Question

If $\overrightarrow{r}\cdot \hat{i}=2\overrightarrow{r}\cdot \hat{j}=4\overrightarrow{r}\cdot \hat{k}$ and $|\overrightarrow{r}|=\sqrt{84}$ , then the value of $\overrightarrow{r}\cdot \left(2\hat{i}-3\hat{j}+\hat{k}\right)$ may be

• A. $0$
• B. $2$
• C. $4$
• D. $6$

Answer 1

Answer: (D)

$\overrightarrow{r}\cdot \hat{i}=2\overrightarrow{r}\cdot \hat{j}=4\overrightarrow{r}\cdot \hat{k}$
Let $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\overrightarrow{k}$
$x=2y=4z$
$|\overrightarrow{r}|=\sqrt{84}\Rightarrow \sqrt{{\left(4z\right)}^2+{\left(2z\right)}^2+z^2}=\sqrt{84}$
$z=2;y=4;x=8$
$\overrightarrow{r}=8\hat{i}+4\hat{j}+2\hat{k}$
$\overrightarrow{r}\cdot \left(2\hat{i}-3\hat{j}+\hat{k}\right)=16-12+2=6$