## 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 $$\Rightarrow \left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = AC = \left| {\overrightarrow {AC} } \right| = \left| {\overrightarrow a + \overrightarrow b } \right|$$