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Giải bài tập 2.4 trang 58 SGK Toán 12 tập 1 - Kết nối tri thức

Cho hình hộp ABCD.A’B’C’D’. Chứng minh rằng: a) \(\overrightarrow {AB} + \overrightarrow {DD'} + \overrightarrow {C'D'} = \overrightarrow {CC'} \); b) \(\overrightarrow {AB} + \overrightarrow {CD'} - \overrightarrow {CC'} = \overrightarrow 0 \); c) \(\overrightarrow {BC} - \overrightarrow {CC'} + \overrightarrow {DC} = \overrightarrow {A'C} \)

Tổng hợp đề thꦚi 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’. Chứng minh rằng:
a) \(\overrightarrow {AB} + \overrightarrow {DD'} + \overrightarrow {C'D'} = \overrightarrow {CC'} \);
b) \(\overrightarrow {AB} + \overrightarrow {CD'} - \overrightarrow {CC'} = \overrightarrow 0 \);
c) \(\overri🦩ghtarrow {BC} - \overrightarrow {CC'} + \ౠoverrightarrow {DC} = \overrightarrow {A'C} \)

Phương pháp giải - Xem chi tiết

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

Lời giải chi tiết

a) Vì ABCD là hình bình hành nên \(\overrightarrow {AB}  = \overrightarrow {DC} \) Vì CDD’C’ là hình bình hành nên \(\overrightarrow {C'D'}  = \overrightarrow {CD} ,\overrightarrow {DD'}  = \overrightarrow {CC'} \) Ta có: \(\overrightarrow {AB}  + \overrightarrow {DD'}  + \overrightarrow {C'D'}  = \overrightarrow {DC}  + \overrightarrow {CC'}  + \overrightarrow {CD}  = \left( {\overrightarrow {CD}  + \overrightarrow {DC} } \right) + \overrightarrow {CC'}  = \overrightarrow {CC'} \) b) Ta có: \(\overrightarrow {AB}  + \overrightarrow {CD'}  - \overrightarrow {CC'}  = \overrightarrow {AB}  + \overrightarrow {C'D'}  = \overrightarrow {AB}  + \overrightarrow {CD}  = \overrightarrow 0 \) c) Vì ABCD là hình bình hành nên \(\overrightarrow {CB}  + \overrightarrow {CD}  = \overrightarrow {CA} \) Vì A’ACC’ là hình bình hành nên \(\overrightarrow {CA}  + \overrightarrow {CC'}  = \overrightarrow {CA'} \) \(\overrightarrow {BC}  - \overrightarrow {CC'}  + \overrightarrow {DC}  =  - \left( {\overrightarrow {CB}  + \overrightarrow {CD} } \right) - \overrightarrow {CC'}  =  - \overrightarrow {CA}  - \overrightarrow {CC'}  =  - \left( {\overrightarrow {CA}  + \overrightarrow {CC'} } \right) =  - \overrightarrow {CA'}  = \overrightarrow {A'C} \)

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