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Giải bài 9 trang 80 sách bài tập toán 12 - Chân trời sáng tạo

Cho ba điểm \(A\left( { - 3;4;2} \right),B\left( { - 5;6;2} \right)\) và \(C\left( { - 4;7; - 1} \right)\). Tìm toạ độ điểm \(D\) thoả mãn \(\overrightarrow {AD} = 2\overrightarrow {AB} + 3\overrightarrow {AC} \).

🉐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 \(A\left( { - 3;4;2} \right),B\left( { - 5;6;2} \right)\) và \(C\left( { - 4;7; - 1} \right)\). Tìm toạ độ điểm \(D\) thoả mãn \(\overrightarrow {AD}  = 2\overrightarrow {AB}  + 3\overrightarrow {AC} \).

Phương pháp giải - Xem chi tiết

‒ Sử dụng toạ độ của vectơ \(\overrightarrow {AB}  = \left( {{x_B} - {x_A};{y_B} - {y_A};{z_B} - {z_A}} \right)\). ‒ Sử dụng tính chất hai vectơ bằng nhau: Với \(\overrightarrow u  = \left( {{x_1};{y_1};{z_1}} \right)\) và \(\overrightarrow v  = \left( {{x_2};{y_2};{z_2}} \right)\), ta có: \(\overrightarrow u  = \overrightarrow v  \Leftrightarrow \left\{ \begin{array}{l}{x_1} = {x_2}\\{y_1} = {y_2}\\{z_1} = {z_2}\end{array} \right.\).

Lời giải chi tiết

Giả sử \(D\left( {x;y;z} \right)\). Ta có: \(\begin{array}{l}\overrightarrow {AB}  = \left( { - 2;2;0} \right),\overrightarrow {AC}  = \left( { - 1;3; - 3} \right),\overrightarrow {AD}  = \left( {x + 3;y - 4;z - 2} \right)\\2\overrightarrow {AB}  + 3\overrightarrow {AC}  = \left( {2.\left( { - 2} \right) + 3.\left( { - 1} \right);2.2 + 3.3;2.0 + 3.\left( { - 3} \right)} \right) = \left( { - 7;13; - 9} \right)\\\overrightarrow {AD}  = 2\overrightarrow {AB}  + 3\overrightarrow {AC}  \Leftrightarrow \left\{ \begin{array}{l}x + 3 =  - 7\\y - 4 = 13\\z - 2 =  - 9\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x =  - 10\\y = 17\\z =  - 7\end{array} \right.\end{array}\) Vậy \(D\left( { - 10;17; - 7} \right)\).

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