Question

The value of $\hat{i}\times (\overline{r}\times \hat{i})+\hat{j}\times (\overline{r}\times \hat{j})+\hat{k}\times (\overline{r}\times \hat{k})$ is

• A. $\overline{r}$
• B. $2\overline{r}$
• C. $3\overline{r}$
• D. $4\overline{r}$

Answer 1

The correct option is B $2\overline{r}$

$\begin{array}{c}\overrightarrow{i}\times \left(\overrightarrow{r}\times \overrightarrow{i}\right)+\overrightarrow{j}\times \left(\overrightarrow{r}\times \overrightarrow{j}\right)+\overrightarrow{k}\times \left(\overrightarrow{r}\times \overrightarrow{k}\right)\\ =\left(\overrightarrow{i}.\overrightarrow{i}\right)\overrightarrow{r}-\left(\overrightarrow{i}.\overrightarrow{r}\right)\overrightarrow{i}+\left(\overrightarrow{i}.\overrightarrow{j}\right)\overrightarrow{r}-\left(\overrightarrow{j}.\overrightarrow{r}\right)\hat{j}+\left(\hat{k}.\hat{k}\right)\overrightarrow{r}-\left(\hat{k}.\overrightarrow{r}\right)\hat{k}\\ =3\overrightarrow{r}-x\overrightarrow{i}-y\overrightarrow{j}-z\overrightarrow{k}\\ =z\overrightarrow{r}\\ \overrightarrow{r}=x\overrightarrow{i}+y\overrightarrow{j}+z\overrightarrow{k}\\ Hence,\\ optionBiscorrectanswer.\end{array}$