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Реферат з математики на тему:

ПлощинаЗагальне рівняння площини та його дослідження

Покажемо, що алгебраїчною поверхнею першого порядку є площина. Для цього
доведемо такі теореми.

Теорема 1. Площина в прямокутній декартовій системі координат
визначається загальним рівнянням першого степеня відносно поточних
координат.

Доведення. Геометрично будь-яку площину в просторі хуz можна задати за
допомогою вектора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,
перпендикулярного до цієї площини, і точки M0 (x0, y0, z0),
Через яку проходить дана площина (рис.1).

Візьмемо довільну точку M (х, у, z) і знайдемо вектор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. Точка M належить заданій
площині тоді і тільки тоді, коли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 Тоді:
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Оскільки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то
скалярний добуток можна записати у вигляді

А(х — х0) + В(у — у0) + C(z — z0) = 0,

або

Ах + By + Cz — (Aх0 + Ву0 + Cz0) = 0. (1)

Позначивши

— (AX0 + Ву0 + Cz0) = D

дістанемо загальне алгебраїчне рівняння першого степеня:

Ах + By + Cz + D = О, (2)

Отже, будь-яка площина в декартових прямокутних координатах може бути
зображена рівнянням першого степеня.

Зауважимо, що рівняння (1) є рівнянням площини, яка проходить

через точку M0 (х0, у0, z0) перпендикулярно до вектора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 = (А, В, С). Доведемо тепер обернену
теорему.

Теорема 2. Загальне рівняння першого степеня

Ax + By + Cz + D = 0, (3)

де А, В, С і D — довільні дійсні числа; х, у, z — поточні координата,
визначає в декартовій прямокутній системі координат площину.

Доведення. Доберемо трійку чисел (х0, y0, z0), які задовольняють
рівняння (3). Це можна зробити таким чином. Два числа х0 і у0 візьмемо
довільно, а третє z0 знайдемо з рівняння (3). Тоді

Ах0 + Ву0 + Cz0 + D = 0.
(4)

Віднімаючи від рівняння (3) рівняння (4), дістаємо

А(х — х0) + В(у — у0) + C(z — z0) = 0.
(5)

Це рівняння є рівнянням площини, перпендикулярної до вектора

picscalex100010009000003c900000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02c00140011200000026060f001a00ffffffff000010000000c0ffffff
c6ffffff00010000860100000b00000026060f000c004d61746854797065000030001c00
0000fb0280fe0000000000009001010000000402001054696d6573204e657720526f6d61
6e001643c7771f43c77720c0c97700003000040000002d01000008000000320a60013a00
010000006e791c000000fb0280fe000000000000900100000002040200104d5420457874
726100000000fa080a831643c7771f43c77720c0c97700003000040000002d0101000400
0000f001000008000000320a6a005f000100000072790a00000026060f000a00ffffffff
0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d
000000000a0021008a0100000000000000009cf312002942c777040000002d0100000400
0000f0010100030000000000000000 = (А, В, С) і такої, що проходить через
точку M0 (х0, у0, z0).

Таким чином, кожна площина є поверхнею першого порядку, і, навпаки,
кожна поверхня першого порядку є площиною. Тому рівняння (l) або (3)
називається загальним рівнянням площини.

Рівняння
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(6)

називається векторним рівнянням площини. Враховуючи, що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 векторне рівняння
площини запишемо у вигляді:
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, або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Якщо у загальному рівнянні площини покласти z — z0 = 0, то дістанемо
рівняння

А(х — х0) + В(у — у0) = 0,

або Ах + By + С = 0,
(7)

де С = — (Ax0 + Ву0). Рівняння ( 7) називається загальним рівнянням
прямої, що лежить у площині хОу.

Дослідження загального рівняння площини

Розглянемо загальне рівняння площини .

Ах + Вy + Cz + D = 0.
(8)

де А, В, С і D — довільні числа, причому хоча б одне з перших трьох
відмінне від нуля.

Дослідимо окремі випадки цього рівняння:
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 Якщо D = 0, то рівняння
(8) набирає вигляду

Ах + By + Cz = 0. (9)

Це рівняння задовольняє точка О (0, 0, 0). Отже, рівняння (9) визначає
площину, яка проходить через початок координат.
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 Якщо А = 0, то рівняння
(8) має вигляд

By + Cz + D = 0
(10)

і визначає площину, нормальний вектор якої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 = (О, В, С) перпендикулярний до осі Ох.
Отже, рівняння (10) визначає площину, паралельну осі абсцис, або
перпендикулярну до площини yOz.

picscalex100010009000003150100000200700000000000050000000201010000000500
00000102ffffff00050000002e0119000000050000000b0200000000050000000c024001
20011300000026060f001c00ffffffff00004e0010000000c0ffffff26000000e0000000
660100000b00000026060f000c004d617468547970650000200005000000090200000002
05000000140200012e001c000000fb0280fe000000000000900100000001000200105379
6d626f6c00774000000076080a55b7c5c677c0c5c67720c0c97700003000040000002d01
000009000000320a0000000001000000b77900037000000026060f00d6004d6174685479
70655555ca00050100050244534d543500011357696e416c6c4261736963436f64655061
67657300110554696d6573204e657720526f6d616e00110353796d626f6c001105436f75
72696572204e65770011044d5420457874726100120008212f458f442f4150f4100f475f
4150f21f1e4150f4150f4100f445f425f48f425f4100f4100f435f4100f48f45f42a5f48
f48f4100f4100f40f48f417f48f4100f412a5f445f45f45f45f45f410f0c010001000102
02020200020001010100030001000400000a01000204862220b700000b00000026060f00
0c00ffffffff01000000000000001c000000fb021000070000000000bc02000000cc0102
022253797374656d000000000a0021008a0100000000ffffffff58e61200b9c1c6770400
00002d01010004000000f0010000030000000000000000 Якщо А = В = 0, а С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, то маємо рівняння площини, паралельної хОу:
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picscalex100010009000003150100000200700000000000050000000201010000000500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 Рівняння х = 0, у = 0, z
= 0 визначають відповідно координатні площини yOz, xOz, хОу.

Різні види рівнянь площини.

Рівняння площини у відрізках на координатних осях.

Розглянемо загальне рівняння площини

Ах + Ву+ Cz + D = 0,
(11)

коли всі його коефіцієнти і вільний член відмінні від нуля. Поділимо
обидві частини рівняння (11) на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і запишемо його у вигляді
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 (12)

Позначимо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 Тоді
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 (13)

Рівняння площини у вигляді (13) називається рівнянням у відрізках.

Знайдемо точки перетину площини (13) з координатними осями:
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 на осі абсцис у = z = 0,
тоді х = а,
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 на осі ординат х = z =0,
тоді у = b,
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 на осі аплікат х = у = 0,
тоді z = с.

Таким чином, площина, задана рівнянням у відрізках, відтинає на
координатних осях відповідно відрізки a, b і с.

Якщо потрібно побудувати площину, задану рівнянням, то зручно це
рівняння записати у відрізках на осях. Тоді по точках M1 (a, 0, 0), М2
(0, b, 0) і М3 (0, 0, c) легко побудувати площину (рис.2).

Рівняння площини, що проходить через три дані точки

Нехай дано три точки М1 (х1 у1, z1), М2 (х2, у2, z2), M3(x3,y3,z3), що
не лежать на одній прямій. Ці точки однозначно визначають площину, яка
проходить через них. Знайдемо рівняння цієї площини.

Візьмемо довільну точку простору M (х, у, z) (рис.3) і побудуємо
вектори:
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picscalex100010009000003540400000400890100000000050000000201010000000500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Точка M (х, у, z) належить шуканій площині тоді і тільки тоді, коли
вектори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 лежать у цій площині, тобто коли вони
компланарні.

Рис.2 Рис.3

Отже, мішаний добуток їх дорівнює нулю:
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(14)

Запишемо цей добуток через координати векторів, які перемножаються.
Маємо:
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(15)

picscalex100010009000003510000000000120000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02200220011200000026060f001a00ffffffff000010000000c0ffffff
a6ffffffe0000000c60100000b00000026060f000c004d61746854797065000050000a00
000026060f000a00ffffffff010000000000030000000000000000Якщо
радіуси-вектори точок М, М1, М2 і М3 відповідно позначити через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 то вектори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 можна зобразити у вигляді
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Тоді рівняння (14) можна записати таким
чином:picscalex100010009000003510000000000120000000000050000000902000000
0005000000020101000000050000000102ffffff00050000002e0118000000050000000b
0200000000050000000c02200220011200000026060f001a00ffffffff000010000000c0
ffffffa6ffffffe0000000c60100000b00000026060f000c004d61746854797065000050
000a00000026060f000a00ffffffff010000000000030000000000000000
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 (16)

Рівняння (15) називається рівнянням площини, що проходить через три дані
точки, у координатній формі, а рівняння (16) — у векторній формі.

Рівняння площини, що проходить

через дану точку паралельно двом даним векторам.

Нехай задано точку M0 (х0, у0, z0) і два неколінеарних (не паралельних)
вектори а і е. Ці умови геометрично однозначно визначають площину, що
проходить через задану точку паралельно заданим векторам. Знайдемо
рівняння площини.

Рівняння площини, що проходить через точку M0, грунтуючись на (1),
запишемо у вигляді

А(х — х0) + В(у — у0) + С(z — z0) =0,
(17)

де
picscalex100010009000003c900000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02c00140011200000026060f001a00ffffffff000010000000c0ffffff
c6ffffff00010000860100000b00000026060f000c004d61746854797065000030001c00
0000fb0280fe0000000000009001010000000402001054696d6573204e657720526f6d61
6e001643c7771f43c77720c0c97700003000040000002d01000008000000320a60013a00
010000006e791c000000fb0280fe000000000000900100000002040200104d5420457874
726100000000bb070afc1643c7771f43c77720c0c97700003000040000002d0101000400
0000f001000008000000320a6a005f000100000072790a00000026060f000a00ffffffff
0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d
000000000a0021008a0100000000000000009cf312002942c777040000002d0100000400
0000f0010100030000000000000000 = (А, В, С) — вектор, перпендикулярний до
даної площини, або нормальний вектор площини (рис. 4).

За умовою площина паралельна векторам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. Отже, нормальний вектор
площини можна виразити через векторний добуток даних векторів
picscalex1000100090000031d01000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c022002a0051200000026060f001a00ffffffff000010000000c0ffffff
b4ffffff60050000d40100000b00000026060f000c004d61746854797065000030001c00
0000fb0280fe0000000000009001010000000402001054696d6573204e657720526f6d61
6e001643c7771f43c77720c0c97700003000040000002d01000008000000320ac0017d04
01000000627908000000320ac001870201000000617908000000320ac0013a0001000000
6e791c000000fb0280fe000000000000900100000002040200104d542045787472610000
000030080abf1643c7771f43c77720c0c97700003000040000002d01010004000000f001
000008000000320a6a00b40401000000727908000000320aca00af020100000072790800
0000320aca005f000100000072791c000000fb0280fe0000000000009001000000020002
001053796d626f6c0077400000009d060a001643c7771f43c77720c0c977000030000400
00002d01000004000000f001010008000000320ac001810301000000b47908000000320a
c0015e01010000003d790a00000026060f000a00ffffffff0100000000001c000000fb02
1000070000000000bc02000000cc0102022253797374656d000000000a0021008a010000
0000010000003cf412002942c777040000002d01010004000000f0010000030000000000
000000.

Якщо позначити радіуси-вектори точок M i M0 відповідно через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, то рівняння (17) можна записати у вигляді
picscalex1000100090000038d01000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02400280081200000026060f001a00ffffffff000010000000c0ffffff
a6ffffff40080000e60100000b00000026060f000c004d61746854797065000060001c00
0000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d61
6e001643c7771f43c77720c0c97700003000040000002d01000008000000320a80017407
01000000307908000000320a80017c0501000000297908000000320a8001c10101000000
28791c000000fb0220ff0000000000009001000000000402001054696d6573204e657720
526f6d616e001643c7771f43c77720c0c97700003000040000002d01010004000000f001
000008000000320ae001db040100000030791c000000fb0280fe00000000000090010000
00020002001053796d626f6c007740000000b6070a1c1643c7771f43c77720c0c9770000
3000040000002d01000004000000f001010008000000320a80015106010000003d790800
0000320a80014e03010000002d7908000000320a8001370101000000d7791c000000fb02
80fe0000000000009001010000000402001054696d6573204e657720526f6d616e001643
c7771f43c77720c0c97700003000040000002d01010004000000f001000008000000320a
8001620401000000727908000000320a80014b0201000000727908000000320a80013a00
010000006e791c000000fb0280fe000000000000900100000002040200104d5420457874
726100000000b6070a1d1643c7771f43c77720c0c97700003000040000002d0100000400
0000f001010008000000320a8a007b0401000000727908000000320a8a00640201000000
727908000000320a8a005f000100000072790a00000026060f000a00ffffffff01000000
00001c000000fb021000070000000000bc02000000cc0102022253797374656d00000000
0a0021008a0100000000010000003cf412002942c777040000002d01010004000000f001
0000030000000000000000,звідки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,але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 Отже, вектори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лежать в одній площині, тобто
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(18)

Вираз (18) є векторною формою рівняння площини, що проходить через
дану точку паралельно двом даним векторам.

Рівняння заданої площини у координатній формі має вигляд :
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 (19)

Рівняння площини, що проходить через

дві дані точки паралельно даному вектору.

Нехай дано дві точки М1(х1, у1, z1), М2 (х2, y2, z2) і вектор
picscalex100010009000003c900000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02c00140011200000026060f001a00ffffffff000010000000c0ffffff
c6ffffff00010000860100000b00000026060f000c004d61746854797065000030001c00
0000fb0280fe0000000000009001010000000402001054696d6573204e657720526f6d61
6e001643c7771f43c77720c0c97700003000040000002d01000008000000320a60013a00
0100000061791c000000fb0280fe000000000000900100000002040200104d5420457874
726100000000c00a0a681643c7771f43c77720c0c97700003000040000002d0101000400
0000f001000008000000320a6a0062000100000072790a00000026060f000a00ffffffff
0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d
000000000a0021008a0100000000000000009cf312002942c777040000002d0100000400
0000f0010100030000000000000000. Знайдемо рівняння площини, що проходить
через дані точки паралельно вектору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. Нехай M (х, у, z) — довільна точка
простору. Позначимо радіуси-вектори точок М, М1, М2 відповідно через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.

За другий вектор, через який проходить задана площина, візьмемо

вектор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Тоді рівняння даної площини, згідно з рівнянням (18), можна
записати у вигляді:
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,
(20)

або, враховуючи, що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,
дістаємо
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.
(21)

Кут між двома площинами.

Умова паралельності та перпендикулярності.

Нехай дві площини задані своїми рівняннями
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(22)

Рис.5

Кутом між двома площинами називають один із суміжних двогранних кутів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або
picscalex100010009000003c900000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c022002a0011200000026060f001a00ffffffff000010000000c0ffffff
a6ffffff60010000c60100000b00000026060f000c004d61746854797065000050001c00
0000fb0220ff0000000000009001000000000402001054696d6573204e657720526f6d61
6e001643c7771f43c77720c0c97700003000040000002d01000008000000320ae0010201
0100000031791c000000fb0280fe0000000000009001010000020402001053796d626f6c
0077400000005a080a3e1643c7771f43c77720c0c97700003000040000002d0101000400
0000f001000008000000320a80011c00010000006a790a00000026060f000a00ffffffff
0100000000001c000000fb021000070000000000bc02000000cc0102022253797374656d
000000000a0021008a0100000000000000009cf312002942c777040000002d0100000400
0000f0010100030000000000000000, утворених цими площинами (рис.5). Якщо
площини не перетинаються, тобто паралельні, то кут між ними дорівнює 0
або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.

Нехай кут між даними площинами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.
Тоді кут між нормальними векторами цих площин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також дорівнюватиме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або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-p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.
Кут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знайдемо за формулою:

picscalex100010009000003a203000004001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02c004e01e1200000026060f001a00ffffffff000010000000c0ffffff
bdffffffa01e00007d0400000b00000026060f000c004d61746854797065000030010800
0000fa0200001000000000000000040000002d0100000500000014024502fb0405000000
13021104fb0405000000140245027606050000001302110476060500000014024502aa06
0500000013021104aa0605000000140245024f0805000000130211044f08050000001402
0002d30405000000130200027d08050000001402ba03400a0500000013029e03710a0800
0000fa0200002000000000000000040000002d010100050000001402a603710a05000000
13026a04b80a040000002d0100000500000014026a04c00a0500000013023f021e0b0500
000014023f021e0b0500000013023f020714050000001402ba032d140500000013029e03
5e14040000002d010100050000001402a6035e140500000013026a04a514040000002d01
00000500000014026a04ad140500000013023f020b150500000014023f020b1505000000
13023f02721e0500000014020002180a0500000013020002921e1c000000fb0220ff0000
000000009001000000000402001054696d6573204e657720526f6d616e001643c7771f43
c77720c0c97700003000040000002d01020008000000320a3203c11d0100000032790800
0000320a5a04331d01000000327908000000320a3203301a01000000327908000000320a
5a04a21901000000327908000000320a3203a91601000000327908000000320a5a041b16
01000000327908000000320a3203561301000000327908000000320a5a04d91201000000
317908000000320a3203ef0f01000000327908000000320a5a04720f0100000031790800
0000320a3203920c01000000327908000000320a5a04150c01000000317908000000320a
c9018f1901000000327908000000320ac901121801000000317908000000320ac9012815
01000000327908000000320ac901b51301000000317908000000320ac901d51001000000
327908000000320ac901630f01000000317908000000320aeb0398070100000032790800
0000320aeb03d00501000000317908000000320ac901c20701000000327908000000320a
c901a7050100000031791c000000fb0280fe000000000000900100000000040200105469
6d6573204e657720526f6d616e001643c7771f43c77720c0c97700003000040000002d01
030004000000f001020009000000320a6002340003000000636f73651c000000fb0280fe
0000000000009001010000000402001054696d6573204e657720526f6d616e001643c777
1f43c77720c0c97700003000040000002d01020004000000f001030008000000320afa03
241c01000000436f08000000320afa03b51801000000426f08000000320afa0341150100
0000416f08000000320afa03e31101000000436f08000000320afa039e0e01000000426f
08000000320afa03540b01000000416f08000000320a6901801801000000436f08000000
320a69011c1701000000436f08000000320a69013b1401000000426f08000000320a6901
e11201000000426f08000000320a6901fb0f01000000416f08000000320a6901a20e0100
0000416f08000000320a8b03d006010000006e6f08000000320a8b032105010000006e6f
08000000320a6901fa06010000006e6f08000000320a6901f804010000006e6f1c000000
fb0280fe0000000000009001000000020002001053796d626f6c0077400000008d070aca
1643c7771f43c77720c0c97700003000040000002d01030004000000f001020008000000
320afa03131b010000002b6f08000000320afa038c17010000002b6f08000000320afa03
d210010000002b6f08000000320afa03750d010000002b6f08000000320a69010b160100
00002b6f08000000320a6901b811010000002b6f08000000320a6002e508010000003d6f
08000000320a6901670601000000d76f08000000320a6002a003010000003d6f1c000000
fb0280fe000000000000900100000002040200104d5420457874726100000000a9060ad1
1643c7771f43c77720c0c97700003000040000002d01020004000000f001030008000000
320a9602f40601000000726f08000000320a9602450501000000726f08000000320a7400
1e0701000000726f08000000320a74001c0501000000726f1c000000fb0280fe00000000
00009001010000020402001053796d626f6c0077400000008d070acb1643c7771f43c777
20c0c97700003000040000002d01030004000000f001020008000000320a600242020100
00006a6f0a00000026060f000a00ffffffff0100000000001c000000fb02100007000000
0000bc02000000cc0102022253797374656d000000000a0021008a010000000002000000
9cf312002942c777040000002d01020004000000f0010300030000000000000000
(23)

Поклавши в цій формулі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, дістанемо умову
перпендикулярності площин:

picscalex1000100090000038101000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c022002600e1200000026060f001a00ffffffff000010000000c0ffffff
a6ffffff200e0000c60100000b00000026060f000c004d61746854797065000050001c00
0000fb0280fe0000000000009001000000000402001054696d6573204e657720526f6d61
6e001643c7771f43c77720c0c97700003000040000002d01000008000000320a8001650d
0100000030791c000000fb0220ff0000000000009001000000000402001054696d657320
4e657720526f6d616e001643c7771f43c77720c0c97700003000040000002d0101000400
0000f001000008000000320ae001450b01000000327908000000320ae001c80901000000
317908000000320ae001de0601000000327908000000320ae0016b050100000031790800
0000320ae0018b0201000000327908000000320ae00119010100000031791c000000fb02
80fe0000000000009001000000020002001053796d626f6c007740000000b8080a4c1643
c7771f43c77720c0c97700003000040000002d01000004000000f001010008000000320a
80013e0c010000003d7908000000320a8001c107010000002b7908000000320a80016e03
010000002b791c000000fb0280fe0000000000009001010000000402001054696d657320
4e657720526f6d616e001643c7771f43c77720c0c97700003000040000002d0101000400
0000f001000008000000320a8001360a01000000437908000000320a8001d20801000000
437908000000320a8001f10501000000427908000000320a800197040100000042790800
0000320a8001b10101000000417908000000320a800158000100000041790a0000002606
0f000a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc0102
022253797374656d000000000a0021008a0100000000000000009cf312002942c7770400
00002d01000004000000f0010100030000000000000000
(24)

Якщо площини (22) паралельні, то і їхні нормальні вектори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 і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 також
паралельні (колінеарні). Із умови паралельності векторів маємо

picscalex1000100090000036501000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02200240051200000026060f001a00ffffffff000010000000c0ffffff
a6ffffff00050000c60100000b00000026060f000c004d61746854797065000050001c00
0000fb0220ff0000000000009001000000000402001054696d6573204e657720526f6d61
6e001643c7771f43c77720c0c97700003000040000002d01000008000000320ae0019804
01000000327908000000320ae001e9000100000031791c000000fb0280fe000000000000
9001010000000402001054696d6573204e657720526f6d616e001643c7771f43c77720c0
c97700003000040000002d01010004000000f001000008000000320a8001d00301000000
6e7908000000320a80013a00010000006e791c000000fb0280fe00000000000090010000
0002040200104d54204578747261000000009b070adf1643c7771f43c77720c0c9770000
3000040000002d01000004000000f001010008000000320a8b00f4030100000072790800
0000320a8b005e000100000072791c000000fb0280fe0000000000009001010000020402
001053796d626f6c00774000000073080ad51643c7771f43c77720c0c977000030000400
00002d01010004000000f001000008000000320a8001f802010000006c791c000000fb02
80fe0000000000009001000000020002001053796d626f6c0077400000009b070ae01643
c7771f43c77720c0c97700003000040000002d01000004000000f001010008000000320a
8001d101010000003d790a00000026060f000a00ffffffff0100000000001c000000fb02
1000070000000000bc02000000cc0102022253797374656d000000000a0021008a010000
00000100000080e412002942c777040000002d01010004000000f0010000030000000000
000000

або
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.

Звідси дістаємо умову паралельності площин:
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.
(25)

Таким чином, у паралельних, площин коефіцієнти при відповідних
координатах пропорційні.

Відстань від точки до площини

Нехай площина задана нормальним рівнянням і дано точку

M0 (х0, у0. z0), що лежить поза площиною. Відстань від точки M0 до
площини позначимо через а. Відхилом точки M0 від даної площини
називається число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якщо точка
M0 і початок координат лежать по різні боки від даної площини, і число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, якщо точка M0 і початок координат лежать по один
бік від площини (рис. 6).

Рис.6

Із точки M0 на дану площину опустимо перпендикуляр М0 М1, де М1 (х1, у1,
z1). Позначимо
picscalex1000100090000030e02000004009a0000000000050000000201010000000500
00000102ffffff00050000002e0119000000050000000b0200000000050000000c020003
60071300000026060f001c00ffffffff00004e0010000000c0ffffffb8ffffff20070000
b80200000b00000026060f000c004d617468547970650000900008000000fa0200001300
000000000002040000002d01000005000000140248004900050000001302b80249000500
0000140248007904050000001302b8027904050000000902000000020500000014024302
e3011c000000fb0222ff0000000000009001000000000002001054696d6573204e657720
526f6d616e00b7c5c677c0c5c67720c0c97700003000040000002d0101000a000000320a
00000000020000003031e401bc01050000001402e0017e001c000000fb0280fe00000000
00009001010000000002001054696d6573204e657720526f6d616e00b7c5c677c0c5c677
20c0c97700003000040000002d01020004000000f00101000c000000320a000000000300
00004d4d64bef901bb030003050000001402e001fa041c000000fb0280fe000000000000
9001000000010002001053796d626f6c00774000000076080abfb7c5c677c0c5c67720c0
c97700003000040000002d01010004000000f001020009000000320a0000000001000000
3d31000305000000140284007e001c000000fb0280fe0000000000009001000000010002
00104d54204578747261000000001a080a85b7c5c677c0c5c67720c0c977000030000400
00002d01020004000000f001010012000000320a00000000070000007575757575757200
8c008c008c008c008c00740000039a00000026060f002a014d6174685479706555551e01
050100050244534d543500001357696e416c6c4261736963436f64655061676573001105
54696d6573204e657720526f6d616e00110353796d626f6c001105436f7572696572204e
65770011044d5420457874726100120008212f458f442f4150f4100f475f4150f21f1e41
50f4150f4100f445f425f48f425f4100f4100f435f4100f48f45f42a5f48f48f4100f410
0f40f48f417f48f4100f412a5f445f45f45f45f45f410f0c010001000102020202000200
01010100030001000400000a01000300040300010003001f020001000200834d0003001b
00000b01000200883000000101000a0200834d0003001b00000b01000200883100000101
00000a020096d720000002009607ec02009608ec000204863d003d020083640000000b00
000026060f000c00ffffffff010000000000000008000000fa0200000000000000000000
040000002d0101001c000000fb021000070000000000bc02000000cc0102022253797374
656d000000000a0021008a0100000000ffffffff58e61200b9c1c677040000002d010300
04000000f0010200030000000000000000.

Розглянемо вектори
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За правилом додавання векторів:

picscalex1000100090000037501000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02800260061200000026060f001a00ffffffff000010000000c0ffffff
adffffff200600002d0200000b00000026060f000c004d61746854797065000060001c00
0000fb0280fe0000000000009001010000020402001053796d626f6c007740000000d807
0ae61643c7771f43c77720c0c97700003000040000002d01000008000000320ac0011a05
0100000064791c000000fb0280fe000000000000900100000002040200104d5420457874
726100000000ca0a0a6f1643c7771f43c77720c0c97700003000040000002d0101000400
0000f001000008000000320a71007a0501000000727908000000320acb00f40201000000
727908000000320add0052000100000076791c000000fb0280fe00000000000090010000
00020002001053796d626f6c007740000000d8070ae71643c7771f43c77720c0c9770000
3000040000002d01000004000000f001010008000000320ac0011504010000002b790800
0000320ac001af01010000003d791c000000fb0220ff0000000000009001000000000402
001054696d6573204e657720526f6d616e001643c7771f43c77720c0c977000030000400
00002d01010004000000f001000008000000320a2002430301000000317908000000320a
2002b6000100000030791c000000fb0280fe000000000000900101000000040200105469
6d6573204e657720526f6d616e001643c7771f43c77720c0c97700003000040000002d01
000004000000f001010008000000320ac001dc0201000000727908000000320ac0013a00
0100000072790a00000026060f000a00ffffffff0100000000001c000000fb0210000700
00000000bc02000000cc0102022253797374656d000000000a0021008a01000000000100
00009cf312002942c777040000002d01010004000000f0010000030000000000000000.

Враховуючи означення відхилу, вектор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 можна записати у вигляді
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де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- одиничний вектор променя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 .

Тоді дістанемо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.

Помножимо обидві частини цього рівняння скалярне на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 :
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.

Оскільки скалярний добуток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, а
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то
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,

або

picscalex100010009000003b801000002001c0000000000050000000902000000000500
0000020101000000050000000102ffffff00050000002e0118000000050000000b020000
0000050000000c02400200161200000026060f001a00ffffffff000010000000c0ffffff
a6ffffffc0150000e60100000b00000026060f000c004d61746854797065000060001c00
0000fb0280fe0000000000009001010000000402001054696d6573204e657720526f6d61
6e001643c7771f43c77720c0c97700003000040000002d01000008000000320a8001f214
01000000707908000000320a8001000f010000007a7908000000320a8001d90801000000
797908000000320a8001b2020100000078791c000000fb0280fe00000000000090010000
00020002001053796d626f6c00774000000009090a6f1643c7771f43c77720c0c9770000
3000040000002d01010004000000f001000008000000320a8001ab13010000002d790800
0000320a8001d10d010000002b7908000000320a80019807010000002b7908000000320a
80017301010000003d791c000000fb0280fe000000000000900101000002040200105379
6d626f6c00774000000012090a5d1643c7771f43c77720c0c97700003000040000002d01
000004000000f001010008000000320a8001871201000000677908000000320a8001890c
01000000627908000000320a8001390601000000617908000000320a8001220001000000
64791c000000fb0280fe0000000000009001000000000402001054696d6573204e657720
526f6d616e001643c7771f43c77720c0c97700003000040000002d01010004000000f001
000009000000320a80016d1003000000636f736509000000320a8001570a03000000636f
736509000000320a80012b0403000000636f73651c000000fb0220ff0000000000009001
000000000402001054696d6573204e657720526f6d616e001643c7771f43c77720c0c977
00003000040000002d01000004000000f001010008000000320ae001a60f01000000306f
08000000320ae001900901000000306f08000000320ae001640301000000306f0a000000
26060f000a00ffffffff0100000000001c000000fb021000070000000000bc02000000cc
0102022253797374656d000000000a0021008a0100000000010000009cf312002942c777
040000002d01010004000000f0010000030000000000000000
(26)

Відхил точки від площини, яку задано нормальним рівнянням, дорівнює
значенню лівої частини цього рівняння у цій точці.

Відстань точки від площини дорівнює модулю відхилу цієї точки від даної
площини:
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 (27)

Якщо площину задано загальним рівнянням

Ах+Ву+Сz+D=0,

то щоб знайти відхил точки M0 (x0,y0,z0) від даної площини, треба
спочатку звести рівняння до нормального вигляду, а потім знайти значення
його лівої частини у точці M0:
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 (28)

Тоді відстань від точки M0 до площини
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Контрольні запитання.

1. Довести, що рівняння площини завжди виражається рівнянням першого
степеня і, навпаки, всяке рівняння першого степеня є рівнянням площини.

2. Який вигляд має загальне рівняння площини? Який зв’язок існує між
нормальним вектором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до площини та коефіцієнтами загального рівняння площини?

3. Дослідіть загальне рівняння площини.

4. Запишіть рівняння площини у відрізках на координатних осях.

5. Як визначається гострий кут між двома площинами, що перетинаються?

6. Запишіть умови паралельності та перпендикулярності площин.

7. Запишіть рівняння площини, що проходить через три задані точки.

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