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Giải bài tập 2.40 trang 84 SGK Toán 12 tập 1 - Cùng khám phá

Hình bình hành ABCD có \(A(1;0;3)\), \(B(2;3; - 4)\), \(C( - 3;1;2)\). Tọa độ điểm \(D\) là: A. \(( - 4; - 2;9)\). B. \((2; - 4;5)\). C. \(( - 2;4; - 5)\). D. \((4;2; - 9)\).
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Đề bài

Hình bình hành ABCD có \(A(1;0;3)\), \(B(2;3; - 4)\), \(C( - 3;1;2)\). Tọa độ điểm \(D\) là: A. \(( - 4; - 2;9)\). B. \((2; - 4;5)\). C. \(( - 2;4; - 5)\). D. \((4;2; - 9)\).

Phương pháp giải - Xem chi tiết

- Sử dụng tính chất của hình bình hành: Vectơ \(\overrightarrow {AB}  = \overrightarrow {DC} \) hoặc \(\overrightarrow {AD}  = \overrightarrow {BC} \). - Tính tọa độ của điểm \(D\) bằng cách sử dụng tọa độ các điểm đã cho.

Lời giải chi tiết

- Tính tọa độ vectơ \(\overrightarrow {AB} \): \(\overrightarrow {AB}  = (2 - 1;3 - 0; - 4 - 3) = (1;3; - 7)\)  - Tính tọa độ vectơ \(\overrightarrow {DC} \): \(\overrightarrow {DC}  = ( - 3 - {x_D};1 - {y_D};2 - {z_D})\) - Bằng cách áp dụng \(\overrightarrow {AB}  = \overrightarrow {CD} \), ta có: \(({x_D} + 3,{y_D} - 1,{z_D} - 2) = (1,3, - 7)\) - Giải hệ phương trình: \( - 3 - {x_D} = 1\quad  \Rightarrow \quad {x_D} =  - 4\) \(1 - {y_D} = 3\quad  \Rightarrow \quad {y_D} =  - 2\) \(2 - {z_D} =  - 7\quad  \Rightarrow \quad {z_D} = 9\) - Tọa độ điểm \(D\) là \(( - 4; - 2;9)\). Chọn A.
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