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Vectors \( \mathbf{A}, \mathbf{B} \) and \( \mathbf{C} \) are such that \( \mathbf{A} \cdot \mathbf{B} = 0 \) and \( \mathbf{A} \cdot \mathbf{C} = 0 \). Then the vector parallel to \( \mathbf{A} \) is

1. \( \mathbf{B} \cdot \mathbf{C} \)
2. \( \mathbf{A} \times \mathbf{B} \)
3. \( \mathbf{B} + \mathbf{C} \)
4. \( \mathbf{B} \times \mathbf{C} \)

Correct answer: \( \mathbf{B} \times \mathbf{C} \)

Solution

Since \( \mathbf{A} \cdot \mathbf{B} = 0 \) and \( \mathbf{A} \cdot \mathbf{C} = 0 \), \( \mathbf{A} \) is perpendicular to both \( \mathbf{B} \) and \( \mathbf{C} \). The cross product \( \mathbf{B} \times \mathbf{C} \) is perpendicular to both \( \mathbf{B} \) and \( \mathbf{C} \), and hence is parallel to \( \mathbf{A} \).

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