Q. The vector equation of the plane passing through the origin and the line of intersection of the planes $\overrightarrow{\mathbf{r}} \bullet \overrightarrow{\mathbf{a}} = \lambda$ and $\overrightarrow{\mathbf{r}} \bullet \overrightarrow{\mathbf{b}} = \mu,$ is

A) $\overrightarrow{\mathbf{r}} \bullet \left( \lambda\ \overrightarrow{\mathbf{a}} - \mu\ \overrightarrow{\mathbf{b}} \right) = 0$
B) $\overrightarrow{\mathbf{r}} \bullet \left( \lambda\ \overrightarrow{\mathbf{b}} - \mu\ \overrightarrow{\mathbf{a}} \right) = 0$
C) $\overrightarrow{\mathbf{r}} \bullet \left( \lambda\ \overrightarrow{\mathbf{a}} + \mu\ \overrightarrow{\mathbf{b}} \right) = 0$
D) $\overrightarrow{\mathbf{r}} \bullet \left( \lambda\ \overrightarrow{\mathbf{b}} + \mu\ \overrightarrow{\mathbf{a}} \right) = 0$
Explanation

(b)
The equation of the plane through the line of intersection of given plane is
$\left( \overrightarrow{\mathbf{r}} \bullet \overrightarrow{\mathbf{a}} - \lambda \right) + k\left( \overrightarrow{\mathbf{r}} \bullet \overrightarrow{\mathbf{b}} - \mu \right)$=0
$or\ \ \overrightarrow{\mathbf{r}} \bullet \left( \overrightarrow{\mathbf{a}} + k\overrightarrow{\mathbf{b}} \right) = \lambda + k\mu\ \ldots..(i)$
this passes through the origin, therefore
$0 \bullet \left( \overrightarrow{\mathbf{a}} + k\overrightarrow{\mathbf{b}} \right) = \lambda + k\mu\$
$\Longrightarrow k = - \frac{\lambda}{\mu}$
On putting the value of $k$ in Eq. (i), we get the equation of the required plane as
$\ \overrightarrow{\mathbf{r}} \bullet \left( \mu\overrightarrow{\mathbf{a}} - \lambda\overrightarrow{\mathbf{b}} \right) = 0$
0$\Longrightarrow \overrightarrow{\mathbf{r}} \bullet \left( \lambda\overrightarrow{\mathbf{b}} - \mu\overrightarrow{\mathbf{a}} \right) = 0$

Dr. Abubacker
Reviewed by Dr. Abubacker

Physics Educator & NEET/JEE Specialist

Ph.D. in Physics with 19+ years of teaching experience.

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