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(D) Given expression: $\vec{a} 	imes (\vec{b} + \vec{c}) + \vec{b} 	imes (\vec{c} + \vec{a}) + \vec{c} 	imes (\vec{a} + \vec{b})$Using the distribu

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$\vec{0}$

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(D) Given expression: $\vec{a} 	imes (\vec{b} + \vec{c}) + \vec{b} 	imes (\vec{c} + \vec{a}) + \vec{c} 	imes (\vec{a} + \vec{b})$Using the distributive property of the cross product: $\vec{a} 	imes \vec{b} + \vec{a} 	imes \vec{c} + \vec{b} 	imes \vec{c} + \vec{b} 	imes \vec{a} + \vec{c} 	imes \vec{a} + \vec{c} 	imes \vec{b}$We know that $\vec{a} 	imes \vec{b} = -(\vec{b} 	imes \vec{a})$,$\vec{a} 	imes \vec{c} = -(\vec{c} 	imes \vec{a})$,and $\vec{b} 	imes \vec{c} = -(\vec{c} 	imes \vec{b})$Substituting these into the expression:$= (\vec{a} 	imes \vec{b}) + (\vec{a} 	imes \vec{c}) + (\vec{b} 	imes \vec{c}) - (\vec{a} 	imes \vec{b}) - (\vec{a} 	imes \vec{c}) - (\vec{b} 	imes \vec{c})$$= \vec{0}$

For any three vectors $\vec{a}, \vec{b}, \vec{c}$,the value of $\vec{a} \times (\vec{b} + \vec{c}) + \vec{b} \times (\vec{c} + \vec{a}) + \vec{c} \times (\vec{a} + \vec{b})$ is equal to:

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