Question

The vector equation of the plane through the point $\hat{i}+2\hat{j}-\hat{k}$ and perpendicular to the line of intersection of the plane $\overline{r}.(3\hat{i}-9\hat{j}+\hat{k})=1$ and $\overline{r}.(\hat{i}+4\hat{j}-2\hat{k})=2$
is:

• A. $\overline{r}.(2\hat{i}+7\hat{j}-13\hat{k})=1$
• B. $\overline{r}.(2\hat{i}-7\hat{j}-13\hat{k})=1$
• C. $\overline{r}.(2\hat{i}-7\hat{j}+13\hat{k})=0$
• D. $\overline{r}.(7\hat{i}-9\hat{j}+13\hat{k})=0$

Answer 1

The correct option is B $\overline{r}.(2\hat{i}-7\hat{j}-13\hat{k})=1$

Equation of plane through the point $(\hat{i}+2\hat{j}-\hat{k})$ is $(\overline{r}-(\hat{i}+2\hat{j}-\hat{k}\left)\right).\overline{n}=0$
Where $\overline{n}$ is normal to the plane. Required plane is perpendicular to the line of intersection of the planes
$\overline{r}.(3\hat{i}-\hat{j}+\hat{k})=1$ and $\overline{r}.(\hat{i}+4\hat{j}-2\hat{k})=2$
$\therefore \overline{n}=(3\hat{i}-\hat{j}+\hat{k})\times \hat{i}+4\hat{j}-2\hat{k}\equiv -2\hat{i}+7\hat{j}+13\hat{k}$
$\therefore$ Required plane is $\overline{r}.(2\hat{i}-7\hat{j}-13\hat{k})=1$