The value of frac{[(vec{a} times vec{b}) time | vedclass

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(B) Let $\vec{u} = \vec{a} 	imes \vec{b}$,$\vec{v} = \vec{b} 	imes \vec{c}$,and $\vec{w} = \vec{c} 	imes \vec{a}$.We know that $[\vec{u}, \vec{v},

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$[\vec{a}, \vec{b}, \vec{c}]^2$

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(B) Let $\vec{u} = \vec{a} 	imes \vec{b}$,$\vec{v} = \vec{b} 	imes \vec{c}$,and $\vec{w} = \vec{c} 	imes \vec{a}$.We know that $[\vec{u}, \vec{v}, \vec{w}] = [\vec{a} 	imes \vec{b}, \vec{b} 	imes \vec{c}, \vec{c} 	imes \vec{a}] = [\vec{a}, \vec{b}, \vec{c}]^2$.The expression given is the scalar triple product of the cross products of these vectors: $[\vec{u} 	imes \vec{v}, \vec{v} 	imes \vec{w}, \vec{w} 	imes \vec{u}]$.Using the property $[\vec{u} 	imes \vec{v}, \vec{v} 	imes \vec{w}, \vec{w} 	imes \vec{u}] = [\vec{u}, \vec{v}, \vec{w}]^2$.Therefore,the expression becomes $\frac{[\vec{u}, \vec{v}, \vec{w}]^2}{[\vec{u}, \vec{v}, \vec{w}]} = [\vec{u}, \vec{v}, \vec{w}] = [\vec{a}, \vec{b}, \vec{c}]^2$.

The value of $\frac{[(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}), (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}), (\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b})]}{[\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}]}$ is equal to

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