Let a, b and c be three vectors such that | a | = 3, | b | = 4, |… - Vidyadip

Question

Let $\vec a,\vec b$ and $\vec c$ be three vectors such that $\left| {\vec a} \right| = 3,\left| {\vec b} \right| = 4,\left| {\vec c} \right| = 5$ and each one of them being perpendicular to the sum of the other two, find $\left| {\vec a + \vec b + \vec c} \right|$.

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Answer

$\vec a.\left( {\vec b + \vec c} \right) = 0,\vec b.\left( {\vec c + \vec a} \right) = 0,\vec c.\left( {\vec a + \vec b} \right) = 0$ (given)
${\left| {\vec a + \vec b + \vec c} \right|^2} = \left( {\vec a + \vec b + \vec c} \right).\left( {\vec a + \vec b + \vec c} \right)$
$= \vec a.\vec a + \vec a\left( {\vec b + \vec c} \right) + \vec b.\vec b + \vec b\left( {\vec a + \vec c} \right) + \vec c.\vec c + \left( {\vec a + \vec b} \right)$
$= {\left| {\vec a} \right|^2} + {\left| {\vec b} \right|^2} + {\left| {\vec c} \right|^2}$
$ = 9 + 16 + 25$
= 50
$\left| {\vec a + \vec b + \vec c} \right| = \sqrt {50} $
$ = 5\sqrt 2 $

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