Question

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

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

Answer 1

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

Let $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$
Then
$\hat{i}\times (\overrightarrow{r}\times \hat{i})+\hat{j}\times (\overrightarrow{r}\times \hat{j})+\hat{k}\times (\overrightarrow{r}\times \hat{k})=(\hat{i}.\hat{i})\overrightarrow{r}-(\hat{i}.\overrightarrow{r})\hat{i}+(\hat{j}.\hat{j})\overrightarrow{r}-\left(\hat{j}\overrightarrow{r}\right)\hat{j}+(\hat{k}.\hat{k})\overrightarrow{r}-\left(\hat{k}\overrightarrow{r}\right)\hat{k}=3\overrightarrow{r}-\hat{i}(\hat{i}.(x\hat{i}+y\hat{j}+z\hat{k}))-\hat{j}(\hat{j}.(x\hat{i}+y\hat{j}+z\hat{k}))-\hat{k}(\hat{k}.(x\hat{i}+y\hat{j}+z\hat{k}))=3\overrightarrow{r}-(x\hat{i}+y\hat{j}+z\hat{k})=3\overrightarrow{r}-\overrightarrow{r}=2\overrightarrow{r}$