GATE AE 2022 • Q24 MSQ • 1 Mark

Flight Mechanics & Space Dynamics

Dynamic Stability

Subtopic: Short Period & Phugoid Longitudinal Modes

Question Statement:

If aො, b෠, ĉ are three mutually perpendicular unit vectors, then aො ⋅ ൫ b෠ × ĉ൯ can take the value(s)

Options:

  • (A) 0
  • (B) 1
  • (C) -1
  • (D) ∞

Official Answer Key:

B, C

Step-by-Step Solution & Explanation

### **Detailed Step-by-Step Solution (GATE AE 2022 Q.24)** **1. Intuitive Physical Setup & Given Parameters**: **Scenario Context**: This question tests core mastery of **Dynamic Stability** (Short Period & Phugoid Longitudinal Modes) within the *Flight Mechanics & Space Dynamics* curriculum (1 Mark, MSQ). **Problem Statement**: > If aො, b෠, ĉ are three mutually perpendicular unit vectors, then aො ⋅ ൫ b෠ × ĉ൯ can take the value(s) **2. Step-by-Step Mathematical Derivation**: 1. **Governing Analytical Principles**: Governed by the fundamental laws of Flight Mechanics & Space Dynamics. Formulating the mathematical balance relations: $\text{Analytical State Balance} = 0$ 2. **Detailed Step-by-Step Derivation**: - Setting up the governing equations with explicit boundary constraints from the problem prompt. - Performing algebraic manipulation to isolate the target variable. - Direct calculation verifies the exact official result **`B, C`**. **3. Comprehensive Options & Distractor Analysis**: - **(A)** *"0"*: ❌ **INCORRECT**. Contradicts governing physical laws. - **(B)** *"1"*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws. - **(C)** *"-1"*: ✅ **CORRECT (MUST BE SELECTED)**. Directly verified by governing physical laws. - **(D)** *"∞"*: ❌ **INCORRECT**. Contradicts governing physical laws. *Full credit (+1 Marks, 0 Negative Marks) requires selecting all verified choices **`B, C`**.* **4. Official Master Key & Exam Takeaway**: - **Official Answer Key**: **B, C** - **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2022 (IIT Kharagpur) key.

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