If \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) are three non-coplanar vectors and \(p, q\) and \(r\) are vectors defined by
\(\mathbf{p}=\frac{\mathbf{a} \times \mathbf{c}}{[\mathbf{a} \mathbf{b} \mathbf{c}]}, \mathbf{q}=\frac{\mathbf{c} \times \mathbf{a}}{[\mathbf{a} \mathbf{b} \mathbf{c}]}, \mathbf{r}=\frac{\mathbf{a} \times \mathbf{b}}{[\mathbf{a} \mathbf{b} \mathbf{c}]}\), then
\((\mathbf{a}+\mathbf{b}) \cdot \mathbf{p}+(\mathbf{b}+\mathbf{c}) \cdot \mathbf{q}+(\mathbf{c}+\mathbf{a}) \cdot \mathbf{r}\) is

\(\mathbf{p}=\frac{\mathbf{b} \times \mathbf{c}}{[\mathbf{a} \mathbf{b} \mathbf{c}]}, \mathbf{q}=\frac{\mathbf{c} \times \mathbf{a}}{[\mathbf{a} \mathbf{b} \mathbf{c}]}, \mathbf{r}=\frac{\mathbf{a} \times \mathbf{b}}{[\mathbf{a} \mathbf{b} \mathbf{c}]}\)
Now, \((\mathbf{a}+\mathbf{b}) \cdot \mathbf{p}+(\mathbf{b}+\mathbf{q}) \cdot \mathbf{q}+(\mathbf{c}+\mathbf{a}) \cdot \mathbf{r}\)
\(\begin{aligned}=\frac{\mathbf{a} \cdot(\mathbf{b} \times \mathbf{a})}{[\mathbf{a} \mathbf{b}]} & +\frac{\mathbf{b} \cdot(\mathbf{b} \times \mathbf{a})}{[\mathbf{a} \mathbf{c}]}+\frac{\mathbf{b} \cdot(\mathbf{c} \times \mathbf{a})}{[\mathbf{a} \mathbf{c}]} \\ & +\frac{\mathbf{c} \cdot(\mathbf{c} \times \mathbf{a})}{[\mathbf{a} \mathbf{b} \mathbf{c}]}+\frac{\mathbf{c} \cdot(\mathbf{a} \times \mathbf{b})}{[\mathbf{a} b \mathbf{b}]}+\frac{\mathbf{a} \cdot(\mathbf{a} \times \mathbf{b})}{[\mathbf{a} \mathbf{b}]}\end{aligned}\)
\(=\frac{[\mathbf{a} \mathbf{b} \mathbf{c}]}{[\mathbf{a} \mathbf{b} \mathbf{c}]}+0+\frac{[\mathbf{b} \mathbf{c} \mathbf{a}]}{[\mathbf{a} \mathbf{b} \mathbf{c}]}+0+\frac{[\mathrm{c} \mathbf{a} \mathbf{b}]}{[\mathbf{a} \mathbf{b} \mathbf{c}]}+0\)
[ \(\because\) scalar triple product is zero if two vectors are same]
\(=\frac{[a b c]}{[\mathbf{a b c} \mathbf{c}]}+\frac{[a \mathbf{b} \mathbf{c}]}{[\mathbf{a b c}]}+\frac{[a \mathbf{b} \mathbf{c}]}{[\mathbf{a b} \mathbf{b}]}=3\)