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Cách tìm m để hai vecto vuông góc trong không gian bằng phương pháp toạ độ - Toán 12

1. Vecto vuông góc là gì?

Trong toán học, hai vecto được gọi là vuông góc khi góc giữa chúng là 90 độ. Điều này có nghĩa là tích vô hướng của chúng bằng 0.

2. Cách tìm m để hai vecto vuông góc trong không gian bằng phương pháp toạ độ

Trong không gian với hệ trục toạ độ Oxyz, cho \(\overrightarrow a \left( {{x_a};{y_a};{z_a}} \right)\) và \(\overrightarrow b \left( {{x_b};{y_b};{z_b}} \right)\). Để \(\overrightarrow a  \bot \overrightarrow b \) thì \(\overrightarrow a .\overrightarrow b  = 0 \Leftrightarrow {x_a}{x_b} + {y_a}{y_b} + {z_a}{z_b} = 0\).

Ví dụ minh hoạ:

1) ꦡCho hai vecto \(\overrightarrow a (1;m; - 1)\), \(\overrightarrow b (2;1;3)\). Tìm m để \(\overrightarrow a  \bot \overrightarrow b \).

Giải:

\(\overrightarrow a .\overrightarrow b  = 0 \Leftrightarrow 1.2 + m.1 - 1.3 = 0 \Leftrightarrow m = 1\).

2) ꦡCho hai vecto \(\overrightarrow a (2;1; - 2)\), \(\overrightarrow b (0;2m; - 4)\). Tìm m để \(\overrightarrow a  \bot \overrightarrow b \).

Giải:

\(\overrightarrow a .\overrightarrow b  = 0 \Leftrightarrow 2.0 + 1.2m - 2.( - 4) = 0 \Leftrightarrow m =  - 4\).
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