Question

The equation of the plane through the line of intersection of $\overline{r}-(\hat{i}-2\hat{j}+2\hat{k})=1$ and $\overline{r}-(2\hat{i}-\hat{j})+2=0$ and $\perp$ to $\overline{r}-(\hat{i}+\hat{j}+2\hat{k})+9=0$

• A. $\overline{r}.-(-5\hat{i}+j+2\hat{k})=7$
• B. $\overline{r}.-(-5\hat{i}-\hat{j}+2\hat{k})=7$
• C. $\overline{r}.-(5\hat{i}-\hat{j}-2\hat{k})=7$
• D. None of these

Answer 1

The correct option is A $\overline{r}.-(-5\hat{i}+j+2\hat{k})=7$

The direction vector of the line of the intersection of the planes will be.

$(n_1\times n_2)$

$=(i-2j+2k)\times (2i-j)$

$=2i+4j+3k$

Hence the normal vector to the plane will be

$(2i+4j+3k)\times (i+2j+2k)$
$=5i-j-2k$.

Thus the required equation of the plane will be

$\overrightarrow{r}.\overrightarrow{n}=7$
$\overrightarrow{r}.(5i-j-2k)=7$