Question

Find the vector equation of plane which is at a distance of $8$ units from the origin and which is normal to the vector $2i+j+2k$.

• A. $\overrightarrow{r}.(2i+j+2k)=24$
• B. $\overrightarrow{r}.(2i+k+2k)=12$
• C. $\overrightarrow{r}.(2i+k+2k)=16$
• D. None of these

Answer 1

The correct option is A $\overrightarrow{r}.(2i+j+2k)=24$

Here, $d=8$ and $\overrightarrow{n}=2i+j+2k$

$\therefore \hat{n}=\frac{\overrightarrow{n}}{\left|\overrightarrow{n}\right|}=\frac{2i+j+2k}{\sqrt{2^2+1^2+2^2}}=\frac{2i+j+2k}{3}$

Hence, the required equation of plane is $\overrightarrow{r}.\hat{n}=d$

$\Rightarrow \overrightarrow{r}\left(\frac{2i+j+2k}{3}\right)=8$

$\Rightarrow \overrightarrow{r}(2i+j+2k)=24$