Let \(\overline{\mathrm{A}}, \overline{\mathrm{B}}, \overline{\mathrm{C}}\) be vectors of lengths 3 units, 4 units, 5 units respectively. let \(\bar{A}\) be perpendicular to \(\overline{\mathrm{B}}+\overline{\mathrm{C}}, \overline{\mathrm{B}}\) be perpendicular to \(\overline{\mathrm{C}}+\overline{\mathrm{A}}\) and \(\overline{\mathrm{C}}\) be perpendicular to \(\overline{\mathrm{A}}+\overline{\mathrm{B}}\), then the length of vector \(\overline{\mathrm{A}}+\overline{\mathrm{B}}+\overline{\mathrm{C}}\) is

\(\begin{aligned}
& |\overline{\mathrm{a}}|=3,|\overline{\mathrm{~b}}|=4 \text { and }|\overrightarrow{\mathrm{c}}|=5 \\
& \overline{\mathrm{a}} \cdot(\overline{\mathrm{~b}}+\overline{\mathrm{c}})=0, \overline{\mathrm{~b}} \cdot(\overline{\mathrm{c}}+\overline{\mathrm{a}})=0 \text { and } \overline{\mathrm{c}} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{b}})=0 \\
& \Rightarrow 2(\overline{\mathrm{a}} \cdot \overline{\mathrm{~b}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{a}})=0
\end{aligned}\)
Now,
\(\begin{aligned}
& |\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|^2=|\overline{\mathrm{a}}|^2+|\overline{\mathrm{b}}|^2+|\overline{\mathrm{c}}|^2+2(\overline{\mathrm{a}} \cdot \overline{\mathrm{~b}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{a}}) \\
& \Rightarrow|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|^2=9+16+25+0 \quad \ldots[\text { From }(\mathrm{i}) \\
& \Rightarrow|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|^2=50 \\
& \Rightarrow|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|=5 \sqrt{2}
\end{aligned}\)