Question

Find the value of λ if the vectors $2\hat{i}+\lambda \hat{j}+3\hat{k}\mathrm{and}3\hat{i}+2\hat{j}-4\hat{k}$ are perpendicular to each other.

Answer 1

$$\begin{gathered} \mathrm{Given}:2\hat{i}+\lambda \hat{j}+3\hat{k}\mathrm{and}3i+2j-4k\mathrm{are}\mathrm{perpendicular}\mathrm{to}\mathrm{each}\mathrm{other}. \\ \text{So, their dot product is zero.} \\ \left(2\hat{i}+\lambda \hat{j}+3\hat{k}\right).\left(3i+2j-4k\right) \\ \Rightarrow 6+2\lambda -12=0 \\ \Rightarrow 2\lambda -6=0 \\ \Rightarrow \lambda =3 \end{gathered}$$