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Giải bài tập 2.36 trang 74 SGK Toán 12 tập 1 - Kết nối tri thức

Cho tứ diện ABCD, lấy hai điểm M, N thỏa mãn (overrightarrow {MB} + 2overrightarrow {MA} = overrightarrow 0 ) và (overrightarrow {NC} = 2overrightarrow {DN} ). Hãy biểu diễn (overrightarrow {MN} ) theo (overrightarrow {AD} ) và (overrightarrow {BC} ).

Tổng hợp đề thi h🎀ọc kì 2 lớp 12 tấtౠ cả các môn - Kết nối tri thức

Toán - Văn - Anh - Hoá - Sinh - Sử - Địa
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Đề bài

Cho tứ diện ABCD, lấy hai điểm M, N thỏa mãn \(\overrightarrow {MB}  + 2\overrightarrow {MA}  = \overrightarrow 0 \) và \(\overrightarrow {NC}  = 2\overrightarrow {DN} \). Hãy biểu diễn \(\overrightarrow {MN} \) theo \(\overrightarrow {AD} \) và \(\overrightarrow {BC} \).

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về quy tắc ba điểm để chứng minh: Nếu A, B, C là ba điểm bất kì thì \(\overrightarrow {AB}  + \overrightarrow {BC}  = \overrightarrow {AC} \)

Lời giải chi tiết

Ta có: \(\overrightarrow {MB}  + 2\overrightarrow {MA}  = \overrightarrow 0  \Rightarrow \overrightarrow {MB}  =  - 2\overrightarrow {MA} ,\overrightarrow {NC}  = 2\overrightarrow {DN}  \Rightarrow \overrightarrow {CN}  =  - 2\overrightarrow {DN} \) Ta có: \(\overrightarrow {MN}  = \overrightarrow {MA}  + \overrightarrow {AD}  + \overrightarrow {DN} \) (1) \(\overrightarrow {MN}  = \overrightarrow {MB}  + \overrightarrow {BC}  + \overrightarrow {CN}  =  - 2\overrightarrow {MA}  + \overrightarrow {BC}  - 2\overrightarrow {DN} \) (2) Cộng vế với vế của (1) và (2) ta có: \(2\overrightarrow {MN}  = \overrightarrow {MA}  + \overrightarrow {AD}  + \overrightarrow {DN}  - 2\overrightarrow {MA}  + \overrightarrow {BC}  - 2\overrightarrow {DN}  =  - \overrightarrow {MA}  - \overrightarrow {DN}  + \overrightarrow {BC}  + \overrightarrow {AD} \) \( = \frac{1}{3}\overrightarrow {AB}  + \frac{1}{3}\overrightarrow {CD}  + \overrightarrow {BC}  + \overrightarrow {AD}  = \frac{1}{3}\left( {\overrightarrow {AC}  + \overrightarrow {CB}  + \overrightarrow {CA}  + \overrightarrow {AD} } \right) + \overrightarrow {BC}  + \overrightarrow {AD}  = \frac{2}{3}\overrightarrow {BC}  + \frac{4}{3}\overrightarrow {AD} \) \(\overrightarrow {MN}  = \frac{1}{3}\overrightarrow {BC}  + \frac{2}{3}\overrightarrow {AD} \)

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