Answer:
Given, \[\overrightarrow{a}=3\hat{i}+2\hat{j}+9\hat{k}\] and \[\overrightarrow{b}=\hat{i}+\lambda \hat{j}+3\hat{k}\] Since, vectors \[\overrightarrow{a}\] and \[\overrightarrow{b}\] are perpendicular to each other. \[\therefore \] \[\vec{a}\cdot \,\,\vec{b}=0\] \[\Rightarrow \] \[(3\hat{i}+2\hat{j}+9\hat{k}).(\hat{i}+\lambda \hat{j}+3\hat{k})=0\] \[\Rightarrow \] \[3\hat{i}+2\lambda +27=0\] \[\Rightarrow \] \[2\lambda =-\,\,30\] \[\Rightarrow \] \[\lambda =-\,\frac{30}{2}=-\,15\] Hence, the required value of \[\lambda \,\,\text{is}\,\,\,-\,15.\]
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