Question

If $\left|\overrightarrow{a}\right|=1$, the projection of $\overrightarrow{r}$ along $\overrightarrow{a}$ is $2$ and $\overrightarrow{a}\times \overrightarrow{r}+\overrightarrow{b}=\overrightarrow{r}$, then $\overrightarrow{r}=$

• A. $\frac{1}{2}[\overrightarrow{a}-\overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{b}]$
• B. $\frac{1}{2}[2\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{b}]$
• C. $\overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{b}$
• D. $\overrightarrow{a}-\overrightarrow{a}\times \overrightarrow{b}$

Answer 1

Answer: (B)

$\frac{1}{2}[2\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{b}]$

Projection of $\overrightarrow{r}$ on $\overrightarrow{a}$ is $2$
$\Rightarrow \overrightarrow{r}.\overrightarrow{a}=2$ -----$\left(1\right)$
$\overrightarrow{a}\times \overrightarrow{r}+\overrightarrow{b}=\overrightarrow{r}$ ------$\left(2\right)$
Dot $\left(1\right)$ with $\overrightarrow{a}$
$\Rightarrow \overrightarrow{a}.\overrightarrow{b}=\overrightarrow{r}.\overrightarrow{a}=2$ by $\left(1\right)$
cross $\left(1\right)$ with $\overrightarrow{a}$
$\Rightarrow \overrightarrow{a}\times (\overrightarrow{a}\times \overrightarrow{r})+\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{r}$
$\Rightarrow 0+\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{a}\times \overrightarrow{r}$
$\therefore 2\overrightarrow{r}-\overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{b}=2\overrightarrow{r}$
$\Rightarrow \overrightarrow{r}=\frac{1}{2}[2\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{b}]$