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

Let \(\mathrm{T}_{1}=(\mathbf{a}+\mathbf{b}) \cdot \mathbf{p}\)
\(=\mathbf{a} \cdot \mathbf{p}+\mathbf{b} \cdot \mathbf{p}\)
\(=\mathbf{a} \cdot \frac{\mathbf{b} \times \mathbf{c}}{[\mathbf{a} \mathbf{b} \mathbf{c}]}+\frac{\mathbf{b} \cdot(\mathbf{b} \times \mathbf{c})}{[\mathbf{a} \mathbf{b} \mathbf{c}]}\)
\(=\frac{[\mathbf{a} \mathbf{b} \mathbf{c}]}{[\mathbf{a} \mathbf{b} \mathbf{c}]}+\frac{[\mathbf{b} \mathbf{b} \mathbf{c}]}{[\mathbf{a} \mathbf{b} \mathbf{c}]}=1+0=1\)
Similarly, \(\mathrm{T}_{2}=(\mathbf{b}+\mathbf{c}) \cdot \mathbf{q}=1\)
and \(\quad \mathrm{T}_{3}=(\mathbf{c}+\mathbf{a}) \cdot \mathbf{r}=1\)
\(\therefore \quad \mathrm{T}_{1}+\mathrm{T}_{2}+\mathrm{T}_{3}=3\)