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Giải bài 5 trang 80 sách bài tập toán 12 - Chân trời sáng tạo

Cho ba điểm (Aleft( {2; - 1;3} right),Bleft( { - 10;5;3} right)) và (Mleft( {2m - 1;2;n + 2} right)). Tìm (m,n) để (A,B,M) thẳng hàng.

☂Tổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Chân trời sáng tạo

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

Cho ba điểm \(A\left( {2; - 1;3} \right),B\left( { - 10;5;3} \right)\) và \(M\left( {2m - 1;2;n + 2} \right)\). Tìm \(m,n\) để \(A,B,M\) thẳng hàng.

Phương pháp giải - Xem chi tiết

Sử dụng tính chất: Ba điểm \(A,B,C\) thẳng hàng nếu hai vectơ \(\overrightarrow {AB} ,\overrightarrow {AC} \) cùng phương.

Lời giải chi tiết

Ta có: \(\overrightarrow {AB}  = \left( { - 12;6;0} \right),\overrightarrow {AM}  = \left( {2m - 3;3;n - 1} \right)\). Để ba điểm \(A,B,M\) thẳng hàng thì tồn tại số \(k\) sao cho \(\overrightarrow {AM}  = k\overrightarrow {AB} \) \( \Leftrightarrow \left\{ \begin{array}{l}2m - 3 = k.\left( { - 12} \right)\\3 = k.6\\n - 1 = k.0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}k = \frac{1}{2}\\m =  - \frac{3}{2}\\n = 1\end{array} \right.\) Vậy với \(m =  - \frac{3}{2},n = 1\) thì ba điểm \(A,B,M\) thẳng hàng.

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