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Giải bài 2 trang 76 sách bài tập toán 12 - Chân trời sáng tạo

Cho hình bình hành \(OABD\) có \(\overrightarrow {OA} = \left( { - 1;1;0} \right)\) và \(\overrightarrow {OB} = \left( {1;1;0} \right)\) với \(O\) là gốc toạ độ. Tìm toạ độ của điểm \(D\).

ওTổng 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 hình bình hành \(OABD\) có \(\overrightarrow {OA}  = \left( { - 1;1;0} \right)\) và \(\overrightarrow {OB}  = \left( {1;1;0} \right)\) với \(O\) là gốc toạ độ. Tìm toạ độ của điểm \(D\).

Phương pháp giải - Xem chi tiết

‒ Sử dụng toạ độ của vectơ: \(\overrightarrow {OM}  = \left( {a;b;c} \right) \Leftrightarrow M\left( {a;b;c} \right)\). ‒ 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

Ta có \(\overrightarrow {OB}  = \left( {1;1;0} \right) \Leftrightarrow B\left( {1;1;0} \right)\) Giả sử \(D\left( {{x_D};{y_D};{z_D}} \right)\). Ta có \(\overrightarrow {DB}  = \left( {1 - {x_D};1 - {y_D}; - {z_D}} \right)\). \(OABD\) là hình bình hành nên \(\overrightarrow {OA}  = \overrightarrow {DB} \). \( \Leftrightarrow \left\{ \begin{array}{l}1 - {x_D} =  - 1\\1 - {y_D} = 1\\ - {z_D} = 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}{x_D} = 2\\{y_D} = 0\\{z_D} = 0\end{array} \right.\). Vậy \(D\left( {2;0;0} \right)\).

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