Question

Write a unit vector perpendicular to $\hat{i}+\hat{j}\mathrm{and}\hat{j}+\hat{k}.$

Answer 1

$$\begin{gathered} \text{Let}\overrightarrow{a}=\hat{i}+\hat{j}+0\hat{k};\overrightarrow{b}=0\hat{i}+\hat{j}+\hat{k} \\ \overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 1& 0\\ 0& 1& 1\end{array}\right| \\ =\hat{i}-\hat{j}+\hat{k} \\ \Rightarrow \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\sqrt{1+1+1} \\ =\sqrt{3} \\ \text{Unit vector perpendicular to}\overrightarrow{a}\text{and}\overrightarrow{b}\text{is,}\frac{\overrightarrow{a}\times \overrightarrow{b}}{\left|\overrightarrow{a}\times \overrightarrow{b}\right|}=\frac{1}{\sqrt{3}}\left(\hat{i}-\hat{j}+\hat{k}\right) \end{gathered}$$