Question

Vector equation of the plane $\overrightarrow{r}=\widehat{i-}\hat{j}+\lambda (\hat{i}-\hat{j}+\hat{k})+\mu (\hat{i}-2\hat{j}+3\hat{k})$

• A. $-\hat{i}-2\hat{j}-\hat{k}$
• B. $\overrightarrow{r}.(5\hat{i}+2\hat{j}-3\hat{k})=7$
• C. $\overrightarrow{r}.(5\hat{i}-2\hat{j}-3\hat{k})=7$
• D. $\overrightarrow{r}.(5\hat{i}+2\hat{j}+3\hat{k})=7$

Answer 1

The correct option is A $-\hat{i}-2\hat{j}-\hat{k}$

$\begin{array}{c}\overrightarrow{r}=\overrightarrow{i}-\overrightarrow{j}+\lambda \left(\overrightarrow{i}-\overrightarrow{j}+\overrightarrow{k}\right)+\mu \left(\hat{i}-2\hat{j}+3\hat{k}\right)\\ planepassesthrough\hat{i}-\hat{j}\\ planeis\parallel to\hat{i}-\hat{j}+\hat{k}and\hat{i}-2\hat{j}+3\hat{k}\\ normalvectortoplaneis:\\ \left(\hat{i}-\hat{j}+\hat{k}\right)\times \left(\hat{i}-2\hat{j}+3\hat{k}\right)\\ \left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -1& 1\\ 1& -2& 3\end{array}\right|=\hat{i}\left(-1\right)-\hat{j}\left(2\right)+\hat{k}\left(-2+1\right)\\ =-\hat{i}-2\hat{j}-\hat{k}\end{array}$

Hence,

option $A$ is correct answer.