If `vec(a) and vec(b)` are two vectors such that `vec(a).vec(b) = 0 and vec(a) xx vec(b) = 0`, then which one of the following is correct?

Options

A. a) `vec(a)` is parallel to `vec(b)`
B. b) `vec(a)` is perpendicular to `vec(b)`
C. c) Either `vec(a) or vec(b)` is a null vector
D. d) None of the above

Answer

To solve the problem, we need to analyze the conditions given for the vectors \( \vec{a} \) and \( \vec{b} \):

1. **Given Conditions**:
- \( \vec{a} \cdot \vec{b} = 0 \)
- \( \vec{a} \times \vec{b} = 0 \)

2. **Understanding the Dot Product**:
- The dot product \( \vec{a} \cdot \vec{b} = 0 \) indicates that the vectors \( \vec{a} \) and \( \vec{b} \) are perpendicular to each other. This is a fundamental property of dot products.

3. **Understanding the Cross Product**:
- The cross product \( \vec{a} \times \vec{b} = 0 \) indicates that the vectors \( \vec{a} \) and \( \vec{b} \) are parallel to each other. This is also a fundamental property of cross products.

4. **Analyzing the Conditions Together**:
- We have established that \( \vec{a} \) and \( \vec{b} \) cannot be both perpendicular and parallel at the same time unless one of them is a zero vector.
- If either \( \vec{a} \) or \( \vec{b} \) is a zero vector, then both the dot product and the cross product would equal zero, satisfying both conditions.

5. **Conclusion**:
- Therefore, the only valid conclusion is that either \( \vec{a} \) or \( \vec{b} \) must be a null vector (zero vector).

Thus, the correct option is **Option C**: either \( \vec{a} \) or \( \vec{b} \) is a null vector.

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