If magnitudes of vectors vec{a}, vec{b}, vec{ | vedclass

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(B) Given that $\vec{a} \perp (\vec{b} + \vec{c})$,therefore $\vec{a} \cdot (\vec{b} + \vec{c}) = 0 \Rightarrow \vec{a} \cdot \vec{b} + \vec{a} \cdot

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$5\sqrt{2}$

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(B) Given that $\vec{a} \perp (\vec{b} + \vec{c})$,therefore $\vec{a} \cdot (\vec{b} + \vec{c}) = 0 \Rightarrow \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = 0$. Similarly,$\vec{b} \perp (\vec{c} + \vec{a}) \Rightarrow \vec{b} \cdot \vec{c} + \vec{b} \cdot \vec{a} = 0$. And $\vec{c} \perp (\vec{a} + \vec{b}) \Rightarrow \vec{c} \cdot \vec{a} + \vec{c} \cdot \vec{b} = 0$. Adding these three equations,we get $2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 0$,which implies $\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = 0$. Now,we know that $|\vec{a} + \vec{b} + \vec{c}|^2 = |\vec{a}|^2 + |\vec{b}|^2 + |\vec{c}|^2 + 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a})$. Substituting the given magnitudes $|\vec{a}| = 3, |\vec{b}| = 4, |\vec{c}| = 5$: $|\vec{a} + \vec{b} + \vec{c}|^2 = 3^2 + 4^2 + 5^2 + 2(0) = 9 + 16 + 25 = 50$. Therefore,$|\vec{a} + \vec{b} + \vec{c}| = \sqrt{50} = 5\sqrt{2}$.

If magnitudes of vectors $\vec{a}, \vec{b}, \vec{c}$ are $3, 4,$ and $5$ respectively,and $\vec{a}$ is perpendicular to $\vec{b} + \vec{c}$,$\vec{b}$ is perpendicular to $\vec{c} + \vec{a}$,and $\vec{c}$ is perpendicular to $\vec{a} + \vec{b}$,then find the value of $|\vec{a} + \vec{b} + \vec{c}|$.

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