Question

If\( \vec{a}=\vec{b}+\vec{c}\), then is it true that |\(\vec{a}\)|=|\(\vec{b}\)|+|\(\vec{c}\)| ? justify your answer.

Answer 1

In \(△ABC\),let \(\overrightarrow{CB}=\vec{a},\overrightarrow{CA}=\vec{b},\)and \(\overrightarrow{AB}=\vec{c}\)(as shown in the following figure).

Now,by the triangle law of vector addition,we have \(\vec{a}=\vec{b}+\vec{c}\).
It is clearly known that |\(\vec{a}\)|,|\(\vec{b}\)|,and |\(\vec{c}\)|represent the sides of \(△ABC.\)
Also,it is known that the sum of the lengths of any two sides of a triangle is greater than the third side.
∴|\(\vec{a}\)|<|\(\vec{b}\)|+|\(\vec{c}\)|
|Hence,it is not true that |\(\vec{a}\)|=|\(\vec{b}\)|+|\(\vec{c}\)|.