If $\vec{a}=\vec{b}+\vec{c},$ then is it true that $|\vec{a}|=|\vec{b}|+|\vec{c}|$ ? Justify your answer.
(N/A) In $\Delta ABC,$ let $\overrightarrow{CB}=\vec{a}, \overrightarrow{CA}=\vec{b},$ and $\overrightarrow{AB}=\vec{c}$ (as shown in the 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 lengths of the sides of $\Delta ABC.$
Also,it is a known geometric property that the sum of the lengths of any two sides of a triangle is always greater than the third side.
Therefore,$|\vec{a}| < |\vec{b}| + |\vec{c}|.$
Hence,it is not true that $|\vec{a}| = |\vec{b}| + |\vec{c}|.$
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 lengths of the sides of $\Delta ABC.$
Also,it is a known geometric property that the sum of the lengths of any two sides of a triangle is always greater than the third side.
Therefore,$|\vec{a}| < |\vec{b}| + |\vec{c}|.$
Hence,it is not true that $|\vec{a}| = |\vec{b}| + |\vec{c}|.$
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