Q. If $P_{1}:\overrightarrow{r} \bullet {\overrightarrow{n}}_{1} - d_{1} = 0,\ P_{2}:\overrightarrow{r} \bullet {\overrightarrow{n}}_{2} - d_{2} = 0$ and $P_{3}:\overrightarrow{r} \bullet {\overrightarrow{n}}_{3} - d_{3} = 0$ are three planes and ${\overrightarrow{n}}_{1},{\overrightarrow{n}}_{2}\ \ and\ {\overrightarrow{n}}_{3}$ are three non-coplanar vectors, then three lines $P_{1} = 0,\ P_{2} = 0;P_{2} = 0,\ P_{3} = 0$ and $P_{3} = 0,\ P_{1} = 0$ are
A) Parallel lines
B) Coplanar lines
C) Coincident lines
D) Concurrent lines
Explanation
(d)
$P_{1} = P_{2} = 0,\ P_{2} = P_{3} = 0$ and $P_{3} = P_{1} = 0$ are lines of intersection of the three planes $P_{1},P_{2}$ and $P_{3}$. As ${\overrightarrow{n}}_{1},\ {\overrightarrow{n}}_{2}$ and ${\overrightarrow{n}}_{3}$ are non-coplanar, planes $P_{1},P_{2}$ and $P_{3}$ will intersect at unique point. So the given liens will pass through a fixed point
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.