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Solve Lesson 41 Page 92 SBT Math 10 – Kite>


Topic

Given two vectors \(\overrightarrow a ,\overrightarrow b \) other than the vector \(\overrightarrow 0 \). Prove that if two vectors have the same direction then \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = \left| {\overrightarrow a + \overrightarrow b } \right| \)

Solution method – See details

Step 1: Construct 2 vectors \(\overrightarrow {AB} = \overrightarrow a ,\overrightarrow {BC} = \overrightarrow b \) satisfying \(\overrightarrow {AB} ,\overrightarrow {BC} \) in the same direction

Step 2: Use the vector addition rule and vector length to transform the hypothesis \(\left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = \left| {\overrightarrow a + \overrightarrow b } \right|\)

Detailed explanation

Take a point A on the flat surface. Construct \(\overrightarrow {AB} = \overrightarrow a ,\overrightarrow {BC} = \overrightarrow b \) so that \(\overrightarrow {AB} ,\overrightarrow {BC} \) is in the same direction

\( \Rightarrow \left| {\overrightarrow a } \right| = AB,\left| {\overrightarrow b } \right| = BC\)

We have: \(\overrightarrow {AB} + \overrightarrow {BC} = \overrightarrow {AC} \Leftrightarrow \overrightarrow a + \overrightarrow b = \overrightarrow {AC} \)

Again there are: AB + BC = AC \( \Rightarrow \left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = AC = \left| {\overrightarrow {AC} } \right| = \left| {\overrightarrow a + \overrightarrow b } \right|\)



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