GATE BT 2024 • Q32 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Fundamentals of Linear Algebra

Question Statement:

Let OR⃗⃗⃗⃗⃗ be the vector that is perpendicular to the vectors OP⃗⃗⃗⃗⃗ = 2î - 3ĵ + k̂ and OQ⃗⃗⃗⃗⃗⃗ = -2î + ĵ + k̂. If the length of the vector OR⃗⃗⃗⃗⃗ is 𝛼 √3, then 𝛼 is ________.

Options:

  • (A) 3
  • (B) 4
  • (C) 5
  • (D) 6

Official Answer Key:

B

Step-by-Step Solution & Explanation

### **Official Solution & Concept Analysis (GATE BT 2024 Q.32)** **1. Governing Concept**: - **Topic**: Vector Calculus & Cross Product - Vector $\vec{OR}$ perpendicular to two non-parallel vectors $\vec{OP} = 2\hat{i} - 3\hat{j} + \hat{k}$ and $\vec{OQ} = -2\hat{i} + \hat{j} + \hat{k}$ is along their cross product $\vec{OP} \times \vec{OQ}$: $\vec{OP} \times \vec{OQ} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -3 & 1 \\ -2 & 1 & 1 \end{vmatrix}$ $= \hat{i}(-3 - 1) - \hat{j}(2 - (-2)) + \hat{k}(2 - 6) = -4\hat{i} - 4\hat{j} - 4\hat{k}$ - Magnitude or specified scalar projection evaluates directly to 4. **2. Option Analysis**: - **Option (A) [3] is INCORRECT**: Calculation error. - **Option (B) [4] is CORRECT**: Exactly matches vector component/magnitude factor 4. - **Option (C) [5] is INCORRECT**: Arithmetic error. - **Option (D) [6] is INCORRECT**: Arithmetic error. **3. Final Answer**: - **Correct Choice**: **B**

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