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Q)

# The equation of the plane containing the lines $\overrightarrow r=\overrightarrow a_1+\lambda\overrightarrow b\:\:and\:\:\overrightarrow r=\overrightarrow a_2+\mu\overrightarrow b$ is ?

$\begin{array}{1 1} (a)\:\overrightarrow r.(\overrightarrow a_1-\overrightarrow a_2)\times\overrightarrow b=[\overrightarrow a_1\:\overrightarrow a_2\:\overrightarrow b]\:\qquad\:(b)\:\overrightarrow r.(\overrightarrow a_2-\overrightarrow a_1)\times\overrightarrow b=[\overrightarrow a_1\:\overrightarrow a_2\:\overrightarrow b]\:\qquad\:(c)\:\overrightarrow r.(\overrightarrow a_1+\overrightarrow a_2)\times\overrightarrow b=[\overrightarrow a_2\:\overrightarrow a_1\:\overrightarrow b]\:\qquad\:(d)\:none \:of\:these \end{array}$</p

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A)
Toolbox:
• Line joining any two points on a plane lie on the plane.
• Every line on the plane is $\perp$ to the normal to the plane.
• A vector $\perp$ to $\overrightarrow a\:\:and\:\:\overrightarrow b$ is given by $\overrightarrow a\times\overrightarrow b$
• $\overrightarrow a.\overrightarrow b\times\overrightarrow c=[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]$
• $-[\overrightarrow a\:\overrightarrow b\:\overrightarrow c]=[\overrightarrow b\:\overrightarrow a\:\overrightarrow c]$
Let the normal to the required plane be $\overrightarrow n$ and any point on the plane be $\overrightarrow a$
It is given that the plane contains the lines,
$\overrightarrow r= \overrightarrow a_1+\lambda\overrightarrow b$.....(i) and
$\overrightarrow r= \overrightarrow a_2+\lambda\overrightarrow b$.....(ii)
$\Rightarrow\:$ the points $\overrightarrow a_1\:\:and\:\:\overrightarrow a_2$ lie on the plane
and $\overrightarrow b$ lies on the plane
$\therefore \:\overrightarrow a=\overrightarrow a_1$ or $\overrightarrow a_2$
$\Rightarrow\:$ the line (vector) $\overrightarrow a_1-\overrightarrow a_2$ also lie on the plane
$\Rightarrow\:$ normal to the plane $\overrightarrow n$ is $\perp$ to both $\overrightarrow b$ and $\overrightarrow a_1-\overrightarrow a_2$
$\Rightarrow\: \overrightarrow n=(\overrightarrow a_1-\overrightarrow a_2)\times\overrightarrow b=\overrightarrow a_1\times\overrightarrow b-\overrightarrow a_2\times\overrightarrow b$
Equation of the required plane is $(\overrightarrow r-\overrightarrow a).\overrightarrow n=0$
$\Rightarrow\:(\overrightarrow r- \overrightarrow a_1).(\overrightarrow a_1-\overrightarrow a_2)\times\overrightarrow b=0$
$i.e.,\:\:\overrightarrow r.(\overrightarrow a_1-\overrightarrow a_2)\times\overrightarrow b=\overrightarrow a_1.[(\overrightarrow a_1-\overrightarrow a_2)\times\overrightarrow b]$
$\Rightarrow\:\overrightarrow r.(\overrightarrow a_1-\overrightarrow a_2)\times\overrightarrow b=-[\overrightarrow a_1\:\overrightarrow a_2\:\overrightarrow b]$
(Since $[\overrightarrow a_1\:\overrightarrow a_1\:\overrightarrow b]=0$.)
Also $- [\overrightarrow a_1\:\overrightarrow a_2\:\overrightarrow b]=[\overrightarrow a_2\:\overrightarrow a_1\:\overrightarrow b]$
$\Rightarrow\:$ The required equation is $\overrightarrow r.(\overrightarrow a_1-\overrightarrow a_2)\times\overrightarrow b=[\overrightarrow a_2\:\overrightarrow a_1\:\overrightarrow b]$