Question

Let $\overrightarrow{p},\overrightarrow{q},\overrightarrow{r}$ be three unit vectors such that $\overrightarrow{p}\times \overrightarrow{q}=\overrightarrow{r}$. If $\overrightarrow{a}$ is any vector such that $\left[\overrightarrow{a}\overrightarrow{q}\overrightarrow{r}\right]=1,\left[\overrightarrow{a}\overrightarrow{r}\overrightarrow{p}\right]=2$ and $\left[\overrightarrow{a}\overrightarrow{p}\overrightarrow{q}\right]=3$ then $\overrightarrow{a}$ is

• A. $\overrightarrow{p}+3\overrightarrow{q}+\overrightarrow{r}$
• B. $3\overrightarrow{p}+2\overrightarrow{q}+\overrightarrow{r}$
• C. $\overrightarrow{p}-2\overrightarrow{q}-3\overrightarrow{r}$
• D. $\overrightarrow{p}+2\overrightarrow{q}+3\overrightarrow{r}$

Answer 1

The correct option is D $\overrightarrow{p}+2\overrightarrow{q}+3\overrightarrow{r}$

$(\overrightarrow{p}\times \overrightarrow{q}).\overrightarrow{r}=|\overrightarrow{r}{|}^2\Rightarrow \left[\overrightarrow{p}\overrightarrow{q}\overrightarrow{r}\right]=1$
Let $\overrightarrow{a}=x\overrightarrow{p}+y\overrightarrow{q}+z\overrightarrow{r}$
Take dot product with $\overrightarrow{p}\times \overrightarrow{q},\overrightarrow{q}\times \overrightarrow{r}$ and $\overrightarrow{r}\times \overrightarrow{p}$
$\left[\overrightarrow{a}\overrightarrow{p}\overrightarrow{q}\right]=1=x$
Similarly $y=2,z=3$
$\Rightarrow \overrightarrow{a}=\overrightarrow{p}+2\overrightarrow{q}+3\overrightarrow{r}$