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If $|\vec{a}|=3,|\vec{b}|=2,|\vec{c}|=1$ then the value of $|\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}|$ is (given that $\vec{a}+\vec{b}+\vec{c}=0$ )
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The correct answer is:
7
$\begin{aligned} & \vec{a}+\vec{b}+\vec{c}=0 \\ & \Rightarrow(\vec{a}+\vec{b}+\vec{c})^{2}=0 \\ & \Rightarrow|\vec{a}|^{2}+|\vec{b}|^{2}+|\vec{c}|^{2}+2(\vec{a} \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0 \\ & \Rightarrow 9+4+1+2(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0 \\ & \Rightarrow \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}=7 \end{aligned}$
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