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If $\bar{a}$ and $\bar{b}$ are vectors such that $|\bar{a}+\bar{b}|=\sqrt{29}$ and $\bar{a} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \bar{b}$, then a possible value of $(\bar{a}+\bar{b}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})$ is
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4
$\begin{aligned} & \vec{a} \times(2 \hat{i}+3 \hat{j}+4 \widehat{k})=(2 \hat{i}+3 \hat{j}+4 \widehat{k}) \times \vec{b}=-\vec{b} \times(2 \hat{i}+3 \hat{j}+4 \widehat{k}) \\ & \Rightarrow(\vec{a}+\vec{b}) \times(2 \hat{i}+3 \hat{j}+4 \widehat{k})=\overrightarrow{0} \\ & \Rightarrow(\vec{a}+\vec{b}) \| 2 \widehat{i}+3 \hat{j}+4 \widehat{k} \\ & \Rightarrow \vec{a}+\vec{b}=\lambda(2 \hat{i}+3 \hat{j}+4 \widehat{k}) \\ & \Rightarrow|\vec{a}+\vec{b}|=\lambda \sqrt{2^2+3^2+4^2} \\ & \Rightarrow \sqrt{29}= \pm \lambda \sqrt{29} \\ & \Rightarrow \lambda= \pm 1 \\ & \Rightarrow \vec{a}+\vec{b}= \pm(2 \hat{i}+3 \hat{j}+4 \widehat{k}) \\ & \Rightarrow(\vec{a}+\vec{b}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})= \pm(2 \hat{i}+3 \hat{j}+4 \widehat{k}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k}) \\ & = \pm(-14+6+12) \\ & = \pm 4\end{aligned}$
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