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

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} $

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A)
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  • 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 $\large\frac{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]}{|\overrightarrow n|}$
$=\large\frac{[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]}{|\overrightarrow c\times \overrightarrow a+\overrightarrow b\times\overrightarrow c|}$
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