Question

Equation of the plane containing the lines $\overrightarrow{r}=\hat{i}+\hat{j}-\hat{k}+\lambda (\hat{i}+2\hat{j}-\hat{k})and\overrightarrow{r}=\hat{i}+2\hat{j}-\hat{k}+\mu (\hat{i}+\hat{j}+3\hat{k})$ is

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

$\overrightarrow{r}(7\hat{i}-4\hat{j}-\hat{k})=0$

$A(1,2,-1)$
$\overrightarrow{AR}.(\overrightarrow{a}\times \overrightarrow{b})=0$

$(\overrightarrow{a}\times \overrightarrow{b})=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 2& -1\\ 1& 1& 3\end{array}\right|=7\hat{i}-4\hat{j}-\hat{k}[\overrightarrow{r}-(\hat{i}+2\hat{j}-\hat{k}\left)\right](7\hat{i}-4\hat{j}-\hat{k})=0\overrightarrow{r}.(7\hat{i}-4\hat{j}-\hat{k})=0$