Question

If $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k},$ then write the value of ${\left|\overrightarrow{r}\times \hat{i}\right|}^2.$

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k} \\ \mathrm{Now}, \\ \overrightarrow{i}=\hat{i}+0\hat{j}+0\hat{k} \\ \overrightarrow{r}\times \overrightarrow{i}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ x& y& z\\ 1& 0& 0\end{array}\right| \\ =0\hat{i}+z\hat{j}-y\hat{k} \\ \Rightarrow \left|\overrightarrow{r}\times \overrightarrow{i}\right|=\sqrt{z^2+y^2} \\ \Rightarrow {\left|\overrightarrow{r}\times \overrightarrow{i}\right|}^2=z^2+y^2 \end{gathered}$$