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Giải bài 75 trang 107 SBT toán 10 - Cánh diều

Cho ba điểm phân biệt I, A, B và số thực k ≠ 1 thoả mãn \(\overrightarrow {IA} = k\overrightarrow {IB} \). Chứng minh rằng với O là điểm bất kì ta có:

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Đề bài

Cho ba điểm phân biệt I, A, B và số thực k ≠ 1 thoả mãn \(\overrightarrow {IA}  = k\overrightarrow {IB} \). Chứng minh rằng với O là điểm bất kì ta có:

\(\overrightarrow {OI}  = \left( {\frac{1}{{1 - k}}} \right)\overrightarrow {OA}  - \left( {\frac{k}{{1 - k}}} \right)\overrightarrow {OB} \) (*)

Phương pháp giải - Xem chi tiết

Tách các vectơ \(\overrightarrow {OA} ,\overrightarrow {OB} \) sao cho xuất hiện vectơ \(\overrightarrow {OI} \) và kết hợp giả thiết để biến đổi vế phải (*)

Lời giải chi tiết

Theo giả thiết, \(\overrightarrow {IA}  = k\overrightarrow {IB} \)Xét vế phải (*) ta có:VT = \(\left( {\frac{1}{{1 - k}}} \right)\overrightarrow {OA}  - \left( {\frac{k}{{1 - k}}} \right)\overrightarrow {OB}  = \left( {\frac{1}{{1 - k}}} \right)\left( {\overrightarrow {OI}  + \overrightarrow {IA} } \right) - \left( {\frac{k}{{1 - k}}} \right)\left( {\overrightarrow {OI}  + \overrightarrow {IB} } \right)\)      \( = \left( {\frac{1}{{1 - k}}} \right)\overrightarrow {OI}  + \left( {\frac{1}{{1 - k}}} \right)\overrightarrow {IA}  - \left( {\frac{k}{{1 - k}}} \right)\overrightarrow {OI}  - \left( {\frac{k}{{1 - k}}} \right)\overrightarrow {IB} \) \( = \left( {\frac{1}{{1 - k}} - \frac{k}{{1 - k}}} \right)\overrightarrow {OI}  + \left( {\frac{1}{{1 - k}}} \right).k\overrightarrow {IB}  - \left( {\frac{k}{{1 - k}}} \right)\overrightarrow {IB} \)      \( = \overrightarrow {OI}  + \left( {\frac{1}{{1 - k}}} \right).k\overrightarrow {IB}  - \left( {\frac{k}{{1 - k}}} \right)\overrightarrow {IB}  = \overrightarrow {OI}  + \left( {\frac{k}{{1 - k}} - \frac{k}{{1 - k}}} \right)\overrightarrow {IB} \) \( = \overrightarrow {OI} \) (ĐPCM)

 

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