Question

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

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

Answer 1

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

$\hat{i}\times (\overrightarrow{r}\times \hat{i})+\hat{j}\times (\overrightarrow{r}\times \hat{j})+\hat{k}\times (\overrightarrow{r}\times \hat{k})$

$\begin{array}{cc}=(i\cdot \hat{i})\overrightarrow{r}-(i\cdot \overrightarrow{r})\hat{i}+(j\cdot \hat{j})\overrightarrow{r}-(\hat{j}\cdot \overrightarrow{r})\hat{j}& \\ & +(\hat{k}\cdot \hat{k})\overrightarrow{r}-(\hat{k}\cdot \overrightarrow{r})\hat{r}\end{array}$

$=\overrightarrow{r}+\overrightarrow{r}+\overrightarrow{r}-\left[\right(\hat{i}\cdot \overrightarrow{r})\hat{i}+(\hat{j}\cdot \overrightarrow{r})\hat{j}+(\hat{k}\cdot \overrightarrow{r}\left)\hat{k}\right]$

$\begin{array}{cc}x& =\left[\right(\hat{i}\cdot \overrightarrow{r})\hat{i}+(\hat{ȷ}\cdot \overrightarrow{r})\hat{j}+(\hat{k}\cdot \overrightarrow{r}\left)\hat{k}\right]\\ & \text{let}\overrightarrow{r}=x\hat{i}+yj+z\hat{k}\end{array}$

$\begin{array}{cc}X=\left[\right(\hat{i}\cdot (x\hat{i}+y\hat{j}+z\hat{k})\hat{i}+\hat{j}\cdot (x\hat{i}+y\hat{j}+z\hat{k})\hat{j}& \\ & +\hat{k}\cdot (x\hat{i}+y\hat{j}+z\hat{k})\hat{k}]\end{array}$

$=[x\hat{i}+0+0+y\hat{j}+0+0+2\hat{k}+0+0]$

$=[x\hat{i}+y\hat{j}+z\hat{k}]$

$=\overrightarrow{r}$

$\begin{array}{cc}\therefore \text{Answer}& =3\overrightarrow{r}-\overrightarrow{r}\\ & =2\overrightarrow{r}\\ \text{option}B\text{is correct}& \end{array}$