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 problem where we need to find the angle between vectors A and B given that \( \bar{A} \times \bar{B} = \bar{B} \times \bar{A} \), we can follow these steps:

### Step-by-Step Solution:

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

2. **Using the Given Condition**:
The problem states that:
\[ \bar{A} \times \bar{B} = \bar{B} \times \bar{A} \]
From the properties of the cross product, we know that:
\[ \bar{B} \times \bar{A} = -(\bar{A} \times \bar{B}) \]
Therefore, we can rewrite the equation as:
\[ \bar{A} \times \bar{B} = -(\bar{A} \times \bar{B}) \]

3. **Setting Up the Equation**:
This implies:
\[ \bar{A} \times \bar{B} + \bar{A} \times \bar{B} = 0 \]
or:
\[ 2(\bar{A} \times \bar{B}) = 0 \]
This means:
\[ \bar{A} \times \bar{B} = 0 \]

4. **Interpreting the Result**:
The cross product of two vectors is zero if and only if the vectors are parallel. This can occur in two cases:
- When the angle \( \theta \) between them is \( 0^\circ \) (they point in the same direction).
- When the angle \( \theta \) between them is \( 180^\circ \) (they point in opposite directions).

5. **Conclusion**:
Since the problem asks for the angle between \( \bar{A} \) and \( \bar{B} \), we can conclude that:
\[ \theta = 0^\circ \text{ or } 180^\circ \]
Thus, the angle between vectors \( \bar{A} \) and \( \bar{B} \) can be expressed as:
\[ \theta = \pi \text{ radians} \text{ (or } 180^\circ\text{)} \]

### Final Answer:
The angle between vectors \( \bar{A} \) and \( \bar{B} \) is \( \pi \) radians (or \( 180^\circ \)).