Let p =2 i + j +3 k and q = i - j + k . If for some real numbers α,...

15 \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+6 \hat{\mathrm{k}}=\alpha(2 \overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}})+\beta(\overrightarrow{\mathrm{p}}-2 \overrightarrow{\mathrm{q}})+\gamma(\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}})$taking dot with$(\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}}) \left|$\begin{array}{ccc}15 & 10 & 6 \\ 2 & 1 & 3 \\ 1 & -1 & 1\end{array}$\right|=0+0+\gamma \left(\mathrm{p}^2\mathrm{q}^2-$($\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{q}})^2\right)$ \quad \left[\because(\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{q}})^2+(\overrightarrow{\mathrm{p}} \cdot \overrightarrow{\mathrm{q}})^2=\mathrm{p}^2\mathrm{q}^2\right]\begin{aligned} & \Rightarrow 52=26 \gamma \\ & \therefore \gamma=2\end{aligned}$$$