If a = b + c, then is it true that | a | = | b | + | c |? Justify… - Vidyadip

Question

If $\vec a = \vec b + \vec c$, then is it true that $\left| {\vec a} \right| = \left| {\vec b} \right| + \left| {\vec c} \right|$? Justify your answer.

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Answer

Given: $\vec a = \vec b + \vec c$
$\therefore$ Either the vectors $\vec a,\vec b,\vec c$ are collinear or form the sides of a triangle.
Case I: Vectors $\vec a,\vec b,\vec c$ are collinear.
Let $\vec a = \overrightarrow {AC} ,\vec b = \overrightarrow {AB} $ and $\vec c = \overrightarrow {BC} $
Then $\vec a = \overrightarrow {AC} = \overrightarrow {AB} + \overrightarrow {BC} = \vec b + \vec c$
Also, $\left| {\vec a} \right| = AC = AB + BC = \left| {\vec b} \right| + \left| {\vec c} \right|$
Case II: Vectors $\vec a,\vec b,\vec c$ form a triangle.
Here also by Triangle Law of vectors, $\vec a = \vec b + \vec c$
But $\left| {\vec a} \right| < \left| {\vec b} \right| + \left| {\vec c} \right|$ [$\because $ Each side of a triangle is less than sum of the other two sides]
$\therefore \,\left| {\vec a = \vec b + \vec c} \right| = \left| {\vec b} \right| + \left| {\vec c} \right|$ is true only when vectors $\vec b$ and $\vec c$are collinear vectors.

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