Question

Let $\overrightarrow{a}=5\hat{i}-\hat{j}+7\hat{k}and\overrightarrow{b}=\hat{i}-\hat{j}+\lambda \hat{k}.$ Find λ such that $\overrightarrow{a}+\overrightarrow{b}$is orthogonal to $\overrightarrow{a}-\overrightarrow{b}$.

Answer 1

$$\begin{gathered} \text{Given that} \\ \overrightarrow{a}=5\hat{i}-\hat{j}+7\hat{k};\overrightarrow{b}=\hat{i}-\hat{j}+\lambda \hat{k} \\ \therefore \overrightarrow{a}+\overrightarrow{b}=5\hat{i}-\hat{j}+7\hat{k}+\hat{i}-\hat{j}+\lambda \hat{k}=6\hat{i}-2\hat{j}+\left(7+\lambda \right)\hat{k} \\ \mathrm{and}\overrightarrow{a}-\overrightarrow{b}=5\hat{i}-\hat{j}+7\hat{k}-\left(\hat{i}-\hat{j}+\lambda \hat{k}\right)=4\hat{i}+0\hat{j}+\left(7-\lambda \right)\hat{k} \\ \text{Given that}\overrightarrow{a}+\overrightarrow{b}\text{is orthogonal to}\overrightarrow{a}\text{-}\overrightarrow{b}. \\ \Rightarrow \left(\overrightarrow{a}+\overrightarrow{b}\right).\left(\overrightarrow{a}-\overrightarrow{b}\right)=0 \\ \Rightarrow \left[6\hat{i}-2\hat{j}+\left(7+\lambda \right)\hat{k}\right].\left[4\hat{i}+0\hat{j}+\left(7-\lambda \right)\hat{k}\right]=0 \\ \Rightarrow 24+0+49-{\lambda }^2=0 \\ \Rightarrow {\lambda }^2=73 \\ \Rightarrow \lambda =\sqrt{73} \end{gathered}$$