Let vec{a}, vec{b}, vec{c} be three vectors i | vedclass

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(A) Given $\vec{a} 	imes \vec{b} = \vec{b} 	imes \vec{c} = \vec{c} 	imes \vec{a} = \vec{v} 
eq 0$.From $\vec{a} 	imes \vec{b} = \vec{b} 	imes \v

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(A) Given $\vec{a} 	imes \vec{b} = \vec{b} 	imes \vec{c} = \vec{c} 	imes \vec{a} = \vec{v} 
eq 0$.From $\vec{a} 	imes \vec{b} = \vec{b} 	imes \vec{c}$,we have $(\vec{a} - \vec{c}) 	imes \vec{b} = 0$,which implies $\vec{a} - \vec{c} = k_1 \vec{b}$ for some scalar $k_1$.Similarly,from $\vec{b} 	imes \vec{c} = \vec{c} 	imes \vec{a}$,we have $(\vec{b} - \vec{a}) 	imes \vec{c} = 0$,which implies $\vec{b} - \vec{a} = k_2 \vec{c}$ for some scalar $k_2$.Adding these equations: $\vec{a} 	imes \vec{b} = \vec{b} 	imes \vec{c} = \vec{c} 	imes \vec{a}$ implies $\vec{a} + \vec{b} + \vec{c} = \vec{0}$ is not necessarily true,but rather $\vec{a} + \vec{b} + \vec{c}$ must be a vector such that the cross products are equal.Actually,$\vec{a} 	imes \vec{b} = \vec{b} 	imes \vec{c} = \vec{c} 	imes \vec{a}$ implies $\vec{a} + \vec{b} + \vec{c} = \vec{k}$ where $\vec{k}$ is a constant vector.However,the condition $\vec{a} 	imes \vec{b} = \vec{b} 	imes \vec{c} = \vec{c} 	imes \vec{a}$ implies $\vec{a} + \vec{b} + \vec{c} = 0$ is not required; instead,it implies that $\vec{a}, \vec{b}, \vec{c}$ are coplanar or satisfy specific relations.Given the symmetry,the only solution for the centroid $\vec{G} = \frac{\vec{a} + \vec{b} + \vec{c}}{3}$ under the constraint $\vec{a} 	imes \vec{b} = \vec{b} 	imes \vec{c} = \vec{c} 	imes \vec{a} 
eq 0$ is that there is only $1$ possible position for the centroid,which is the origin $\vec{0}$.

List $I$	List $II$
$P$. Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $2$. Then the volume of the parallelepiped determined by vectors $2(\vec{a} \times \vec{b}), 3(\vec{b} \times \vec{c})$ and $(\vec{c} \times \vec{a})$ is	$1$. $100$
$Q$. Volume of parallelepiped determined by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $5$. Then the volume of the parallelepiped determined by vectors $3(\vec{a}+\vec{b}), (\vec{b}+\vec{c})$ and $2(\vec{c}+\vec{a})$ is	$2$. $30$
$R$. Area of a triangle with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $20$. Then the area of the triangle with adjacent sides determined by vectors $(2\vec{a}+3\vec{b})$ and $(\vec{a}-\vec{b})$ is	$3$. $24$
$S$. Area of a parallelogram with adjacent sides determined by vectors $\vec{a}$ and $\vec{b}$ is $30$. Then the area of the parallelogram with adjacent sides determined by vectors $(\vec{a}+\vec{b})$ and $\vec{a}$ is	$4$. $60$

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