Question

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

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have} \\ \overrightarrow{a}=2\hat{i}+\lambda \hat{j}+\hat{k}\mathrm{and}\overrightarrow{b}=\hat{i}-2\hat{j}+3\hat{k} \\ \text{The given vectors are perpendicular. So, their dot product is zero.} \\ \left(2\hat{i}+\lambda \hat{j}+\hat{k}\right).\left(\hat{i}-2\hat{j}+3\hat{k}\right) \\ \Rightarrow 2-2\lambda +3=0 \\ \Rightarrow 5-2\lambda =0 \\ \Rightarrow -2\lambda =-5 \\ \Rightarrow \lambda =\frac{5}{2} \end{gathered}$$