Let $\vec{a}, \vec{b}$ be two distinct vectors that are not parallel. The vector $\vec{c} = \vec{a} \times \vec{b}$ is:
Options:
(A) zero.
(B) orthogonal to $\vec{a}$ alone.
(C) orthogonal to $\vec{a} + \vec{b}$.
(D) orthogonal to $\vec{b}$ alone.
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2018 Q.11 / AE Q.1)**
**1. Intuitive Physical Setup & Given Parameters**:
- Let $\vec{a}$ and $\vec{b}$ be two distinct non-zero, non-parallel vectors in $\mathbb{R}^3$.
- Vector $\vec{c} = \vec{a} \times \vec{b}$.
**2. Step-by-Step Mathematical Derivation**:
1. **Properties of Cross Product**:
By definition of the vector cross product, $\vec{c} = \vec{a} \times \vec{b}$ is perpendicular (orthogonal) to the plane containing both $\vec{a}$ and $\vec{b}$.
Therefore:
$\vec{c} \cdot \vec{a} = 0 \quad \text{and} \quad \vec{c} \cdot \vec{b} = 0$
2. **Orthogonality to the Sum $\vec{a} + \vec{b}$**:
Consider the dot product of $\vec{c}$ with $(\vec{a} + \vec{b})$:
$\vec{c} \cdot (\vec{a} + \vec{b}) = (\vec{c} \cdot \vec{a}) + (\vec{c} \cdot \vec{b}) = 0 + 0 = 0$
Since the dot product is identically zero, $\vec{c}$ is orthogonal to $\vec{a} + \vec{b}$.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *zero*: ❌ Incorrect. Since $\vec{a}$ and $\vec{b}$ are not parallel, $\vec{a} \times \vec{b} \neq 0$.
- **(B)** *orthogonal to $\vec{a}$ alone*: ❌ Incorrect. The word 'alone' is false; $\vec{c}$ is orthogonal to *both* $\vec{a}$ and $\vec{b}$.
- **(C)** *orthogonal to $\vec{a} + \vec{b}$*: ✅ **CORRECT**. Directly proved via linearity of dot product: $\vec{c}\cdot(\vec{a}+\vec{b}) = 0$.
- **(D)** *orthogonal to $\vec{b}$ alone*: ❌ Incorrect. Again, 'alone' is incorrect.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **C**
- **Exam Rule**: If a vector is perpendicular to two vectors, it is perpendicular to any linear combination of those two vectors.
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