Three vectors satisfy the relation A.B =0 and A.C=0 then A is parallel to

Options

A. C
B. B
C. `BxxC`
D. B.C

Explanation

To solve the problem, we need to determine which vector \( \mathbf{A} \) is parallel to, given the conditions \( \mathbf{A} \cdot \mathbf{B} = 0 \) and \( \mathbf{A} \cdot \mathbf{C} = 0 \).

### Step-by-Step Solution:

1. **Understand the Dot Product Condition:**
- Given \( \mathbf{A} \cdot \mathbf{B} = 0 \), this implies that vector \( \mathbf{A} \) is perpendicular to vector \( \mathbf{B} \).
- Given \( \mathbf{A} \cdot \mathbf{C} = 0 \), this implies that vector \( \mathbf{A} \) is perpendicular to vector \( \mathbf{C} \).

2. **Visualize the Perpendicularity:**
- Since \( \mathbf{A} \) is perpendicular to both \( \mathbf{B} \) and \( \mathbf{C} \), \( \mathbf{A} \) must lie along the direction that is perpendicular to the plane formed by \( \mathbf{B} \) and \( \mathbf{C} \).

3. **Use the Cross Product:**
- The cross product \( \mathbf{B} \times \mathbf{C} \) gives a vector that is perpendicular to both \( \mathbf{B} \) and \( \mathbf{C} \).
- Therefore, \( \mathbf{A} \) must be parallel to \( \mathbf{B} \times \mathbf{C} \).

4. **Conclusion:**
- Since \( \mathbf{A} \) is perpendicular to both \( \mathbf{B} \) and \( \mathbf{C} \), and the vector \( \mathbf{B} \times \mathbf{C} \) is also perpendicular to both \( \mathbf{B} \) and \( \mathbf{C} \), \( \mathbf{A} \) is parallel to \( \mathbf{B} \times \mathbf{C} \).

### Final Answer:
\( \mathbf{A} \) is parallel to \( \mathbf{B} \times \mathbf{C} \).