# Question 82dbe

Jan 20, 2017

Given absvec(a)=3;absvec(b)=4 and absvec(c)=5#

Again

$\vec{a} \bot \left(\vec{b} + \vec{c}\right)$

$\implies \vec{a} . \left(\vec{b} + \vec{c}\right) = 0. \ldots \ldots \left[1\right]$

$\vec{b} \bot \left(\vec{c} + \vec{a}\right)$

$\implies \vec{b} . \left(\vec{c} + \vec{a}\right) = 0. \ldots \ldots \ldots \left[2\right]$

$\vec{c} \bot \left(\vec{a} + \vec{b}\right)$

$\implies \vec{c} . \left(\vec{a} + \vec{b}\right) = 0. \ldots \ldots \ldots . . \left[3\right]$

Adding [1],[2] and [3] we get

$2 \left(\vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a}\right) = 0$

$\implies 2 \left\mid \vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a} \right\mid = 0$

Now
${\left\mid \vec{a} + \vec{b} + \vec{c} \right\mid}^{2} = {\left\mid \vec{a} \right\mid}^{2} + {\left\mid \vec{b} \right\mid}^{2} + {\left\mid \vec{c} \right\mid}^{2} + 2 \left\mid \vec{a} . \vec{b} + \vec{b} . \vec{c} + \vec{c} . \vec{a} \right\mid$

$\implies {\left\mid \vec{a} + \vec{b} + \vec{c} \right\mid}^{2} = {3}^{2} + {4}^{2} + {5}^{2} + 0$

$\implies {\left\mid \vec{a} + \vec{b} + \vec{c} \right\mid}^{2} = 50$

$\implies \left\mid \vec{a} + \vec{b} + \vec{c} \right\mid = \sqrt{50} = 5 \sqrt{2}$