The angle between Vectors `(vec(A)xxvec(B))` and `(vec(B)xxvec(A))` is

Options

A. zero
B. `pi`
C. `(pi)//(4)`
D. `(pi)/(2)`

Explanation

To find the angle between the vectors \( \vec{A} \times \vec{B} \) and \( \vec{B} \times \vec{A} \), we can follow these steps:

### Step-by-Step Solution:

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

2. **Direction of the Cross Products**:
- The vector \( \vec{A} \times \vec{B} \) is perpendicular to both \( \vec{A} \) and \( \vec{B} \) and follows the right-hand rule.
- Conversely, the vector \( \vec{B} \times \vec{A} \) is also perpendicular to both vectors but points in the opposite direction. This is because:
\[ \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \]

3. **Finding the Angle Between the Two Cross Products**:
Since \( \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \), it implies that the two vectors \( \vec{A} \times \vec{B} \) and \( \vec{B} \times \vec{A} \) are in opposite directions.

4. **Calculating the Angle**:
The angle \( \phi \) between two vectors \( \vec{X} \) and \( \vec{Y} \) can be calculated using the dot product:
\[ \cos(\phi) = \frac{\vec{X} \cdot \vec{Y}}{|\vec{X}| |\vec{Y}|} \]
However, since \( \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \), we know that:
\[ \phi = 180^\circ \text{ or } \pi \text{ radians} \]

5. **Conclusion**:
Therefore, the angle between the vectors \( \vec{A} \times \vec{B} \) and \( \vec{B} \times \vec{A} \) is:
\[ \phi = 180^\circ \text{ or } \pi \text{ radians} \]

### Final Answer:
The angle between \( \vec{A} \times \vec{B} \) and \( \vec{B} \times \vec{A} \) is \( \pi \) radians (or 180 degrees).

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