Question

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

Answer 1

Hint: As equation of plane is $ \overrightarrow{r}.\hat{n}=d $ in which $ \overrightarrow{r} $ is position vector on place, $ \hat{n} $ is unit vector of normal and d is the distance. So we will simply put these values to find the equation of the plane.

Complete step by step answer:
Moving ahead with the question in step-to-step manner;
Distance of plane from origin $ =8 $
Normal vector $ =2\hat{i}+\hat{j}+2\hat{k} $
So unit vector along normal, we know that unit vector of any vector is that vector upon vector mode i.e. unit vector $ =\dfrac{\overrightarrow{a}}{|a|} $ , so unit vector of normal vector is equal to;
 $ \hat{n}=\dfrac{\overrightarrow{n}}{|n|} $ . So by putting the value we will get;
 $ \begin{align}
  & \hat{n}=\dfrac{\overrightarrow{n}}{|n|} \\
 & \hat{n}=\dfrac{2\hat{i}+\hat{j}+2\hat{k}}{\sqrt{{{2}^{2}}+{{1}^{2}}+{{2}^{2}}}} \\
 & \hat{n}=\dfrac{2\hat{i}+\hat{j}+2\hat{k}}{\sqrt{9}} \\
 & \hat{n}=\dfrac{2\hat{i}+\hat{j}+2\hat{k}}{3} \\
\end{align} $
And $ \overrightarrow{r} $ is position vector of any point on the place i.e. $ x\hat{i}+y\hat{j}+z\hat{k} $
So as we know, the equation of the plane when the normal vector and distance of the plane from the origin is given is $ \overrightarrow{r}.\hat{n}=d $ . So to find the equation put the value, from where we will get;
 $ \begin{align}
  & \overrightarrow{r}.\hat{n}=d \\
 & \overrightarrow{r}.\left( \dfrac{2\hat{i}+\hat{j}+2\hat{k}}{3} \right)=8 \\
\end{align} $
On further simplifying it, we will get;
 $ \overrightarrow{r}.\left( 2\hat{i}+\hat{j}+2\hat{k} \right)=24 $
So the equation of the plane is $ \overrightarrow{r}.\left( 2\hat{i}+\hat{j}+2\hat{k} \right)=24 $ .
Hence the answer is $ \overrightarrow{r}.\left( 2\hat{i}+\hat{j}+2\hat{k} \right)=24 $ .

Note: There are many other forms of equation of plane rather than being $ \overrightarrow{r}.\hat{n}=d $ , based on the given condition in questions. Moreover we can write $ \overrightarrow{r}.\left( 2\hat{i}+\hat{j}+2\hat{k} \right)=24 $ equation as $ \left( x\hat{i}+y\hat{j}+z\hat{k} \right).\left( 2\hat{i}+\hat{j}+2\hat{k} \right)=24 $ both are correct.