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Giải bài 4.61 trang 70 sách bài tập toán 10 - Kết nối tri thức với cuộc sống

Cho tam giác ABC có AB = 4,AC = 5

🤡Tổng hợp đề thi học kì 2 lớp 10 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Lí - Hóa - Sinh - Sử - Địa...
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Đề bài

Cho tam giác \(ABC\) có \(AB = 4,\,\,AC = 5\) và \(\widehat {CAB} = {60^ \circ }.\) a) Tính tích vô hướng \(\overrightarrow {AB} .\overrightarrow {AC} ,\,\,\overrightarrow {AB} .\overrightarrow {BC} .\) b) Lấy các điểm \(M,\,\,N\) thỏa mãn \(2\overrightarrow {AM}  + 3\overrightarrow {MC}  = \overrightarrow 0 \) và \(\overrightarrow {NB}  + x\overrightarrow {NC}  = \overrightarrow 0 \,\,\left( {x \ne  - 1} \right).\) Xác định \(x\) sao cho \(AN\) vuông góc với \(BM.\)

Lời giải chi tiết

a)      Ta có: \(\overrightarrow {AB} .\overrightarrow {AC}  = AB.AC.\cos \widehat {CAB} = 4.5.\cos {60^ \circ } = 10\)\(\overrightarrow {AB} .\overrightarrow {BC}  = \overrightarrow {AB} \left( {\overrightarrow {AC}  - \overrightarrow {AB} } \right) = \overrightarrow {AB} .\overrightarrow {AC}  - {\overrightarrow {AB} ^2} = 10 - {4^2} =  - 6\)b)     Ta có: \(2\overrightarrow {AM}  + 3\overrightarrow {MC}  = \overrightarrow 0 \)\( \Leftrightarrow \) \(2\left( {\overrightarrow {AB}  + \overrightarrow {BM} } \right) + 3\left( {\overrightarrow {BC}  - \overrightarrow {BM} } \right) = \overrightarrow 0 \)\( \Leftrightarrow \) \(\overrightarrow {BM}  =  - 2\overrightarrow {AB}  - 3\overrightarrow {BC}  = 2\overrightarrow {AB}  + 3\left( {\overrightarrow {AC}  - \overrightarrow {AB} } \right) =  - \overrightarrow {AB}  + 3\overrightarrow {AC} \)     (1)Ta có: \(\overrightarrow {NB}  + x\overrightarrow {NC}  = \overrightarrow 0 \)\( \Leftrightarrow \) \(\left( {\overrightarrow {AB}  - \overrightarrow {AN} } \right) + x\left( {\overrightarrow {AC}  - \overrightarrow {AN} } \right) = \overrightarrow 0 \)
\( \Leftrightarrow \) \(\left( {1 + x} \right)\overrightarrow {AN}  = \overrightarrow {AB}  + x\overrightarrow {AC} \)            (2)Từ (1) và (2) \( \Rightarrow \) \(\left( {1 + x} \right)\overrightarrow {AN} .\overrightarrow {BM}  = \left( {\overrightarrow {AB}  + x\overrightarrow {AC} } \right)\left( { - \overrightarrow {AB}  + 3\overrightarrow {AC} } \right)\)\( \Leftrightarrow \) \(\left( {1 + x} \right)\overrightarrow {AN} .\overrightarrow {BM}  =  - {\overrightarrow {AB} ^2} + 3\overrightarrow {AB} .\overrightarrow {AC}  - x\overrightarrow {AC} .\overrightarrow {AB}  + 3x{\overrightarrow {AC} ^2}\)\( \Leftrightarrow \) \(\left( {1 + x} \right)\overrightarrow {AN} .\overrightarrow {BM}  =  - 16 + 3.10 - x.10 + 3x.25 = 65x + 14\)Để \(AN \bot BM\) \( \Leftrightarrow \) \(\overrightarrow {AN} .\overrightarrow {BM}  = 0\)\( \Leftrightarrow \) \(65x + 14 = 0\) \( \Leftrightarrow \) \(x =  - \frac{{14}}{{64}}\) (thỏa mãn)Vậy \(x =  - \frac{{14}}{{64}}\) thì \(AN \bot BM\)

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