If overrightarrow{a}+2 overrightarrow{b}+3 ov | vedclass

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(D) Given,$\overrightarrow{a}+2 \overrightarrow{b}+3 \overrightarrow{c}=\overrightarrow{0} \quad \dots(i)$Taking the cross product of equation $(i)$ w

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$6(\overrightarrow{b} 	imes \overrightarrow{c})$

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(D) Given,$\overrightarrow{a}+2 \overrightarrow{b}+3 \overrightarrow{c}=\overrightarrow{0} \quad \dots(i)$Taking the cross product of equation $(i)$ with $\overrightarrow{b}$:$\overrightarrow{a} 	imes \overrightarrow{b} + 2(\overrightarrow{b} 	imes \overrightarrow{b}) + 3(\overrightarrow{c} 	imes \overrightarrow{b}) = \overrightarrow{0} 	imes \overrightarrow{b}$Since $\overrightarrow{b} 	imes \overrightarrow{b} = \overrightarrow{0}$ and $\overrightarrow{c} 	imes \overrightarrow{b} = -(\overrightarrow{b} 	imes \overrightarrow{c})$:$\overrightarrow{a} 	imes \overrightarrow{b} - 3(\overrightarrow{b} 	imes \overrightarrow{c}) = \overrightarrow{0} \implies \overrightarrow{a} 	imes \overrightarrow{b} = 3(\overrightarrow{b} 	imes \overrightarrow{c})$Taking the cross product of equation $(i)$ with $\overrightarrow{c}$:$\overrightarrow{a} 	imes \overrightarrow{c} + 2(\overrightarrow{b} 	imes \overrightarrow{c}) + 3(\overrightarrow{c} 	imes \overrightarrow{c}) = \overrightarrow{0} 	imes \overrightarrow{c}$Since $\overrightarrow{c} 	imes \overrightarrow{c} = \overrightarrow{0}$ and $\overrightarrow{a} 	imes \overrightarrow{c} = -(\overrightarrow{c} 	imes \overrightarrow{a})$:$-(\overrightarrow{c} 	imes \overrightarrow{a}) + 2(\overrightarrow{b} 	imes \overrightarrow{c}) = \overrightarrow{0} \implies \overrightarrow{c} 	imes \overrightarrow{a} = 2(\overrightarrow{b} 	imes \overrightarrow{c})$Now,substituting these into the expression $\overrightarrow{a} 	imes \overrightarrow{b} + \overrightarrow{b} 	imes \overrightarrow{c} + \overrightarrow{c} 	imes \overrightarrow{a}$:$= 3(\overrightarrow{b} 	imes \overrightarrow{c}) + (\overrightarrow{b} 	imes \overrightarrow{c}) + 2(\overrightarrow{b} 	imes \overrightarrow{c})$$= 6(\overrightarrow{b} 	imes \overrightarrow{c})$

If $\overrightarrow{a}+2 \overrightarrow{b}+3 \overrightarrow{c}=\overrightarrow{0}$,then $\overrightarrow{a} \times \overrightarrow{b}+\overrightarrow{b} \times \overrightarrow{c}+\overrightarrow{c} \times \overrightarrow{a}$ is equal to

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