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Giải bài 4 trang 103 sách bài tập toán 10 - Chân trời sáng tạo

Chứng minh rằng với hai vectơ không cùng phương

ꦫTổng hợp đề thi học kì 2 lớp 10 tất cả các môn - Chân trời sáng tạo

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

Chứng minh rằng với hai vectơ không cùng phương \(\overrightarrow a \) và \(\overrightarrow b \), ta có:    \(\left| {\overrightarrow a } \right| - \left| {\overrightarrow b } \right| \le \left| {\overrightarrow a  + \overrightarrow b } \right| \le \left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right|\)

Lời giải chi tiết

TH1: \(\overrightarrow a  = \overrightarrow 0 \)\( \Rightarrow \left| {\overrightarrow a } \right| - \left| {\overrightarrow b } \right| = \left| {\overrightarrow a  + \overrightarrow b } \right| = \left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = \left| {\overrightarrow b } \right|\)TH2: \(\overrightarrow b  = \overrightarrow 0 \)\( \Rightarrow \left| {\overrightarrow a } \right| - \left| {\overrightarrow b } \right| = \left| {\overrightarrow a  + \overrightarrow b } \right| = \left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = \left| {\overrightarrow a } \right|\)TH3: \(\overrightarrow a  \ne \overrightarrow 0 \) và \(\overrightarrow b  \ne \overrightarrow 0 \)Lấy A bất kì, vẽ \(\overrightarrow {AB}  = \overrightarrow a ,\overrightarrow {AD}  = \overrightarrow b \). Dựng hình bình hành ABCD, đặt \(\overrightarrow c  = \overrightarrow {AC} \)

 Ta có: \(\overrightarrow a  + \overrightarrow b  = \overrightarrow {AB}  + \overrightarrow {AD}  = \overrightarrow {AC}  = \overrightarrow c \)
Xét tam giác ABC, theo bất đẳng thức tam giác ta có:\(AB - BC < AC < AB + BC\)Mà \(\left| {\overrightarrow a } \right| = \left| {\overrightarrow {AB} } \right| = AB;\left| {\overrightarrow b } \right| = \left| {\overrightarrow {AD} } \right| = AD = BC;\left| {\overrightarrow c } \right| = \left| {\overrightarrow {AC} } \right| = AC;\)\( \Rightarrow \left| {\overrightarrow a } \right| - \left| {\overrightarrow b } \right| < \left| {\overrightarrow a  + \overrightarrow b } \right| < \left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right|\)Vậy \(\left| {\overrightarrow a } \right| - \left| {\overrightarrow b } \right| \le \left| {\overrightarrow a  + \overrightarrow b } \right| \le \left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right|\)

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