Question

Find the Cartesian equation of the following planes:
$\overrightarrow{r}\cdot (2\hat{i}+3\hat{j}-4\hat{k})=1$.

Answer 1

Given, $\overrightarrow{r}\cdot (2\hat{i}+3\hat{j}-4\hat{k})=\mathrm{1........}\left(1\right)$

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

$(x\hat{i}+y\hat{j}+z\hat{k}).(2\hat{i}+3\hat{j}-4\hat{k})=1\Rightarrow 2x+3y-4z=1$

This is the Cartesian equation of the plane.