Question

The equation of the plane which contains the lines $\overrightarrow{r}=\hat{i}+2\hat{j}-\hat{k}+\lambda (\hat{i}+2\hat{j}-\hat{k})$ and $\overrightarrow{r}=\hat{i}+2\hat{j}-\hat{k}+\mu (\hat{i}+\hat{j}+3\hat{k})$ must be

• A. $\overrightarrow{r}.(7\hat{i}-4\hat{j}-\hat{k})=0$
• B. $7(x-1)-4(y-2)-(z+1)=0$
• C. $\overrightarrow{r}.(\hat{i}+2\hat{j}-\hat{k})=0$
• D. $\overrightarrow{r}.(\hat{i}+\hat{j}+3\hat{k})=0$

Answer 1

The correct option is B $7(x-1)-4(y-2)-(z+1)=0$

Clearly lines are parallel to $\hat{i}+2\hat{j}-\hat{k}$ and $\hat{i}+\hat{j}+3\hat{k}$
Thus normal vector of required plane is,
$\overrightarrow{n}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& -1\\ 1& 1& 3\end{array}\right|=7\hat{i}-4\hat{j}-\hat{k}$
Hence, required plane is
$[\overrightarrow{r}-(\hat{i}+2\hat{j}-\hat{k}\left)\right]\cdot \overrightarrow{n}=0$
$\Rightarrow [\overrightarrow{r}-(\hat{i}+2\hat{j}-\hat{k}\left)\right]\cdot (7\hat{i}-4\hat{j}-\hat{k})=0$
$\Rightarrow 7(x-1)-4(y-2)-(z+1)=0$