If each of  $\overrightarrow{a}$, $\overrightarrow{b}$ , $\overrightarrow{c}$  is orthogonal to sum of other two vectors and $\left|\overrightarrow{a}\right|=3,\left|\overrightarrow{b}\right|=4,\left|\overrightarrow{c}\right|=5$ then 

${\sin }^2\theta +{\cos }^2\theta +{\sin }^2\theta =1$
$\Rightarrow 3{\sin }^2\theta =1\Rightarrow {\sin }^2\theta =\frac{1}{3}\Rightarrow \sin \theta =\frac{1}{\sqrt{3}}$
$\Rightarrow Tan\theta =\sqrt{2}$
$a\cdot (b+c)=0,b\cdot (c+a)=0,c\cdot (a+b)=0$

$\Rightarrow a.b+b.c+c.a=0\Rightarrow 1a-b{1}^2=25-24\sin \theta$
$\Rightarrow 1a-b1=\sqrt{25-24\sin \theta }=\sqrt{25+24}=7$
$\Rightarrow 1a-b1=\sqrt{25-24}=1$