# The equation of the line through the point $(1,2,3)$ and $\perp$ to the plane $\overrightarrow r.(\hat i+2\hat j-5\hat k)+9=0$ is ?

$\begin{array}{1 1} (a)\:\overrightarrow r=(\hat i+2\hat j-5\hat k)+\lambda(\hat i+2\hat j+3\hat k) & (b)\: \overrightarrow r=(\hat i+2\hat j+3\hat k)+\lambda(\hat i+2\hat j-5\hat k) \\ (c)\:\overrightarrow r=(-\hat i-2\hat j+5\hat k)+\lambda(\hat i+2\hat j+3\hat k) & (d)\:\overrightarrow r=(-\hat i-2\hat j-3\hat k)+\lambda(\hat i+2\hat j-5\hat k) \end{array}$

Given eqn. of the plane is $\overrightarrow r.(\hat i+2\hat j-5\hat k)+9=0$
Since the line is $\perp$ to the plane, it is along the normal to the plane.
$\overrightarrow n=(1,2,-5)$
$\therefore$ Eqn. of the line through the point $(1,2,3)$ and $\perp$ to the given plane is
$\overrightarrow r= (\hat i+2\hat j+3\hat k)+\lambda(\hat i+2\hat j-5\hat k)$
answered Dec 31, 2013