Question

$(\overline{r}\cdot \hat{i})(\overline{r}\times \hat{i})+(\overline{r}\cdot \hat{j})(\overline{r}\times \hat{j})+(\overline{r}\cdot \hat{k})(\overline{r}\times \hat{k})$ is equal to

• A. $2\overline{r}$
• B. $\overline{r}$
• C. $4\overline{r}$
• D. $\overline{0}$

Answer 1

Answer: (D)

$\overline{0}$

Let $\overline{r}=a\hat{i}+b\hat{j}+c\hat{k}$
$\therefore \overrightarrow{r}\cdot \hat{i}=a,\overrightarrow{r}\times \hat{j}=b\hat{k}-c\hat{i}$
$\therefore (\overline{r}\cdot \hat{i})(\overline{r}\times \hat{i})=ab\hat{k}+ac\hat{j}$
Similarly $(\overrightarrow{r}\cdot \hat{j})(\overline{r}\times \hat{j})=ab\hat{k}-bc\hat{i}$
$(\overrightarrow{r}\cdot \hat{k})(\overline{r}\times \hat{k})=-ac\hat{j}+bc\hat{i}$
$\therefore (\overline{r}.\hat{i})(\overline{r}\times \hat{i})+(\overline{r}.\hat{j})(\overline{r}\times \hat{j})+(\overline{r}.\hat{k})(\overline{r}\times \hat{k})=\overline{0}$