Question

Assertion :

Consider planes $P_1:(\overrightarrow{r}-\hat{i}).(\hat{i}-\hat{j}-\hat{k})=0$ and $P_2:(\overrightarrow{r}-(2\hat{i}-\hat{j}-\hat{k}\left)\right).(\hat{i}-2\hat{k})\times (2\hat{i}-\hat{j}-3\hat{k})=0$ and line $L:\overrightarrow{r}=5\hat{i}+\lambda (\hat{i}-\hat{j}-\hat{k})$

$P_1$ and $P_2$ are parallel planes Reason: $L$ is parallel to both $P_1$ and $P_2$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Given planes

$P1:(\overrightarrow{r}-\hat{i}).(\hat{i}-\hat{j}-\hat{k})=0$

$P2:(\overrightarrow{r}-(2\hat{i}-\hat{j}-\hat{k}\left)\right).(\hat{i}-2\hat{k})\times (2\hat{i}-\hat{j}-3\hat{k})=0$

comparing both eq with general eq of plane

$(\overrightarrow{r}-\overrightarrow{a})\cdot \overrightarrow{n}=0$

$\overrightarrow{n_1}=\hat{i}-\hat{j}-\hat{k}$

$\overrightarrow{n_2}=(\hat{i}-2\hat{k})\times (2\hat{i}-\hat{j}-3\hat{k})$

$\overrightarrow{n_2}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& 0& -2\\ 2& -1& -3\end{array}\right|$

$\overrightarrow{n_2}=\hat{k}+3\hat{j}-4\hat{j}+2\hat{i}$

$\overrightarrow{n_2}=-2\hat{i}-\hat{j}+\hat{k}$

Here both normal vector is not equal so not parallel

$line:\overrightarrow{r}=5\hat{i}+\lambda (\hat{i}-\hat{j}-\hat{k})$

normal vector of line $\overrightarrow{n}=\hat{i}-\hat{j}-\hat{k}$

$\overrightarrow{n_1}\times \overrightarrow{n}=(\hat{i}-\hat{j}-\hat{k})\times (\hat{i}-\hat{j}-\hat{k})=0$

it is parallel to plane P1

$\overrightarrow{n_2}\times \overrightarrow{n}=(-2\hat{i}-\hat{j}+\hat{k})\times (\hat{i}-\hat{j}-\hat{k})\ne 0$