Let \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) be three non-coplanar vectors and \(\overline{\mathrm{p}}, \overline{\mathrm{q}}, \overline{\mathrm{r}}\) defined by the relations
\(\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}, \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}, \overline{\mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{~b}} \overline{\mathrm{c}}]}\)
then the value of the expression \((\bar{a}+\bar{b}) \cdot \bar{p}+(\bar{b}+\bar{c}) \cdot \bar{q}+(\bar{c}+\bar{a}) \cdot \bar{r}\) is equal to

\(\overline{\mathrm{p}} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{b}}) =\overline{\mathrm{p}} \cdot \overline{\mathrm{a}}+\overline{\mathrm{p}} \cdot \overline{\mathrm{b}} \)
\( =\frac{(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \cdot \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}+\frac{(\overline{\mathrm{b}} \times \overline{\mathrm{c}}) \cdot \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]} \)
\( =\frac{[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{a}}]}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}+\frac{[\overline{\mathrm{b}} \overline{\mathrm{c}} \overline{\mathrm{b}}]}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]} \)
\( =1+0=1\)
Similarly, \(\overline{\mathrm{q}} \cdot(\overline{\mathrm{b}}+\overline{\mathrm{c}})=1\) and \(\overline{\mathrm{r}} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{c}})=1\)
\(\begin{aligned}
& (\overline{\mathrm{a}}+\overline{\mathrm{b}}) \cdot \overline{\mathrm{p}}+(\stackrel{\rightharpoonup}{\mathrm{b}}+\overline{\mathrm{c}}) \cdot \overline{\mathrm{q}}+(\overline{\mathrm{c}}+\overline{\mathrm{a}}) \cdot \overline{\mathrm{r}} \\
& =1+1+1 \\
& =3
\end{aligned}\)