Relation of Perpendicular Lines in a Plane
📺 Video Explanation
📝 Question
Let \( R \) be a relation on the set of all straight lines in a plane such that:
\[ L_1 \, R \, L_2 \iff L_1 \perp L_2 \]
Then, \( R \) is:
- (a) symmetric
- (b) reflexive
- (c) transitive
- (d) an equivalence relation
✅ Solution
We check reflexive, symmetric, and transitive properties.
🔹 Reflexive
A line cannot be perpendicular to itself.
\[ L \not\perp L \]
❌ Not reflexive.
🔹 Symmetric
If line \( L_1 \) is perpendicular to line \( L_2 \), then line \( L_2 \) is also perpendicular to line \( L_1 \).
\[ L_1 \perp L_2 \Rightarrow L_2 \perp L_1 \]
✔ Symmetric.
🔹 Transitive
Suppose:
\[ L_1 \perp L_2 \quad \text{and} \quad L_2 \perp L_3 \]
Then \( L_1 \) and \( L_3 \) are parallel, not perpendicular.
❌ Not transitive.
🎯 Final Answer
\[ \boxed{\text{R is symmetric}} \]
✔ Correct option: (a)
🚀 Exam Shortcut
- Perpendicular relation is always symmetric
- No line is perpendicular to itself → not reflexive
- Two lines perpendicular to same line become parallel → not transitive