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Cách xác định vecto pháp tuyến của mặt phẳng - Toán 12

1. Vecto pháp tuyến của mặt phẳng là gì?

Nếu vecto \(\overrightarrow n  \ne \overrightarrow 0 \) và có giá vuông góc với \(\left( \alpha  \right)\) thì \(\overrightarrow n \) được gọi là vecto pháp tuyến của \(\left( \alpha  \right)\).

2. Cách xác định vecto pháp tuyến của mặt phẳng

Trong không gian với hệ trục toạ độ Oxyz, cho mặt phẳng (P): Ax + By + Cz + D = 0. Một vecto pháp tuyến của (P) là \(\overrightarrow n  = (A;B;C)\). Khi đó, với số thực \(k \ne 0\), \(k\overrightarrow n  = (kA;kB;kC)\) cũng là một vecto pháp tuyến của (P).

3. Ví dụ minh hoạ

1) Mặt phẳng (P): -2x + 5y + z – 3 = 0 có một vecto pháp tuyến là \(\overrightarrow n = \left( { - 2;5;1} \right)\).

2) Mặt phẳng (Q): x – y + z – 1 = 0 có một vecto pháp tuyến là \(\overrightarrow m  = \left( {1; - 1;1} \right)\).

Ta có \(2\overrightarrow m = \left( {2; - 2;2} \right)\) cũng là một vecto pháp tuyến của (Q).

4. Bài tập vận dụng

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