Question

The vector equation of a plane which is at a distance of 9 units from the origin and which is normal to the
vector $2\hat{i}-\hat{j}+2\hat{k}$ is

• A. $\overrightarrow{r}.(2\hat{i}-\hat{j}+2\hat{k})=27$
• B. $\overrightarrow{r.}(2\hat{i}-\hat{j}+2\hat{k})=9$
• C. $\overrightarrow{r}.(2\hat{i}-\hat{j}+2\hat{k})=3$

Answer 1

The correct option is A $\overrightarrow{r}.(2\hat{i}-\hat{j}+2\hat{k})=27$

The vector equation of a plane is given in the form $\overrightarrow{r}.\hat{n}=d$
Here d = 9 and

$\hat{n}=\frac{2\hat{i}-\hat{j}+2\hat{k}}{\sqrt{2^2+{(-1)}^2+2^2}}=\frac{2}{3}\hat{i}-\frac{1}{3}\hat{j}+\frac{2}{3}\hat{k}$
So,
$\overrightarrow{r.}\left(\frac{2}{3}\hat{i}-\frac{1}{3}\hat{j}+\frac{2}{3}\hat{k}\right)=9$
$\overrightarrow{r}.(2\hat{i}-\hat{j}+2\hat{k})=27.$