If $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are vectors of lengths 4, 4, 5 respectively, and $\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are perpendicular to $\overline{\mathrm{b}} + \overline{\mathrm{c}}, \overline{\mathrm{c}}+\overline{\mathrm{a}}, \overline{\mathrm{a}} + \overline{\mathrm{b}},$ then $|\overline{\mathrm{a}} + \overline{\mathrm{b}} + \overline{\mathrm{c}}| =$

$$\begin{gathered} \left|\overline{\mathrm{a}}\right|=4 \left|\overline{\mathrm{b}}\right|=4 \left|\overline{\mathrm{c}}\right|=5 \\ \overline{\mathrm{a}}.(\overline{\mathrm{b}}+\overline{\mathrm{c}})=0 \Rightarrow \overline{\mathrm{a}}.\overline{\mathrm{b}}+\overline{\mathrm{a}}.\overline{\mathrm{c}}=0 \\ \overline{\mathrm{b}}.(\overline{\mathrm{c}}.\overline{\mathrm{a}})=0 \Rightarrow \overline{\mathrm{b}}.\overline{\mathrm{c}}+\overline{\mathrm{b}}.\overline{\mathrm{a}}=0 \\ \overline{\mathrm{c}}.(\overline{\mathrm{a}}+\overline{\mathrm{b}})=0 \Rightarrow \overline{\mathrm{c}}.\overline{\mathrm{a}}+\overline{\mathrm{c}}.\overline{\mathrm{b}}=0 \\ \mathrm{adding} \Rightarrow 2(\overline{\mathrm{a}}.\overline{\mathrm{b}}+\overline{\mathrm{b}}.\overline{\mathrm{c}}+\overline{\mathrm{c}}.\overline{\mathrm{a}})=0 \\ {\left|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}\right|}^2=16+16+25+0=57 \\ \left|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}\right|=\sqrt{57} \end{gathered}$$