Question

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

• A. $\overrightarrow{r}\cdot (\lambda \overrightarrow{a}-\mu \overrightarrow{b})=0$
• B. $\overrightarrow{r}\cdot (\lambda \overrightarrow{b}-\mu \overrightarrow{a})=0$
• C. $\overrightarrow{r}\cdot (\lambda \overrightarrow{a}+\mu \overrightarrow{b})=0$
• D. $\overrightarrow{r}\cdot (\lambda \overrightarrow{b}+\mu \overrightarrow{a})=0$

Answer 1

The correct option is B $\overrightarrow{r}\cdot (\lambda \overrightarrow{b}-\mu \overrightarrow{a})=0$

The equation of a plane through the line of intersection of the planes $\overrightarrow{r}.\overrightarrow{a}=\lambda$ and $\overrightarrow{r}.\overrightarrow{b}=\mu$ is
$(\overrightarrow{r}.\overrightarrow{a}-\lambda )+k(\overrightarrow{r}.\overrightarrow{b}-\mu )=0$
$\Rightarrow \overrightarrow{r}.(\overrightarrow{a}+k\overrightarrow{b})=\lambda +k\mu \dots \left(1\right)$
This passes through the origin, therefore
$\overrightarrow{O}(\overrightarrow{a}+k\overrightarrow{b})=\lambda +\mu k$
$\Rightarrow k=\frac{-\lambda }{\mu }$
Putting the value of $k$ in $\left(1\right)$, we get the equation of the required plane as
$\overrightarrow{r}.(\mu \overrightarrow{a}-\lambda \overrightarrow{b})=0$ or $\overrightarrow{r}.(\lambda \overrightarrow{b}-\mu \overrightarrow{a})=0$