Three vector `vec(A)`,`vec(B)`, `vec(C )` satisfy the relation `vec(A)*vec(B)=0`and `vec(A).vec(C )=0`. The vector `vec(A)` is parallel to

Options

A. `vec(b)`
B. `vec(c )`
C. `vec(b).vec(c )`
D. `vec(B) xx vec(C )`

Answer

To solve the problem, we need to analyze the relationships between the vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) given the conditions \(\vec{A} \cdot \vec{B} = 0\) and \(\vec{A} \cdot \vec{C} = 0\).

### Step-by-Step Solution:

1. **Understanding the Dot Product**:
The dot product of two vectors \(\vec{P}\) and \(\vec{Q}\) is given by:
\[ \vec{P} \cdot \vec{Q} = |\vec{P}| |\vec{Q}| \cos(\theta) \]
where \(\theta\) is the angle between the two vectors. If the dot product is zero, it implies that \(\cos(\theta) = 0\), which occurs when \(\theta = 90^\circ\). This means the vectors are perpendicular.

2. **Applying the Conditions**:
- From \(\vec{A} \cdot \vec{B} = 0\), we conclude that \(\vec{A}\) is perpendicular to \(\vec{B}\).
- From \(\vec{A} \cdot \vec{C} = 0\), we conclude that \(\vec{A}\) is also perpendicular to \(\vec{C}\).

3. **Analyzing the Relationships**:
Since \(\vec{A}\) is perpendicular to both \(\vec{B}\) and \(\vec{C}\), it cannot be parallel to either of these vectors.

4. **Finding the Direction of \(\vec{A}\)**:
The vector \(\vec{A}\) is perpendicular to the plane formed by \(\vec{B}\) and \(\vec{C}\). The direction of \(\vec{A}\) can be represented as being parallel to the cross product of \(\vec{B}\) and \(\vec{C}\):
\[ \vec{A} \parallel \vec{B} \times \vec{C} \]
The cross product \(\vec{B} \times \vec{C}\) gives a vector that is perpendicular to both \(\vec{B}\) and \(\vec{C}\).

5. **Conclusion**:
Therefore, the vector \(\vec{A}\) is parallel to \(\vec{B} \times \vec{C}\).

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

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