Chat with tutor

Ask Questions, Get Answers

Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class11  >>  3-D Geometry

The equation of the plane that contains the line $\overrightarrow r=2\hat i+\lambda(\hat j-\hat k)$ and $\perp$ to the plane $\overrightarrow r.(\hat i+\hat k)=3$ is ?

$\begin{array}{1 1} (a)\:\overrightarrow r.(\hat i-\hat j-\hat k)=2\:\qquad\:(b)\:\overrightarrow r.(\hat i+\hat j-\hat k)=2\:\qquad\:(c)\:\overrightarrow r.(\hat i+\hat j+\hat k)=2\:\qquad\:(d)\:\overrightarrow r.(\hat i-\hat j+\hat k)=0 \end{array} $

1 Answer

Let the normal to the required plane be $\overrightarrow n$ and a point on the plane be $\overrightarrow a$
Given that the line $\overrightarrow r=2\hat i+\lambda(\hat j-\hat k)$.......(i) lies on the required plane.
$\therefore$ all the points on the line should lie on the plane.
$\therefore (2,0,0)$ is a point on the plane.
$i.e.,$ $ \overrightarrow a=2\hat i$
and  since the line lies on the plane,
the line (i) is $\perp$ to the normal to the plane $ (\overrightarrow n)$
Also given that the required plane is $\perp $ to the plane $\overrightarrow r.(\hat i+\hat k)=3$
$i.e.,$ $\overrightarrow n$ is $\perp$ to $ (0,1,-1) \:\:and\:\:(1,0,1)$
$\Rightarrow \overrightarrow n=\left |\begin {array}{ccc} \hat i &\hat j & \hat k\\0 & 1 & -1\\ 1 & 0 & 1\end {array} \right|=(1,-1,-1)$
The equation of the required plane is $(\overrightarrow r-\overrightarrow a).\overrightarrow n=0$
$\therefore$ Eqn. of the required plane is $(\overrightarrow r-2\hat i).(\hat i-\hat j-\hat k)=0$
$\Rightarrow\:\overrightarrow r .(\hat i-\hat j-\hat k)=2$


Help Clay6 to be free
Clay6 needs your help to survive. We have roughly 7 lakh students visiting us monthly. We want to keep our services free and improve with prompt help and advanced solutions by adding more teachers and infrastructure.

A small donation from you will help us reach that goal faster. Talk to your parents, teachers and school and spread the word about clay6. You can pay online or send a cheque.

Thanks for your support.
Please choose your payment mode to continue
Home Ask Homework Questions
Your payment for is successful.
Clay6 tutors use Telegram* chat app to help students with their questions and doubts.
Do you have the Telegram chat app installed?
Already installed Install now
*Telegram is a chat app like WhatsApp / Facebook Messenger / Skype.