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Giải mục II trang 89, 90 SGK Toán 10 tập 1 - Cánh diều

Cho ba điểm A, B, C. Chứng minh

🐈Tổng hợp đề thi học kì 2 lớp 10 tất cả các môn - Cánh diều

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

Luyện tập – vận dụng 2 trang 89 Sách giáo khoa Toán 10 – Cánh Diều

Cho ba điểm A, B, C. Chứng minh \(3\left( {\overrightarrow {AB}  + 2\overrightarrow {BC} } \right) - 2\left( {\overrightarrow {AB}  + 3\overrightarrow {BC} } \right) = \overrightarrow {AB} \)

Phương pháp giải - Xem chi tiết

Áp dụng các tính chất sau: \(\begin{array}{l}k\left( {\overrightarrow a  \pm \overrightarrow b } \right) = k\overrightarrow a  \pm k\overrightarrow b \\\left( {h + k} \right)\overrightarrow a  = h\overrightarrow a  + k\overrightarrow a \\h\left( {k\overrightarrow a } \right) = \left( {hk} \right)\overrightarrow a \end{array}\)

Lời giải chi tiết

Ta có: \(3\left( {\overrightarrow {AB}  + 2\overrightarrow {BC} } \right) - 2\left( {\overrightarrow {AB}  + 3\overrightarrow {BC} } \right)\)\( = 3\overrightarrow {AB}  + 3.\left( {2\overrightarrow {BC} } \right) - \left[ {2\overrightarrow {AB}  + 2.\left( {3\overrightarrow {BC} } \right)} \right]\)\[ = 3\overrightarrow {AB}  + 6.\overrightarrow {BC}  - \left( {2\overrightarrow {AB}  + 6.\overrightarrow {BC} } \right)\]\[ = 3\overrightarrow {AB}  + 6.\overrightarrow {BC}  - 2\overrightarrow {AB}  - 6.\overrightarrow {BC} \]\[ = \left( {3\overrightarrow {AB}  - 2\overrightarrow {AB} } \right) + \left( {6.\overrightarrow {BC}  - 6.\overrightarrow {BC} } \right) = \overrightarrow {AB} .\]

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