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Lý thuyết Tọa độ của vecto trong không gian Toán 12 Chân trời sáng tạo

Bài 2. Tọa độ của vecto trong không gian 1. Hệ trục tọa độ trong không gian

🏅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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1. Hệ trục tọa độ trong không gian

Trong không gian, cho ba trục Ox, Oy, Oz đôi một vuông góc. Gọi \(\overrightarrow i ,\overrightarrow j ,\overrightarrow k \) lần lượt là ba vecto đơn vị trên các trục Ox, Oy, Oz. Hệ ba trục như vậy được gọi là hệ trục tọa độ Descartes vuông góc Oxyz, hay đơn giản gọi là hệ tọa độ Oxyz. 

2. Tọa độ của điểm và vecto

a) Tọa độ của điểm

Trong không gian Oxyz, cho điểm M. Nếu \[\overrightarrow {OM}  = x\overrightarrow i  + y\overrightarrow j  + z\overrightarrow k \] thì ta gọi bộ ba số (x;y;z) là tọa độ điểm M đối với hệ trục tọa độ Oxyz và viết M = (x;y;z) hoặc M (x;y;z); x là hoành độ, y là tung độ, z là cao độ của điểm M.

b) Tọa độ của vecto

Trong không gian Oxyz, cho \(\overrightarrow a \). Nếu \(\overrightarrow a  = {a_1}\overrightarrow i  + {a_2}\overrightarrow j  + {a_3}\overrightarrow k \) thì ta gọi bộ ba số \(\left( {{a_1};{a_2};{a_3}} \right)\) là tọa độ của \(\overrightarrow a \) đối với hệ tọa độ Oxyz và viết \(\overrightarrow a  = \left( {{a_1};{a_2};{a_3}} \right)\) hoặc \(\overrightarrow a \left( {{a_1};{a_2};{a_3}} \right)\).

Ví dụ: 𒊎Trong không gian Oxyz, cho hình lăng trụ tam giác ABC.A’B’C có A(1;0;2), B(3;2;5), C(7;-3;9)

a) Tìm tọa độ của \(\overrightarrow {AA'} .\)b) Tìm tọa độ của các điểm B’, C’.

Lời giải

a) Ta có: \(\overrightarrow {AA'}  = ({x_{A'}} - {x_A};{y_{A'}} - {y_A};{z_{A'}} - {z_A}) = (4;0; - 1)\).b) Gọi tọa độ của điểm B’ là (x,y,z) thì \(\overrightarrow {BB'} \) = (x - 3; y - 2; z - 5). Vì ABC.A’B’C’ là hình lăng trụ nên ABB’A’ là hình bình hành, suy ra \(\overrightarrow {AA'} \) = \(\overrightarrow {BB'} .\)Do đó \(\left\{ \begin{array}{l}x - 3 = 4\\y - 2 = 0\\z - 5 =  - 1\end{array} \right.\) hay x = 7, y = 2, z = 4.Vậy B’(7;2;4).Lập luận tương tự suy ra C’ (11;-3;8).

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