$\bar{a},\bar{b}$ are vectors such that $|\bar{a}+\bar{b}|=\sqrt{29}$ and $\bar{a}\times (2\bar{i}+3\bar{j}+4\bar{k})=(2\bar{i}+3\bar{j}+4\bar{k})\times \bar{b}$ then possible value of  $(\bar{a}+\bar{b})\cdot (-7\bar{i}+2\bar{j}+3\bar{k})=$

Let $\stackrel{\mathrm{‾}}{\mathrm{x}}=2\stackrel{\mathrm{‾}}{\mathrm{i}}+3\stackrel{\mathrm{‾}}{\mathrm{j}}+4\stackrel{\mathrm{‾}}{\mathrm{K}}$
$$\begin{gathered} \bar{a}\times (2\bar{i}+3\bar{j}+4\bar{k})=(2\bar{i}+3\bar{j}+4\bar{k})\times \bar{b} \\ \Rightarrow \bar{a}\times \overline{x}=\overline{x}\times \bar{b} \\ \Rightarrow (\stackrel{\mathrm{‾}}{\mathrm{a}}+\stackrel{\mathrm{‾}}{\mathrm{b}})\times \stackrel{\mathrm{‾}}{\mathrm{x}}=0\mathrm{ } \\ \Rightarrow \stackrel{\mathrm{‾}}{\mathrm{x}}=\mathrm{\lambda }(\stackrel{\mathrm{‾}}{\mathrm{a}}+\stackrel{\mathrm{‾}}{\mathrm{b}}) \end{gathered}$$

$$\begin{gathered} \begin{array}{ll}\left|\stackrel{\mathrm{‾}}{\mathrm{x}}\right|=\mathrm{\lambda }|\stackrel{\mathrm{‾}}{\mathrm{a}}+\stackrel{\mathrm{‾}}{\mathrm{b}}|& \\ \end{array} \\ \Rightarrow \sqrt{29}=\lambda \sqrt{29}\Rightarrow \lambda =1 \\ \Rightarrow \overline{x}= \stackrel{\mathrm{‾}}{\mathrm{a}}+\stackrel{\mathrm{‾}}{\mathrm{b}}=2\stackrel{\mathrm{‾}}{\mathrm{i}}+3\stackrel{\mathrm{‾}}{\mathrm{j}}+4\stackrel{\mathrm{‾}}{\mathrm{k}} \end{gathered}$$

$\begin{array}{ll}now & (\stackrel{\mathrm{‾}}{\mathrm{a}}+\stackrel{\mathrm{‾}}{\mathrm{b}})\cdot (-7\stackrel{\mathrm{‾}}{\mathrm{i}}+2\stackrel{\mathrm{‾}}{\mathrm{j}}+3\stackrel{\mathrm{‾}}{\mathrm{k}})\\ \Rightarrow & -14+6+12=4\end{array}$