Question
$\text{If}\ \ \vec{\text{a}}=\vec{\text{b}}+\vec{\text{c}},$ then is it true that $\big|\vec{\text{a}}\big|=\big|\vec{\text{b}}\big|+\big|\vec{\text{c}}\big|?$ Justify your answer.
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Answer
$\text{If}\ \triangle\text{ABC},\ \text{let}\ \overrightarrow{\text{CB}}=\vec{\text{a}},\ \overrightarrow{\text{CA}}=\vec{\text{b}},\text{and}\ \overrightarrow{\text{AB}}=\vec{\text{c}}$ (as shown in the following figure).
Now, by the triangle law of vector additio, we have $\vec{\text{a}}=\vec{\text{b}}+\vec{\text{c}}$It is clearly known that $|\vec{\text{a}}|,\big|\vec{\text{b}}\big|,\ \text{and}\ \big|\vec{\text{c}}\big|$ represent the sides of $\triangle\text{ABC}.$
Also, it is known that the sum of the lengths of any two sides of a triangle is greater than the third side. $\therefore\ |\vec{\text{a}}|<\big|\vec{b}\big|+|\vec{c}|$ Hence, it is not true that $|\vec{\text{a}}|=\big|\vec{\text{b}}\big|+\big|\vec{\text{c}}\big|$
Now, by the triangle law of vector additio, we have $\vec{\text{a}}=\vec{\text{b}}+\vec{\text{c}}$It is clearly known that $|\vec{\text{a}}|,\big|\vec{\text{b}}\big|,\ \text{and}\ \big|\vec{\text{c}}\big|$ represent the sides of $\triangle\text{ABC}.$
Also, it is known that the sum of the lengths of any two sides of a triangle is greater than the third side. $\therefore\ |\vec{\text{a}}|<\big|\vec{b}\big|+|\vec{c}|$ Hence, it is not true that $|\vec{\text{a}}|=\big|\vec{\text{b}}\big|+\big|\vec{\text{c}}\big|$
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