The equation of the plane which passes through the line of intersection of planes $\vec{r} \cdot \vec{n_{1}} = q_{1}, \vec{r} \cdot \vec{n_{2}} = q_{2}$ and parallel to the line of intersection of planes $\vec{r} \cdot \vec{n_{3}} = q_{3} and \vec{r} \cdot \vec{n_{4}} = q_{4}$ is

Moderate

A

$[\vec{n_{2}} \vec{n_{3}} \vec{n_{4}}] (\vec{r} \cdot \vec{n_{1}} - q_{1}) = [\vec{n_{1}} \vec{n_{3}} \vec{n_{4}}] (\vec{r} \cdot \vec{n_{2}} - q_{2})$
B

$[\vec{n_{1}} \vec{n_{2}} \vec{n_{3}}] (\vec{r} \cdot \vec{n_{4}} - q_{4}) = [\vec{n_{4}} \vec{n_{3}} \vec{n_{1}}] (\vec{r} \cdot \vec{n_{2}} - q_{2})$
C

$[{\vec{n}}_{4} {\vec{n}}_{3} {\vec{n}}_{1}] (\vec{r} \cdot \vec{n_{4}} - q_{4}) = [\vec{n_{1}} \vec{n_{2}} \vec{n_{3}}] (\vec{r} \cdot \vec{n_{2}} - q_{2})$
D

none of these

Solution

$\vec{r} \cdot \vec{n_{1}} + \vec{λr} \cdot \vec{n_{2}} = q_{1} + {λq}_{2} - - - - (i)$
So $\vec{n_{1}} + λ \vec{n_{2}}$ is normal to plane (i). Now, any plane
parallel to the line of intersection of the planes $\vec{r .} \vec{n_{3}} = q_{3} and \vec{r} \cdot \vec{n_{4}} = q_{4}$ is of the form $\vec{r} \cdot (\vec{n_{3}} \times \vec{n_{4}}) = d$
Hence, we must have
$\begin{array}{l} [{\vec{n}}_{1} + λ \vec{n_{2}}] \cdot [\vec{n_{3}} \times \vec{n_{4}}] = 0 \\ or [\vec{n_{1}} {\vec{n}}_{3} {\vec{n}}_{4}] + λ [{\vec{n}}_{2} {\vec{n}}_{3} {\vec{n}}_{4}] = 0 \\ or λ = \frac{- [{\vec{n}}_{1} {\vec{n}}_{3} {\vec{n}}_{4}]}{[\vec{n_{2}} \vec{n_{3}} \vec{n_{4}}]} \end{array}$
On putting this value in Eq. (i), we have the equation of the required plane as
$\begin{array}{l} \vec{r} \cdot \vec{n_{1}} - q_{1} = \frac{[{\vec{n}}_{1} {\vec{n}}_{3} \vec{n_{4}}]}{[\vec{n_{2}} \vec{n_{3}} \vec{n_{4}}]} (r \cdot \vec{n_{2}} - q_{2}) \\ or [\vec{n_{2}} \vec{n_{3}} \vec{n_{4}}] (\vec{r} \cdot {\vec{n}}_{1} - q_{1}) \\ = [{\vec{n}}_{1} {\vec{n}}_{3} n_{4}] (\vec{r} \cdot \vec{n_{2}} - q_{2}) \end{array}$

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