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 \( \Rightarrow \left| {\overrightarrow a } \right| + \left| {\overrightarrow b } \right| = AC = \left| {\overrightarrow {AC} } \right| = \left| {\overrightarrow a + \overrightarrow b } \right|\)
