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Let $\vec{a}, \vec{b}, \vec{c}$ be vectors of lengths $3,4,5$ respectively. Let $\vec{a}$ be perpendicular to $\vec{b}+\vec{c}, \vec{b}$ be perpendicular to $\vec{c}+\vec{a}$ and $\vec{c}$ be perpendicular to $\vec{a}+\vec{b}$, then the length of vector $\vec{a}+\vec{b}+\vec{c}$ is
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$5 \sqrt{2}$
$\begin{aligned} & \text { Now }|\vec{a}+\vec{b}+\vec{c}|=\sqrt{|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2+2 \vec{a} \cdot \vec{b}+2 \vec{b} \cdot \vec{c}+2 \vec{c} \cdot \vec{a}} \\ & =\sqrt{3^2+4^2+5^2+0} \quad[\because|\vec{a}|=3,|\vec{b}|=4,|\vec{c}|=5 \text { given and from(iv) } \\ & =\sqrt{50} \\ & =5 \sqrt{2}\end{aligned}$
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