If $(\mathbf{p} \times \mathbf{q}) \times \mathbf{r} = u\mathbf{p} + v\mathbf{q} + w\mathbf{r}$…

The key is recognizing that the vector triple product $(\mathbf{p} \times \mathbf{q}) \times \mathbf{r}$ can be rewritten using the BAC-CAB rule: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b})$. Once you expand the left side into a linear combination of $\mathbf{p}$, $\mathbf{q}$, and $\mathbf{r}$, you can match coefficients with the right side to find $u$, $v$, and $w$—then sum them.

The key is recognizing that the vector triple product $(\mathbf{p} \times \mathbf{q}) \times \mathbf{r}$ can be rewritten using the BAC-CAB rule: $\mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = \mathbf{b}(\mathbf{a} \cdot \mathbf{c}) - \mathbf{c}(\mathbf{a} \cdot \mathbf{b})$. Once you expand the left side into a linear combination of $\mathbf{p}$, $\mathbf{q}$, and $\mathbf{r}$, you can match coefficients with the right side to find $u$, $v$, and $w$—then sum them.