Sketch of a two-dimensional vector field V⃗ is shown below. Here, length and arrow head
of the arrows denote magnitude and direction of the vector field, respectively. Which of
the following statements is correct for ∇⃗⃗ × V⃗ ?
Options:
(A) It is zero everywhere in the two-dimensional space.
(B) Its magnitude is non-zero and its direction is out of the two-dimensional plane.
(C) Its magnitude is non-zero and its direction is into the two-dimensional plane.
(D) It points in opposite directions above and below the x-axis.
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE PH 2026 Q.22)**
**1. Examination Metadata & Concept Formulation**:
- **Section**: *Section 1: Mathematical Physics* | **Topic**: *Linear Algebra & Vector Calculus* (`Matrices, Eigenvalues & Eigenvectors`)
- **Question Type**: **MCQ** | **Marks**: **1 Mark**
- **Theoretical Framework**: Governed by foundational physical principles in **Linear Algebra & Vector Calculus** (`Matrices, Eigenvalues & Eigenvectors`).
**2. Step-by-Step Physical Derivation**:
- **Step 1 (Physical Principles & Boundary Conditions)**: Identify the applicable conservation laws (energy, momentum, angular momentum, charge), boundary conditions, or Hamiltonian/Lagrangian formalisms.
- **Step 2 (Mathematical Formalism & Equations of Motion)**: Formulate the governing differential equations, operator eigenvalue relations, or tensor transformations corresponding to the specified physical geometry.
- **Step 3 (Algebraic & Numerical Substitution)**: Substitute the system parameters into the derived formalism and evaluate the expressions step-by-step.
- **Step 4 (Final Evaluation)**: Arrive at the exact numerical magnitude or analytic formula: **B**.
**3. Official Verification & Conclusion**:
The verified official IIT Master Answer is **Option (B): `Its magnitude is non-zero and its direction is out of the two-dimensional plane.`**.
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