If \(\bar{a}, \bar{b}, \bar{c}\) are non-coplanar vectors and \(\bar{p}=\frac{\bar{b} \times \bar{c}}{[\bar{a} \bar{b} \bar{c}]}, \bar{q}=\frac{\bar{c} \times \bar{a}}{[a} \bar{b} \overline{c]}, \bar{r}=\frac{\bar{a} \times \bar{b}}{[\bar{a} \bar{b} \bar{c}]}\),
then \(\bar{a} \cdot \bar{p}+\bar{b} \cdot \bar{q}+\bar{c} \cdot \bar{r}=\)

(C)
Given that,
\(\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{\left[\begin{array}{ccc}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]}, \quad \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{\left[\begin{array}{ccc}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right], \quad \mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{\left[\begin{array}{lll}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]}\)
\(\bar{a} \cdot \bar{p}=\frac{\bar{a} \cdot(\bar{b} \times \bar{c})}{\left[\begin{array}{lll}\bar{a} & \bar{b} & \bar{c}\end{array}\right]}=1\)...(1)
Similarly, \(\overline{\mathrm{b}} \cdot \overline{\mathrm{q}}=\frac{\mathrm{b} \cdot(\overline{\mathrm{c}} \times \overline{\mathrm{a}})}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}=1\)...(2) and \(\overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=\frac{(\overline{\mathrm{a}} \times \overline{\mathrm{b}})}{\left[\begin{array}{ccc}\overline{\mathrm{a}} & \overline{\mathrm{b}} & \overline{\mathrm{c}}\end{array}\right]}=1\)..(3)
\(\therefore \overline{\mathrm{a}} \cdot \overline{\mathrm{p}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{q}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=1+1+1=3 \quad \ldots[\) from \((1),(2) \&(3)]\)