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Lý thuyết Tọa độ của vecto Toán 12 Cánh Diều

1. Tọa độ của một điểm a) Hệ trục tọa độ trong không gian

ꦬTổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Cánh diều

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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1. Tọa độ của một điểm

a) Hệ trục tọa độ trong không gian

Hệ gồm ba trục Ox, Oy, Oz đôi một vuông góc được gọi là hệ trục tọa độ vuông góc Oxyz trong không gian, hay đơn giản gọi là hệ tọa độ Oxyz.

b) Tọa độ của một điểm

Trong không gian với hệ tọa độ Oxyz, cho điểm M. - Xác định hình chiếu \({M_1}\) của điểm M trên mặt phẳng (Oxy). Trong mặt phẳng tọa độ (Oxy), tìm hoành độ a, tung độ b của điểm \({M_1}\) - Xác định hình chiếu P của điểm M trên trục cao Oz, điểm P ứng với số c trên trục Oz. Số c là cao độ của điểm M. Bộ số (a;b;c) là tọa độ của điểm M trong không gian với hệ tọa độ Oxyz, kí hiệu là M(a;b;c)

2. Tọa độ của một vecto

Tọa độ của điểm M được gọi là tọa độ của vecto \(\overrightarrow {OM} \) Trong không gian với hệ tọa độ Oxyz, tọa độ của một vecto \(\overrightarrow u \) là tọa độ của điểm A, trong đó A là điểm sao cho \(\overrightarrow {OA}  = \overrightarrow u \) Trong không gian với hệ tọa độ Oxyz, nếu \(\overrightarrow u \) = (a;b;c) thì \[\overrightarrow u  = a\overrightarrow i  + b\overrightarrow j  + c\overrightarrow k \] . Ngược lại, nếu \[\overrightarrow u  = a\overrightarrow i  + b\overrightarrow j  + c\overrightarrow k \] thì \(\overrightarrow u \) = (a;b;c) Trong không gian với hệ tọa độ Oxyz, cho hai điểm \(M({x_M};{y_M};{z_M})\) và \(N({x_N};{y_N};{z_N})\). Khi đó: \(\overrightarrow {MN}  = ({x_N} - {x_M};{y_N} - {y_M};{z_N} - {z_M})\)

Ví dụ: ♓Trong 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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