Let the position vectors of the three points be,
$\overrightarrow{\mathit{a}}$ = $\hat{\mathit{i}}+\hat{\mathit{j}}-2\hat{\mathit{k}}$ , $\overrightarrow{\mathit{b}}$ = $2\hat{\mathit{i}}-\hat{\mathit{j}}+\hat{\mathit{k}}$ and $\overrightarrow{\mathit{c}}$ = $\hat{\mathit{i}}+2\hat{\mathit{j}}+\hat{\mathit{k}}$.
So, the equation of the plane passing through the points $\overrightarrow{\mathit{a}},\overrightarrow{\mathit{b}}$ and $\overrightarrow{\mathit{c}}$ is
$\left(\overrightarrow{\mathit{r}}-\overrightarrow{\mathit{a}}\right).\left[\left(\overrightarrow{\mathit{b}}-\overrightarrow{\mathit{c}}\right)\times \left(\overrightarrow{\mathit{c}}-\overrightarrow{\mathit{a}}\right)\right]$ = 0
⇒ $\left[\overrightarrow{\mathit{r}}-\hat{\mathit{i}}+\mathit{j}^\to +2\hat{\mathit{k}}\right]$ . $\left[\hat{\mathit{i}}-3\hat{\mathit{j}}\times \hat{\mathit{j}}+3\hat{\mathit{k}}\right]$ = 0
⇒ $\left[\overrightarrow{\mathit{r}}-\hat{\mathit{i}}+\mathit{j}^\to +2\hat{\mathit{k}}\right]$ . $\hat{\mathit{k}}-3\hat{\mathit{j}}-9\hat{\mathit{i}}$ = 0
⇒ $\overrightarrow{\mathit{r}}.\left(9\hat{\mathit{i}}+3\hat{\mathit{j}}-\hat{\mathit{k}}\right)$ = 14 ... (1)
So, the vector equation of the required plane is $\overrightarrow{\mathit{r}}.\left(9\hat{\mathit{i}}+3\hat{\mathit{j}}-\hat{\mathit{k}}\right)$ = 14
The equation of the given line is $\overrightarrow{\mathit{r}}$ = $3\hat{\mathit{i}}-\hat{\mathit{j}}-\hat{\mathit{k}}+\mathit{\lambda }\left(2\hat{\mathit{i}}-2\hat{\mathit{j}}+\hat{\mathit{k}}\right)$
Position vector of any point on the given line is
$\overrightarrow{\mathit{r}}$ = 3 + 2λ $\hat{\mathit{i}}$ + (- 1 - 2λ) $\hat{\mathit{j}}$ + (- 1 + λ) $\hat{\mathit{k}}$ ... (2)
The point (2) lies on plane (1) if,
= 14
⇒ 9 (3) + 2λ + 3 - 1 - 2λ - (-1) + λ = 14
⇒ 11λ + 25 = 14
⇒ λ = - 1
Putting λ = - 1 in (2), we have
$\overrightarrow{\mathit{r}}$ = (3 + 2λ) $\hat{\mathit{i}}$ + (- 1 - 2λ) $\hat{\mathit{j}}$ + (- 1 + λ) $\hat{\mathit{k}}$
= (3 + 2 - 1) $\hat{\mathit{i}}$ + (- 1 - 2 - 1) $\hat{\mathit{j}}$ + (- 1 + (- 1)) $\hat{\mathit{k}}$
= $\hat{\mathit{i}}+\hat{\mathit{j}}-2\hat{\mathit{k}}$
Thus, the position vector of the point of intersection of the given line and plane (1) is
$\hat{\mathit{i}}+\hat{\mathit{j}}-2\hat{\mathit{k}}$ and its co-ordinates are 1, 1, - 2 .