Question

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

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

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

Answer 1

(1) Vector equation of the plane is :
$\overrightarrow{r}.(\hat{i}+\hat{j}-\hat{k})=2$

Substituting $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$, we get

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

$\Rightarrow x+y-z=2$

Hence, the Cartesian equation of the plane is
$x+y-z=2$

(2) Vector equation of the plane is :
$\overrightarrow{r}.(2\hat{i}+3\hat{j}-4\hat{k})=1$

Substituting $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$, we get

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

$\Rightarrow 2x+3y-4z=1$

Hence, the Cartesian equation of the plane is
$2x+3y-4z=1$

(3) Vector equation of the plane is :
$\overrightarrow{r}.\left[\right(s-2t)\hat{i}+(3-t)\hat{j}+(2s+t\left)\hat{k}\right]=15$

Substituting $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$, we get

$(x\hat{i}+y\hat{j}+z\hat{k})\overrightarrow{r}.\left[\right(s-2t)\hat{i}+(3-t)\hat{j}+(2s+t\left)\hat{k}\right]=15$

$\Rightarrow x(s-2t)+y(3-t)+z(2s+t)=15$

Hence, the Cartesian equation of the plane is
$(s-2t)x+(3-t)y+(2s+t)z=15$.