§ 22: 22.18

22.18 Складіть рівняння площини, яка проходить через точки

A (1; 2; 3), B (4; 1; 2) і C (2; –1; 1).

A (1; 2; 3); B (4; 1; 2); C (2; –; 1).

α: ax + by + cz + d = 0.a (a; b; c) ⊥ α.AB (3; –1; –1).

AB ⊥ a ⇒ AB • a = 0; 3a – b – c = 0.

BC (–2; –2; –1).

BC ⊥ a ⇒ AC • a = 0; a – 3b – 2c = 0.

3a – b – c = 0;–2a – 2b – c = 0;a – 3b – 2c = 0;	3 • (3b + 2c) – b – c = 0;–2(3b + 2c) – 2b – c = 0; a = 3b + 2c.
9b + 6c – b – c = 0;–6b – 4c – 2b – c = 0;a = 3b + 2c;	8b + 5c = 0;–8b – 5c = 0;a = 3b + 2c;	8b = –5c;a = 3b + 2c.

c = 8, тоді

8b = –40; a = 3 • (–5) + 2 • 8 = 1.

x – 5y + 8z + d = 0.

A (1; 2; 3) ⊂ α.

1 – 5 • 2 + 8 • 3 + d = 0;

1 – 10 + 24 + d = 0;

d = –15.

α: x – 5y + 8z – 15 = 0.