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Giải bài tập 2.12 trang 59 SGK Toán 12 tập 1 - Kết nối tri thức

Cho tứ diện ABCD. Chứng minh rằng: a) (overrightarrow {AB} .overrightarrow {CD} = overrightarrow {AC} .overrightarrow {CD} + overrightarrow {BC} .overrightarrow {DC} ); b) (overrightarrow {AB} .overrightarrow {CD} + overrightarrow {AC} .overrightarrow {DB} + overrightarrow {AD} .overrightarrow {BC} = 0).

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) \(\overrightarrow {AB} .\overrightarrow {CD} = \overrightarrow {AC} .\overrightarrow {CD} + \overrightarrow {BC} .\overrightarrow {DC} \);
b) \(\overrightarrow {AB} .\overrightarrow {CꦏD} + \overrightarrow {AC} .\overrightarrow {✅DB} + \overrightarrow {AD} .\overrightarrow {BC} = 0\).

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

a) Ta có: \(\overrightarrow {AC} .\overrightarrow {CD}  + \overrightarrow {BC} .\overrightarrow {DC}  = \overrightarrow {AC} .\overrightarrow {CD}  - \overrightarrow {BC} .\overrightarrow {CD}  = \overrightarrow {CD} \left( {\overrightarrow {AC}  + \overrightarrow {CB} } \right) = \overrightarrow {CD} .\overrightarrow {AB} \) (đpcm) b) \(\overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {AC} .\overrightarrow {DB}  + \overrightarrow {AD} .\overrightarrow {BC}  = \overrightarrow {AB} .\overrightarrow {CD}  + \left( {\overrightarrow {AB}  + \overrightarrow {BC} } \right).\overrightarrow {DB}  + \left( {\overrightarrow {AB}  + \overrightarrow {BD} } \right).\overrightarrow {BC} \) \( = \overrightarrow {AB} .\overrightarrow {CD}  + \overrightarrow {AB} .\overrightarrow {DB}  + \overrightarrow {BC} .\overrightarrow {DB}  + \overrightarrow {AB} .\overrightarrow {BC}  + \overrightarrow {BD} .\overrightarrow {BC} \) \( = \overrightarrow {AB} .\left( {\overrightarrow {CD}  + \overrightarrow {DB}  + \overrightarrow {BC} } \right) + \left( {\overrightarrow {BC} .\overrightarrow {DB}  + \overrightarrow {BD} .\overrightarrow {BC} } \right) = \overrightarrow {AB} .\left( {\overrightarrow {CB}  + \overrightarrow {BC} } \right) + \overrightarrow {BC} \left( {\overrightarrow {DB}  + \overrightarrow {BD} } \right) = 0\)

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