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Giải bài tập 2.3 trang 64 SGK Toán 12 tập 1 - Cùng khám phá

Cho hình chóp S.ABCD có đáy ABCD là hình bình hành. Chứng minh rằng: \(\overrightarrow {SA} + \overrightarrow {SC} = \overrightarrow {SB} + \overrightarrow {SD} \)
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Đề bài

Cho hình chóp S.ABCD có đáy ABCD là hình bình hành. Chứng minh rằng: \(\overrightarrow {SA}  + \overrightarrow {SC}  = \overrightarrow {SB}  + \overrightarrow {SD} \)

Phương pháp giải - Xem chi tiết

Sử dụng tính chất của hình bình hành và phép biến đổi vectơ.

Lời giải chi tiết

Ta có thể viết: \(\overrightarrow {SA}  + \overrightarrow {SC}  = (\overrightarrow {SB}  + \overrightarrow {BA} ) + (\overrightarrow {SD}  + \overrightarrow {DC} )\) Thay \(\overrightarrow {BA}  =  - \overrightarrow {AB} \) và \(\overrightarrow {DC}  =  - \overrightarrow {CD} \) vào biểu thức trên, ta được: \(\overrightarrow {SA}  + \overrightarrow {SC}  = (\overrightarrow {SB}  - \overrightarrow {AB} ) + (\overrightarrow {SD}  - \overrightarrow {CD} )\) Sử dụng tính chất của hình bình hành: \(\overrightarrow {AB}  = \overrightarrow {DC} \quad {\rm{và}}\quad \overrightarrow {AD}  = \overrightarrow {BC} \) Nên ta có: \(\overrightarrow {SA}  + \overrightarrow {SC}  = \overrightarrow {SB}  - \overrightarrow {AB}  + \overrightarrow {SD}  + \overrightarrow {DC}  = \overrightarrow {SB}  + \overrightarrow {SD} \) Vậy đẳng thức \(\overrightarrow {SA}  + \overrightarrow {SC}  = \overrightarrow {SB}  + \overrightarrow {SD} \) đã được chứng minh.

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