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The line through the point $\hat i+3\hat j+2\hat k$ and $\perp$ to the lines $\overrightarrow r=(\hat i+2\hat j-\hat k)+\lambda(2\hat i+\hat j+\hat k)$ and $\overrightarrow r=(2\hat i+6\hat j+\hat k)+\mu (\hat i+2\hat j+3\hat k)$ is ?

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

• $\overrightarrow a\times\overrightarrow b$ is along the direction $\perp$ to both $\overrightarrow a$ and $\overrightarrow b$.
Since the required line is $\perp$ to the given two lines,
$\overrightarrow r=(\hat i+2\hat j-\hat k)+\lambda(2\hat i+\hat j+\hat k)$ and $\overrightarrow r=(2\hat i+6\hat j+\hat k)+\mu(\hat i+2\hat j+3\hat k)$,
It will be along $(2\hat i+\hat j+\hat k)\times (\hat i+2\hat j+3\hat k)$
$i.e.,$ $d.r.$ of the required line is $\left|\begin {array}{ccc}\hat i & \hat j &\hat k \\ 2 & 1 & 1 \\ 1 & 2 & 3\end {array} \right|=(1,-5,3)$
$\therefore\:$ Eqn. of the required line through $(1,3,2)$ is
$\overrightarrow r=(\hat i+3\hat j+2\hat k)+\beta(\hat i-5\hat j+3\hat k)$