Q. The line of intersection of the planes $\overrightarrow{\mathbf{r}} \bullet \left( \widehat{\dot{\mathbf{i}}} - 3\widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) = 1$ and $\overrightarrow{\mathbf{r}} \bullet \left( 2\widehat{\dot{\mathbf{i}}} + 5\widehat{\dot{\mathbf{j}}} - 3\widehat{\mathbf{k}} \right) = 2$ is parallel to the vector
Explanation
(b)
Given planes are
$\ \overrightarrow{\mathbf{r}} \bullet \left( \widehat{\dot{\mathbf{i}}} - 3\widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) = 1\ \ldots..(i)$
$and\ \ \overrightarrow{\mathbf{r}} \bullet \left( 2\widehat{\dot{\mathbf{i}}} + 5\widehat{\dot{\mathbf{j}}} - 3\widehat{\mathbf{k}} \right) = 2\ \ \ldots\ldots.(ii)$
Now,
$\left( \widehat{\dot{\mathbf{i}}} - 3\widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}} \right) \times \left( 2\widehat{\dot{\mathbf{i}}} + 5\widehat{\dot{\mathbf{j}}} - 3\widehat{\mathbf{k}} \right) = \left| \begin{matrix}
\widehat{\dot{\mathbf{i}}} & \widehat{\dot{\mathbf{j}}} & \widehat{\mathbf{k}} \\
1 & - 3 & 1 \\
2 & 5 & - 3
\end{matrix} \right|$
$= 4\widehat{\dot{\mathbf{i}}} + 5\widehat{\dot{\mathbf{j}}} + 11\widehat{\mathbf{k}}$
Hence, line of intersection of the planes is parallel to the vector $4\widehat{\dot{\mathbf{i}}} + 5\widehat{\dot{\mathbf{j}}} + 11\widehat{\mathbf{k}}.$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.