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# 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 ?

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

Can you answer this question?

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 $(\overrightarrow r-2\hat i).(\hat i-\hat j-\hat k)=0$
$\Rightarrow\:\overrightarrow r .(\hat i-\hat j-\hat k)=2$

answered Jan 4, 2014