Question

Find the equation of the plane through the point $2\hat{i}+\hat{j}-\hat{k}$ and passing through the line of intersection of the planes $\overrightarrow{r}·\left(\hat{i}+3\hat{j}-\hat{k}\right)=0\mathrm{and}\overrightarrow{r}·\left(\hat{j}+2\hat{k}\right)=0.$

Answer 1

$$\begin{gathered} \text{The equation of the plane passing through the line of intersection of the given planes is} \\ \overrightarrow{r}.\left(\hat{i}+3\hat{j}-\hat{k}\right)+\lambda \left(\overrightarrow{r}.\left(\hat{j}+2\hat{k}\right)\right)=0 \\ \overrightarrow{r}.\left[\hat{i}+\left(3+\lambda \right)\hat{j}+\left(-1+2\lambda \right)\hat{k}\right]=0...\left(1\right) \\ \text{This passes through 2}\hat{i}\text{+}\hat{j}\text{-}\hat{k}\text{. So,} \\ \left(2\hat{i}+\hat{j}-\hat{k}\right)\left[\hat{i}+\left(3+\lambda \right)\hat{j}+\left(-1+2\lambda \right)\hat{k}\right]=0 \\ \Rightarrow 2+3+\lambda +1-2\lambda =0 \\ \Rightarrow \lambda =6 \\ \text{Substituting this in (1), we get} \\ \overrightarrow{r}.\left[\hat{i}+\left(3+6\right)\hat{j}+\left(-1+12\right)\hat{k}\right]=0 \\ \Rightarrow \overrightarrow{r}.\left(\hat{i}+9\hat{j}+11\hat{k}\right)=0 \end{gathered}$$