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Giải bài 41 trang 92 SBT toán 10 - Cánh diều

Cho hai vectơ \(\overrightarrow a ,\overrightarrow b \) khác vectơ \(\overrightarrow 0 \). Chứng minh rằng nếu hai vectơ cùng hướng thì \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = \left| {\overrightarrow a + \overrightarrow b } \right|\)

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

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Đề bài

Cho hai vectơ \(\overrightarrow a ,\overrightarrow b \) khác vectơ \(\overrightarrow 0 \). Chứng minh rằng nếu hai vectơ cùng hướng thì \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = \left| {\overrightarrow a  + \overrightarrow b } \right|\)

Phương pháp giải - Xem chi tiết

Bước 1: Dựng 2 vectơ \(\overrightarrow {AB}  = \overrightarrow a ,\overrightarrow {BC}  = \overrightarrow b \) thỏa mãn \(\overrightarrow {AB} ,\overrightarrow {BC} \)cùng hướng Bước 2: Sử dụng quy tắc cộng vectơ và độ dài vectơ để biến đổi giả thiết \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = \left| {\overrightarrow a  + \overrightarrow b } \right|\)

Lời giải chi tiết

Lấy một điểm A trên mặt phẳng. Dựng \(\overrightarrow {AB}  = \overrightarrow a ,\overrightarrow {BC}  = \overrightarrow b \) sao ♈cho \(\overrightarrow {AB} ,\overrightarrow {BC} \)cùng hướng

\( \Rightarrow \left| {\overrightarrow a } \right| = AB,\left| {\overrightarrow b } \right| = BC\)Ta có: \(\overrightarrow {AB}  + \overrightarrow {BC}  = \overrightarrow {AC}  \Leftrightarrow \overrightarrow a  + \overrightarrow b  = \overrightarrow {AC} \)

Lại có: AB + BC = AC \( \Rightarrow \left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = AC = \left| {\🌟overrightarrow {AC} } \right| = \left| {\overrightarrow a  + \overrightarrow b } \right|\)  (ĐPCM)

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