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(D) Given that $\overrightarrow{p}=\frac{\overrightarrow{b} 	imes \overrightarrow{c}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}, \o

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$3$

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(D) Given that $\overrightarrow{p}=\frac{\overrightarrow{b} 	imes \overrightarrow{c}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}, \overrightarrow{q}=\frac{\overrightarrow{c} 	imes \overrightarrow{a}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}, \overrightarrow{r}=\frac{\overrightarrow{a} 	imes \overrightarrow{b}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}$.We need to calculate $\overrightarrow{a} \cdot \overrightarrow{p}+\overrightarrow{b} \cdot \overrightarrow{q}+\overrightarrow{c} \cdot \overrightarrow{r}$.Substituting the values,we get:$\overrightarrow{a} \cdot \overrightarrow{p} = \overrightarrow{a} \cdot \frac{\overrightarrow{b} 	imes \overrightarrow{c}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]} = \frac{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]} = 1$.$\overrightarrow{b} \cdot \overrightarrow{q} = \overrightarrow{b} \cdot \frac{\overrightarrow{c} 	imes \overrightarrow{a}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]} = \frac{[\overrightarrow{b} \overrightarrow{c} \overrightarrow{a}]}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]} = \frac{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]} = 1$.$\overrightarrow{c} \cdot \overrightarrow{r} = \overrightarrow{c} \cdot \frac{\overrightarrow{a} 	imes \overrightarrow{b}}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]} = \frac{[\overrightarrow{c} \overrightarrow{a} \overrightarrow{b}]}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]} = \frac{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]}{[\overrightarrow{a} \overrightarrow{b} \overrightarrow{c}]} = 1$.Therefore,$\overrightarrow{a} \cdot \overrightarrow{p}+\overrightarrow{b} \cdot \overrightarrow{q}+\overrightarrow{c} \cdot \overrightarrow{r} = 1+1+1 = 3$.

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