Question

For any vector $\overline{r}$,
$\overline{i}\times (\overline{r}\times \overline{i})+\overline{j}\times (\overline{r}\times \overline{j})+\overline{k}\times (\overline{r}\times \overline{k})=$

• A. $\overline{0}$
• B. $2\overline{r}$
• C. $3\overline{r}$
• D. $4\overline{r}$

Answer 1

Answer: (B)

$2\overline{r}$

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

$=(\overline{i}.\overline{i})\overline{r}-(\overline{i}.\overline{r})\overline{i}+(\overline{j}.\overline{j})\overline{r}-(\overline{j}.\overline{r})\overline{j}+(\overline{k}.\overline{k})\overline{r}-(\overline{k}.\overline{r})\overline{k}$

$=3\overline{r}-\left[\right(\overline{i}.\overline{r})\overline{i}+(\overline{j}.\overline{r})\overline{j}+(\overline{k}.\overline{r}\left)\overline{k}\right]$

$=2\overline{r}$