Question

Find the Cartesian equation of the following planes:
$\overrightarrow{r}\cdot \left[\right(s-2t)\hat{i}+(3-t)\hat{j}+(2s+t\left)\hat{k}\right]=15$.

Answer 1

$\overrightarrow{r}.\left[\right(s-2t)\hat{i}+(3-t)\hat{j}+(2s+t\left)\hat{k}\right]=\mathrm{15.....}\left(1\right)$
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
$(x\hat{i}+y\hat{j}+z\hat{k}).\left[\right(s-2t)\hat{i}+(3-t)\hat{j}+(2s+t\left)\hat{k}\right]=15$
$\Rightarrow$ $(s-2t)x+(3-t)y+(2s+t)z=15$
This is the Cartesian equation of the given plane.