Relation of Perpendicular Lines in a Plane
📺 Video Explanation
📝 Question
Let \(L\) denote the set of all straight lines in a plane. A relation \(R\) is defined by:
\[ lRm \iff l \perp m \]
Then, \(R\) is:
- A. reflexive
- B. symmetric
- C. transitive
- D. none of these
✅ Solution
🔹 Reflexive Check
A line is never perpendicular to itself.
\[ l \not\perp l \]
❌ Not reflexive.
🔹 Symmetric Check
If:
\[ l\perp m \]
then:
\[ m\perp l \]
✔ Symmetric.
🔹 Transitive Check
Suppose:
\[ l\perp m \quad \text{and} \quad m\perp n \]
Then:
\[ l \parallel n \]
not necessarily perpendicular.
❌ Not transitive.
🎯 Final Answer
\[ \boxed{\text{R is symmetric}} \]
✔ Correct option: B
🚀 Exam Shortcut
- Perpendicular relation is always symmetric
- No line is perpendicular to itself
- Two lines perpendicular to same line become parallel