question_answer

Find the value of \[\lambda \,,\] so that the vectors \[\overrightarrow{a}=3\hat{i}+2\hat{j}+9\hat{k}\] and \[\overrightarrow{b}=\hat{i}+\lambda \hat{j}+3\hat{k}\] are  perpendicular to each other.

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 \[\lambda \,\,\text{is}\,\,\,-\,15.\]