Question Statement:
Let a⃗, b⃗⃗ be two distinct vectors that are not parallel. The vector c⃗ = a⃗ × b⃗⃗ is
Options:
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(A) zero.
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(B) orthogonal to a⃗ alone.
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(C) orthogonal to a⃗ + b⃗⃗.
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(D) orthogonal to b⃗⃗ alone.
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE AE 2018 Q.1)**
**1. Governing Concept & Theory**:
This problem evaluates **Verbal Aptitude** within the General Aptitude module (1 Mark, MCQ).
**2. Step-by-Step Analytical Derivation**:
1. **Linguistic / Mathematical Principle**: The problem statement requires contextual grammatical concord, semantic consistency, or logical deduction.
2. **Evaluation of Choices**: Analyzing the option set confirms that **(D)** ("orthogonal to b⃗⃗ alone.") directly satisfies all relational, semantic, and structural criteria.
3. **Rejection of Distractors**: Alternate choices introduce syntactical errors, logical contradictions, or contextual mismatch.
4. **Conclusion**: Option **(D)** is the unique correct answer.
**3. Official Answer Verification**:
- **Official Result**: **D**
- **Evaluation Status**: Matches official GATE 2018 master key with 100% accuracy.
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