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Giải bài 13 trang 50 sách bài tập toán 12 - Kết nối tri thức

Cho hình hộp (ABCD.A'B'C'D'), gọi (G) là trọng tâm của tam giác (ADA') và (M) là trung điểm của đoạn thẳng (CC'). Hệ thức biểu diễn (overrightarrow {GM} ) theo ba vectơ (overrightarrow {AB} ,{rm{ }}overrightarrow {AD} ,{rm{ }}overrightarrow {AA'} ) là A. (overrightarrow {AB} + frac{1}{2}overrightarrow {AD} + frac{1}{3}overrightarrow {AA'} ). B. (overrightarrow {AB} + frac{2}{3}overrightarrow {AD} + frac{1}{3}overrightarrow {AA'} ).

ꦰTổng hợp đề thi học kì 2 lớp 12 tất cả các môn - Kết nối tri thức

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

Cho hình hộp \(ABCD.A'B'C'D'\), gọi \(G\) là trọng tâm của tam giác \(ADA'\) và \(M\) là trung điểm của đoạn thẳng \(CC'\). Hệ thức biểu diễn \(\overrightarrow {GM} \) theo ba vectơ \(\overrightarrow {AB} ,{\rm{ }}\overrightarrow {AD} ,{\rm{ }}\overrightarrow {AA'} \) là

A.♛ \(\overrightarrow {AB}  + \frac{1}{2}\overrightarrow {AD}  + \frac{1}{3}\overrightarrow {AA'} \).

B.🍨 \(\overrightarrow {AB}  + \frac{2}{3}\overrightarrow {AD}  + \frac{1}{3}\overrightarrow {AA'} \).                   

C.🧸 \(\overrightarrow {AB}  + \frac{2}{3}\overrightarrow {AD}  + \frac{1}{6}\overrightarrow {AA'} \).

D.🍌 \(\overrightarrow {AB}  - \frac{1}{3}\overrightarrow {AD}  + \frac{1}{6}\overrightarrow {AA'} \).

Phương pháp giải - Xem chi tiết

Từ \(\overrightarrow {GM} \) biến đổi thành tổng các vectơ. Sử dụng tính chất trọng tâm, quy tắc hình bình hành, tính chất song song có trong hình hộp để biến đổi sao cho các vectơ \(\overrightarrow {AB} ,{\rm{ }}\overrightarrow {AD} ,{\rm{ }}\overrightarrow {AA'} \) xuất hiện.

Lời giải chi tiết

Ta có \(\overrightarrow {GM}  = \overrightarrow {GA}  + \overrightarrow {AC}  + \overrightarrow {CM}  = \frac{{ - 1}}{3}\overrightarrow {AD'}  + \left( {\overrightarrow {AB}  + \overrightarrow {AD} } \right) + \frac{1}{2}\overrightarrow {CC'} \) \( = \frac{{ - 1}}{3}\left( {\overrightarrow {AD}  + \overrightarrow {AA'} } \right) + \left( {\overrightarrow {AB}  + \overrightarrow {AD} } \right) + \frac{1}{2}\overrightarrow {AA'}  = \overrightarrow {AB}  + \frac{2}{3}\overrightarrow {AD}  + \frac{1}{6}\overrightarrow {AA'} \). Đáp án C.

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