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Giải bài tập 23 trang 92 SGK Toán 12 tập 2 - Kết nối tri thức

Cho tứ diện ABCD, chứng minh rằng: a) \(\overrightarrow {AB} .\overrightarrow {CD} + \overrightarrow {AC} .\overrightarrow {DB} + \overrightarrow {AD} .\overrightarrow {BC} = 0\); b) Nếu \(AB \bot CD\) và \(AC \bot BD\) thì \(AD \bot BC\).

🌟Tổng 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 tứ diện ABCD, chứng minh rằng: a) \(\overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {AC} .\overrightarrow {DB}  + \overrightarrow {AD} .\overrightarrow {BC}  = 0\); b) Nếu \(AB \bot CD\) và \(AC \bot BD\) thì \(AD \bot BC\).
 

Phương pháp giải - Xem chi tiết

Sử dụng kiến thức về quy tắc ba điểm để tính: Nếu A, B, C là ba điểm bất kì thì \(\overrightarrow {AB}  + \overrightarrow {BC}  = \overrightarrow {AC} \) Sử dụng kiến thức về tích vô hướng của hai vectơ trong không gian để tính: Cho hai vectơ \(\overrightarrow a \), \(\overrightarrow b \) đều khác \(\overrightarrow 0 \). Khi đó, \(\overrightarrow a  \bot \overrightarrow b  \Leftrightarrow \overrightarrow a  \cdot \overrightarrow b  = 0\)
 

Lời giải chi tiết

a) Ta có: \(\overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {AC} .\overrightarrow {DB}  + \overrightarrow {AD} .\overrightarrow {BC} \) \(= \overrightarrow {AB} .\overrightarrow {CD}  + \left( {\overrightarrow {AB}  + \overrightarrow {BC} } \right)\left( {\overrightarrow {DC}  + \overrightarrow {CB} } \right) + \left( {\overrightarrow {AB}  + \overrightarrow {BD} } \right)\overrightarrow {BC} \) \( = \overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {AB} .\overrightarrow {DC}  + \overrightarrow {AB} .\overrightarrow {CB}  + \overrightarrow {BC} .\overrightarrow {DC}  + \overrightarrow {BC} .\overrightarrow {CB}  + \overrightarrow {AB} .\overrightarrow {BC}  + \overrightarrow {BC} .\overrightarrow {BD} \) \( = \overrightarrow {AB} \left( {\overrightarrow {CD}  + \overrightarrow {DC} } \right) + \overrightarrow {AB} .\left( {\overrightarrow {CB}  + \overrightarrow {BC} } \right) + \overrightarrow {BC} .\left( {\overrightarrow {DC}  + \overrightarrow {CB} } \right) + \overrightarrow {BC} .\overrightarrow {BD} \) \( = \overrightarrow {BC} .\overrightarrow {DB}  + \overrightarrow {BC} .\overrightarrow {BD}  = \overrightarrow {BC} .\left( {\overrightarrow {DB}  + \overrightarrow {BD} } \right) = 0\) Vậy \(\overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {AC} .\overrightarrow {DB}  + \overrightarrow {AD} .\overrightarrow {BC}  = 0\). b) Vì \(AB \bot CD\) nên \(\overrightarrow {AB} .\overrightarrow {CD}  = 0\), \(AC \bot BD\) nên \(\overrightarrow {AC} .\overrightarrow {DB}  = 0\) Mà \(\overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {AC} .\overrightarrow {DB}  + \overrightarrow {AD} .\overrightarrow {BC}  = 0\) nên \(\overrightarrow {AD} .\overrightarrow {BC}  = 0\). Do đó, \(AD \bot BC\). 
 

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