Question

If $\overline{r}$ satisfies the equation $\overline{r}\times (\hat{i}+2\hat{j}+\hat{k})=\hat{i}-\hat{k},$ then for any scalar $t$, $\overline{r}$ is equal to

• A. $\hat{i}+t(\hat{i}+2\hat{j}+\hat{k})$
• B. $\hat{j}+t(\hat{i}+2\hat{j}+\hat{k})$
• C. $\hat{k}+t(\hat{i}+2\hat{j}+\hat{k})$
• D. None of these

Answer 1

Answer: (B)

$\hat{j}+t(\hat{i}+2\hat{j}+\hat{k})$

Given $\overline{r}$ satisfies the equation $\overline{r}\times (\hat{i}+2\hat{j}+\hat{k})=\hat{i}-\hat{k},$
let $\overline{r}=a\hat{i}+b\hat{j}+\hat{k}$
$\Rightarrow \overline{r}\times (\hat{i}+2\hat{j}+\hat{k})=(b-2c)\hat{i}-(a-c)\hat{j}+(2a-b)\hat{k}=\hat{i}-\hat{k}$
$\Rightarrow b-2c=1;a=c$ and $2a-b=-1$
let $a=t$
$\Rightarrow \overline{r}=t\hat{i}+(1+2t)\hat{j}+t\hat{k}$
$\therefore \overline{r}=\hat{j}+t(\hat{i}+2\hat{j}+\hat{k})$
Hence, option B.