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# The length of $\perp$ from origin to the plane passing through the point $\overrightarrow a$ and containing the line $\overrightarrow r=\overrightarrow b+\lambda \overrightarrow c$ is ?

$\begin{array}{1 1} (a)\:\large\frac{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]}{|\overrightarrow a\times \overrightarrow b+\overrightarrow b\times\overrightarrow c+\overrightarrow c\times\overrightarrow a|}\: & \:(b)\:\large\frac{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]}{|\overrightarrow a\times \overrightarrow b+\overrightarrow b\times\overrightarrow c||} \\ (c)\:\large\frac{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]}{|\overrightarrow b\times\overrightarrow c+\overrightarrow c\times\overrightarrow a|}\: & \:(d)\:\large\frac{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]}{|\overrightarrow a\times \overrightarrow b+\overrightarrow c\times\overrightarrow a|} \end{array}$

• Vector equation of a plane through the point $\overrightarrow a$ and with normal $\overrightarrow n$ is $(\overrightarrow r-\overrightarrow a).\overrightarrow n=0$
Since the given line $\overrightarrow r=\overrightarrow b+\lambda \overrightarrow c$ and the point $\overrightarrow a lie on the plane, the vectors$\overrightarrow a-\overrightarrow b$and$\overrightarrow c$lie on the plane.$\Rightarrow\:$Normal vector to the plane$(\overrightarrow n)$is$\perp $to both the vectors$\overrightarrow a-\overrightarrow b\:\:and\:\:\overrightarrow c\therefore \overrightarrow n=\overrightarrow c\times (\overrightarrow a-\overrightarrow b)\therefore$Equation of the plane through the point$\overrightarrow a$and the line$\overrightarrow r=\overrightarrow b+\lambda \overrightarrow c$is$(\overrightarrow r-\overrightarrow a).\overrightarrow n=0\Rightarrow\:\overrightarrow r.\overrightarrow n=\overrightarrow a.\overrightarrow n\Rightarrow\:\overrightarrow r.\overrightarrow n=\overrightarrow a.(\overrightarrow c\times\overrightarrow a-\overrightarrow c\times \overrightarrow b)=[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]$Lenth of$\perp$from origin to this plane is given by$\large\frac{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]}{|\overrightarrow n|}=\large\frac{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]}{|\overrightarrow c\times \overrightarrow a+\overrightarrow b\times\overrightarrow c|}\$