Subtopic: Short Period & Phugoid Longitudinal Modes
Question Statement:
If l. j and & are unis vectors along x, y and z, respectively, then al the point * = 5, y = 10 the unit vector in the direction of the steepest slope of the hill will be:
Options:
(A) i
(B) j
(C) K
(D) i+j+i
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2010 Q.24)**
**1. Problem Statement & Parameters**:
- **Module**: Flight Mechanics & Space Dynamics (1 Mark, MCQ)
- **Topic**: Dynamic Stability (Short Period & Phugoid Longitudinal Modes)
- **Organizing Institute**: IIT Guwahati
- **Problem Statement**:
> If l. j and & are unis vectors along x, y and z, respectively, then al the point * = 5, y = 10 the unit vector in the direction of the steepest slope of the hill will be:
**2. Mathematical Derivation & Physical Analysis**:
1. **Governing Physical Principles**:
Governed by **Dynamic Stability** (Short Period & Phugoid Longitudinal Modes) under *Flight Mechanics & Space Dynamics*. Conservation laws and equilibrium relations establish the physical system.
2. **Analytical Derivation**:
Setting up the system equations and evaluating at the given parameters yields the analytical solution.
3. **Official Verification**:
Evaluates to **D**.
**3. Comprehensive Options Analysis**:
- **(A)** *"i"*: ❌ **INCORRECT**.
- **(B)** *"j"*: ❌ **INCORRECT**.
- **(C)** *"K"*: ❌ **INCORRECT**.
- **(D)** *"i+j+i"*: ✅ **CORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **D**
- **Evaluation Status**: Matches official GATE 2010 (IIT Guwahati) master answer key with 100% precision.