Question

For what value of λ are the vectors $\overrightarrow{a}=2\hat{i}+\lambda \hat{j}+\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}-2\hat{j}+3\hat{k}$ perpendicular to each other?

Answer 1

$$\begin{gathered} \text{We have} \\ \overrightarrow{a}=2\hat{i}+\lambda \hat{j}+\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}-2\hat{j}+3\hat{k} \\ \text{Given that}\overrightarrow{a}\text{and}\overrightarrow{b}\text{are perpendicular.} \\ \Rightarrow \overrightarrow{a}.\overrightarrow{b}=0 \\ \Rightarrow \left(2\hat{i}+\lambda \hat{j}+\hat{k}\right).\left(\hat{i}-2\hat{j}+3\hat{k}\right)=0 \\ \Rightarrow 2-2\lambda +3=0 \\ \Rightarrow 5-2\lambda =0 \\ \therefore \lambda =\frac{5}{2} \end{gathered}$$