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(C) Given that $\vec{a} \perp (\vec{b}+\vec{c}) \implies \vec{a} \cdot (\vec{b}+\vec{c}) = 0 \implies \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} =

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$5 \sqrt{2}$

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(C) Given that $\vec{a} \perp (\vec{b}+\vec{c}) \implies \vec{a} \cdot (\vec{b}+\vec{c}) = 0 \implies \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0 \quad (i)$$\vec{b} \perp (\vec{c}+\vec{a}) \implies \vec{b} \cdot (\vec{c}+\vec{a}) = 0 \implies \vec{b} \cdot \vec{c} + \vec{b} \cdot \vec{a} = 0 \quad (ii)$$\vec{c} \perp (\vec{a}+\vec{b}) \implies \vec{c} \cdot (\vec{a}+\vec{b}) = 0 \implies \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0 \quad (iii)$Adding $(i), (ii),$ and $(iii)$,we get:$2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0 \implies \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = 0 \quad (iv)$Now,the length of $\vec{a}+\vec{b}+\vec{c}$ is given by:$|\vec{a}+\vec{b}+\vec{c}| = \sqrt{|\vec{a}|^2+|\vec{b}|^2+|\vec{c}|^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a})}$Substituting the given values $|\vec{a}|=3, |\vec{b}|=4, |\vec{c}|=5$ and the result from $(iv)$:$|\vec{a}+\vec{b}+\vec{c}| = \sqrt{3^2+4^2+5^2 + 2(0)}$$|\vec{a}+\vec{b}+\vec{c}| = \sqrt{9+16+25} = \sqrt{50} = 5\sqrt{2}$

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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