If vec{a}, vec{b}, vec{c} are three non-copla | vedclass

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(B) Using the vector identity $(\vec{u} 	imes \vec{v}) 	imes \vec{w} = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{v} \cdot \vec{w})\vec{u}$,we expand ea

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

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(B) Using the vector identity $(\vec{u} 	imes \vec{v}) 	imes \vec{w} = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{v} \cdot \vec{w})\vec{u}$,we expand each term:$(\vec{a} 	imes \vec{b}) 	imes (\vec{r} 	imes \vec{c}) = ((\vec{a} 	imes \vec{b}) \cdot \vec{c})\vec{r} - ((\vec{a} 	imes \vec{b}) \cdot \vec{r})\vec{c} = [\vec{a}, \vec{b}, \vec{c}]\vec{r} - [\vec{a}, \vec{b}, \vec{r}]\vec{c}$Similarly:$(\vec{b} 	imes \vec{c}) 	imes (\vec{r} 	imes \vec{a}) = [\vec{b}, \vec{c}, \vec{a}]\vec{r} - [\vec{b}, \vec{c}, \vec{r}]\vec{a} = [\vec{a}, \vec{b}, \vec{c}]\vec{r} - [\vec{b}, \vec{c}, \vec{r}]\vec{a}$$(\vec{c} 	imes \vec{a}) 	imes (\vec{r} 	imes \vec{b}) = [\vec{c}, \vec{a}, \vec{b}]\vec{r} - [\vec{c}, \vec{a}, \vec{r}]\vec{b} = [\vec{a}, \vec{b}, \vec{c}]\vec{r} - [\vec{c}, \vec{a}, \vec{r}]\vec{b}$Summing these,we get:$3[\vec{a}, \vec{b}, \vec{c}]\vec{r} - ([\vec{b}, \vec{c}, \vec{r}]\vec{a} + [\vec{c}, \vec{a}, \vec{r}]\vec{b} + [\vec{a}, \vec{b}, \vec{r}]\vec{c})$Since $\vec{r} = x\vec{a} + y\vec{b} + z\vec{c}$,we know that $[\vec{b}, \vec{c}, \vec{r}]\vec{a} + [\vec{c}, \vec{a}, \vec{r}]\vec{b} + [\vec{a}, \vec{b}, \vec{r}]\vec{c} = [\vec{a}, \vec{b}, \vec{c}]\vec{r}$.Thus,the expression equals $3[\vec{a}, \vec{b}, \vec{c}]\vec{r} - [\vec{a}, \vec{b}, \vec{c}]\vec{r} = 2[\vec{a}, \vec{b}, \vec{c}]\vec{r}$.

If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors and $\vec{r}$ is an arbitrary vector,then $(\vec{a} \times \vec{b}) \times (\vec{r} \times \vec{c}) + (\vec{b} \times \vec{c}) \times (\vec{r} \times \vec{a}) + (\vec{c} \times \vec{a}) \times (\vec{r} \times \vec{b}) = \dots$

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