. 5.2. .5.3. ....5.3.1. ...5.3.2. ...5.3.3. ...5.4. . 5.5. . .5.6. .5.7. .....5.8. .5.9. ........5.10. ..5.11. ..5.11.1. ...5.11.2. .. .............43.............44.............46.............48.............49.............50.............51.............53.............57.............59.............61............636............65............66............66............67............68

  • , . , . , , . . 7

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    . : i Power Point i -.i i i , i .i ii ii i i ( ). ii i, i. i i - i , . i , , i i, i . 8

  • i ii, i . i - . ii i Esc i i, , "i ". I - ( , i). ii . , i i. , : vm_kolosov@ksame.kharkov.ua

    9

  • 10

  • Ox, Oy Oz O (. 1). yxzOM(x;y;z). 1 M(x;y;z), , M M1M2 , = 1. M1(x1;y1;z1) M2(x2;y2;z2) Ox , Oy , Oz M M(x; y; z) .11xyz

  • ABC A(2;1;4), B(3;2;6), C(5;0;2). ABC . AM (. 2).

    M BC:12

  • xyzO. 2A(2;1;4)B(3; 2; 6)C(5; 0;2)M

  • 2.1.1. , 2.1.2. 2.1.3. , 2.1.4. 2.2. . 2.3. 132.1.

  • 2.1.1. , 0(x0;y0;z0) M(x;y;z) (.3). M , ,

    , . M0M. 3 14 2.1.

  • 2.1.2. A(xx0)+B(yy0)+C(zz0)=0 AxAx0+ByBy0+CzCz0 = 0. D= Ax0By0Cz0. , x, y, z, A, B, C , . . . . - x, y, z. x, y, z . 15

  • , :

    A = 0, By + Cz + D = 0 Ox;B = 0, Ax + Cz + D=0 Oy;C = 0, Ax + By + D=0 Oz (. 4);D = 0, Ax + By + Cz = 0 (. 5);A = 0 B = 0, Cz + D = 0 Oz (. 6);A = 0 i C = 0, By + D = 0 Oy;B = 0 i C = 0, Ax + D = 0 Ox;A = 0 i D = 0, By + Cz = 0 Ox;B = 0 i D = 0, Ax + Cz = 0 Oy;C = 0 i D = 0, Ax + By = 0 Oz; z = 0 Oxy; y = 0 Oxz; x = 0 Oyz.16

  • Ax + By + D = 0Ax + By + Cz = 0 Cz + D = 0xxyyzzxyz

  • M(2; 1; 3), N( 5; 6; 0) P(1; 3; 1). , M

    . 17

  • 2.1.3. , M1(x1; y1; z1), M2(x2; y2; z2) M3(x3; y3; z3), (.7). M(x; y; z) , . , . M1M2M3M. 7 , M1. M(x; y; z) 18

  • M(1; 1; 2), N(5; 6; 0) P(1; 3; 1). , . (x 1) (15 4) (y + 1) ( 12 0) + (z 2) ( 8 0) = 0;11 (x 1) + 12 (y + 1) 8 (z 2) = 0;11x 11 + 12y + 12 8z + 16 = 0;11x + 12y 8z + 17 = 0.( ) , A(3; 1; 2), B(4; 1; 1) C(2; 0; 2). 19

  • (x 3) (0 + 3) (y + 1) (0 + 3) + (z 2) (1 0) = 0;3 (x 3) 3 (y + 1) + (z 2) = 0;3x 3y + z 14 = 0.3x 9 3y 3 + z 2 = 0;

  • 2.1.4. Ox, Oy Oz M1(a; 0; 0), M2(0; b; 0) M3(0; 0; c) (.8). . abcM1M2M3Oxyz. 8 , , 20

  • 6x y + 4z + 12 = 0 . : 3x 2y + 6z 12 = 0 , (. 9). M1M2M3Oxyz. 921

  • 2.2. . 1 2 : 1 2 (. 10). , . , , 22

  • 1: 2x 3y + 2z + 1 = 0 Oxy. , M(3; 2; 4) x 2y + z 4 = 0.A(x x0) + B(y y0) + C(z z0) = 0;A(x 3) + B(y 2) + C(z + 4) = 0; : A = t, B = 2t, C = t;t(x 3) 2t(y 2) + t(z + 4) = 0;tx 3t 2ty + 4t + tz + 4t = 0 | (:t) ;x 3 2y + 4 + z + 4 = 0;x 2y + z + 5 = 0.23

  • 2.3. i: Aix + Biy + Ciz + Di = 0 (i=1,2,3) , , , . , : 2x 4y + 3z 1 = 0, 3x y + 5z 2 = 0, 4x + 3y + 4z = 0. . . 24

  • ( 1; 0; 1)

  • 3.1.1. ii i 3.1.2. i i 3.1.3. i , i i 3.1.4. i i 3.2. i . i i i 3.3. . i i i 253.1.

  • 3.1.1. ii i l M0(x0; y0; z0) , (.11). M0(x0; y0; z0)M (x; y; z). 11 M(x; y; z) M , () . , . 26 3.1.

  • 3.1.2. x, y z, : , t . . ( ). 27

  • 3.1.3. , l M1(x1; y1; z1) M2(x2; y2; z2). , . , M1(1; 2; 0) M2(2; 4; 3). 28

  • 3.1.4. . , l 1 2. 1: A1x + B1y + C1z + D1 = 0 i 2: A2x + B2y + C2z + D2 = 0, . 1. . 2. A1x + B1y + C1z + D1 + (A2x + B2y + C2z + D2) = 0, , , l. 29

  • l 1) ; 2) . 1) M0 . x = 0 , 2) . 30

  • 3.2. i . i i i l1 l2 i , . 31

  • 3.3. . . i l1 l2 , i ( ). : 1. l1 l2 . , l1 l2 . l1 l2, . 32

  • 2. l1 l2, . , d = 0. : i i l1 il2 i. d l1 l2 , l1 l2 . 33

  • 4.1. i . i i 4.2. 4.3. i i 344.4. i i

  • 4.1. . l 90 (.12). , , , . 35

  • 4.2. l . ( ), x, y, z . t(Am + Bn + Cp)t = (Ax0 + By0 + Cz0 + D).1) Am + Bn + Cp 0, , , 2) Am + Bn + Cp = 0, , Ax0 + By0 + Cz0 + D 0, M0(x0; y0; z0) l , t . . 3) Am + Bn + Cp = 0, , Ax0 + By0 + Cz0 + D = 0 , M0(x0; y0; z0) l , t . . 36

  • N M0(2; 5; 4) : 3x + 2y z 6 = 0 . N , M0 (.13). NM0. 13 M0N . M0N: x = 3t + 2; y = 2t 5; z = t + 4. , t, N 3(3t + 2) + 2(2t 5) ( t + 4) 6 = 0;t = 1. , N(1; 7; 5). 37

  • 4.3. Ax + By + Cz + D = 0 M0(x0; y0; z0) (.14). M1(x1; y1; z1)dM0(x0; y0; z0). 14 M1(x1; y1; z1) d M0

    Ax1 By1 Cz1 = D , d M0(2; 4; 3) : 3x 2y 6z 1 = 0. ( ). 38

  • 4.4. d M1(x1; y1; z1) l, : x = mt + x0 ; y = nt + y0 ; z = pt + z0 . . 1. M0(x0; y0; z0) (.15). 2. M1 , l (.16). M0lM1d. 15M1dlN. 16 N . , M1 l. , d = M1N. S l. . 39

  • 3. u = d 2(t), M1 l. d M1 l . , . tm u'(t) = 0: d = d(tm).40

  • d M1 l: M1(2; 3; 2), 1: ( 2 3 ). 41

  • 2: 1 , : 1(x 2) 2(y 3) + 2(z ( 2)) = 0;x + 2y 2z 12 = 0. : t 1 + 2( 2t + 2) 2(2t) 12 = 0;t = 1; x = 0; y = 4; z = 2. N(0; 4; 2)

  • 3:

  • 5.1. 5.3.1. 5.3. 5.4. . 5.5. . 5.11. 5.2. 5.3.2. 5.3.3. 5.6. 5.7. 5.8. 5.9. 5.10. 5.11.1. 5.11.2. 42

  • 5.1. , z : 2 + By2 + Cz2 + Dxy + z + Fz + Gx + Hy + z + L = 0, , ,..., L , , , , D, E, F . . : , , . .43

  • , , . . Oxyz (0; 0; z0) R (. 17). 5.2. i i i : i:x2 + 2 + z2 2x0x 20 2z0z + 02 + 02 + z02 R2 = 0. i , i i i : x2 + y2 + z2 = R2. (.18) ( x0)2 + ( 0)2 + (z z0)2 = R2. 17. 18 , (x; ; z) i i, i , = R 44

  • , , : x2 + y2 + z2 6x + 4y + 5z + 3 = 0.x2 + y2 + z2 6x + 4y + 5z + 3 = 0;(x2 6x) + (y2 + 4y) + (z2 + 5z) + 3 = 0; C(3; 2; 5/2) .45

  • 5.3. ii i () , () l, () l0, a0, l0 a0 . , , . , (, , , ..). 1. . , l0. Oxy (. 19). 1. Oxyz F(x,y)=0, z, S, Oz, . 1946

  • M(x; y; z) S N(x; y; 0) Oxy l0, , , M(x; y; z) F(x; y) = 0, z. , , S, , Oxy l0. 2. F(y; z) = 0, x, , Ox. F(x; z) = 0, y, , Oy. 47

  • 5.3.1. i i (.20), a = b = R, x2 + y2 + z2 = R2 . . 20 48

  • 5.3.2. ii i i i ii i i, i Oz, i i (. 21).. 2149

  • 5.3.3. i i i i i, i Oz, i (. 22).. 2250

  • 5.4. . () , () l, C(x0; y0; z0) () () l0, C l0. l0 . M(x; y; z) . , , , C(x0; y0; z0) N(X; Y; Z) : M(x; y; z) ( ) N(X; Y; Z), 51

  • ( ) (.23) O(0;0;0), a b, z = c, Oz. Oz , . , a =

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