If `barA xx barB =barB xx barA` then the angle between A and B is :

Options

A. `pi`
B. `pi//3`
C. `pi//2`
D. `pi//4`

Explanation

To solve the question "If \( \bar{A} \times \bar{B} = \bar{B} \times \bar{A} \), then the angle between \( \bar{A} \) and \( \bar{B} \) is:", we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Cross Product**:
The cross product of two vectors \( \bar{A} \) and \( \bar{B} \) is given by:
\[ \bar{A} \times \bar{B} = |\bar{A}| |\bar{B}| \sin(\theta) \hat{n} \]
where \( \theta \) is the angle between the two vectors and \( \hat{n} \) is the unit vector perpendicular to the plane formed by \( \bar{A} \) and \( \bar{B} \).

2. **Using the Property of Cross Products**:
The property of cross products states that:
\[ \bar{A} \times \bar{B} = -(\bar{B} \times \bar{A}) \]
This implies that \( \bar{A} \times \bar{B} \) is equal to \( \bar{B} \times \bar{A} \) only when the sine component is zero.

3. **Setting Up the Equation**:
Given \( \bar{A} \times \bar{B} = \bar{B} \times \bar{A} \), we can write:
\[ |\bar{A}| |\bar{B}| \sin(\theta) = -|\bar{B}| |\bar{A}| \sin(\theta) \]

4. **Simplifying the Equation**:
Since \( |\bar{A}| \) and \( |\bar{B}| \) are non-zero (assuming the vectors are not zero vectors), we can cancel these terms from both sides:
\[ \sin(\theta) = -\sin(\theta) \]

5. **Solving for \( \theta \)**:
This leads to:
\[ 2\sin(\theta) = 0 \]
Therefore, \( \sin(\theta) = 0 \).

6. **Finding the Angles**:
The sine function is zero at:
\[ \theta = n\pi \quad (n \in \mathbb{Z}) \]
For the angle between two vectors, we consider \( \theta = 0 \) or \( \theta = \pi \).

7. **Conclusion**:
Since the question asks for the angle between \( \bar{A} \) and \( \bar{B} \), the possible angles are \( 0 \) or \( \pi \). However, since the vectors must be in opposite directions for the cross product to be zero, the angle between them is:
\[ \theta = \pi \]

### Final Answer:
The angle between \( \bar{A} \) and \( \bar{B} \) is \( \pi \) radians.

---