If \( |\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=5 \) each one of \( \vec{a}, \vec{b} \& \vec{c} \) is perpendicular to the sum of the
remaining then \( |\vec{a}+\vec{b}+\vec{c}| \) is equal to

Given that, each one of \(\vec{a}, \vec{b}\) and \(\vec{c}\) is perpendicular to the sum of the remaining. So,
\(\vec{a} \cdot(\vec{b}+\vec{c})=0 \rightarrow(1)\)
\(\vec{b} \cdot|\vec{c}+\vec{a}|=0 \rightarrow(2)\)
\(\vec{c} \cdot \mid \vec{a}+\vec{b})=0 \rightarrow(3)\)
Now, adding Eqs. (1), (2) and (3), we get
\(2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0\)
Now,
\(|a+\vec{b}+\vec{c}|^{2}=|\vec{a}|^{2}+|\vec{b}|^{2}+|\vec{c}|^{2}+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})\)
\(\Rightarrow|a+\vec{b}+\vec{c}|^{2}=9+16+25\)
\(\Rightarrow|a+\vec{b}+\vec{c}|^{2}=50 \Rightarrow|a+\vec{b}+\vec{c}|^{2}=5 \sqrt{2}\)