Question

Find the Cartesian equation of the following planes:
$\overrightarrow{r}\cdot (\hat{i}+\hat{j}-\hat{k})=2$.

Answer 1

It is given that equation of the plane is
$\overrightarrow{r}.(\hat{i}+\hat{j}-\hat{k})=\mathrm{2.....}\left(1\right)$
For any arbitrary point $P(x,y,z)$ on the plane, position vector $\overrightarrow{r}$ is given by,
$\overrightarrow{r}=x\hat{i}+y\hat{j}-z\hat{k}$
Substituting the value of $\overrightarrow{r}$ in equation (1), we obtain
$(x\hat{i}+y\hat{j}-z\hat{k}).(x\hat{i}+y\hat{j}-z\hat{k})=2$
$\Rightarrow$ $x+y-z=2$
This is the Cartesian equation of the plane.