Question

Let $\overline{r}$ be a unit vector satisfying $\overline{r}\times \overline{a}=\overline{b},\left|\overline{a}\right|=\sqrt{3},$ and $\left|\overline{b}\right|=\sqrt{2},$ then $\overline{r}$ is

• A. $\frac{2}{3}(\overline{a}+(\overline{a}\times \overline{b}\left)\right)$
• B. $\frac{1}{3}(\overline{a}+(\overline{a}\times \overline{b}\left)\right)$
• C. $\frac{2}{3}(\overline{a}-(\overline{a}\times \overline{b}\left)\right)$
• D. $\frac{1}{3}(-\overline{a}+(\overline{a}\times \overline{b}\left)\right)$

Answer 1

Answer: (B), (D)

The correct options are
B $\frac{1}{3}(\overline{a}+(\overline{a}\times \overline{b}\left)\right)$
D $\frac{1}{3}(-\overline{a}+(\overline{a}\times \overline{b}\left)\right)$

$\overline{a}\times (\overline{r}\times \overline{a})=\overline{a}\times \overline{b}$
$(\overline{a}.\overline{a})\overline{r}-(\overline{a}.\overline{r})\overline{a}=\overline{a}\times \overline{b}$
$3\overline{r}=(\overline{a}.\overline{r})\overline{a}+\overline{a}\times \overline{b}$ ... $\left(1\right)$
Now $|\overline{r}\times \overline{a}{|}^2=2$
$|\overline{r}{|}^2|\overline{a}{|}^2{\sin }^2\theta =2$
$|\overline{r}{|}^2|\overline{a}{|}^2(1-{\cos }^2\theta )=2$
$|\overline{r}{|}^2|\overline{a}{|}^2-(\overline{r}.\overline{a}{)}^2=2$
$1.3-(\overline{r}.\overline{a}{)}^2=2$
$(\overline{r}.\overline{a}{)}^2=1$

$\Rightarrow \overline{r}.\overline{a}=\pm 1$
$\therefore$ from $\left(1\right)$
$\overline{r}=\frac{1}{3}\{\overline{a}+(\overline{a}\times \overline{b}\left)\right\}$ or $\frac{1}{3}\{-\overline{a}+(\overline{a}\times \overline{b}\left)\right\}$