The angle between vector `(overset(rarr)Axxoverset(rarr)B)` and `(overset(rarr)B xx overset(rarr)A)` is :

Options

A. zero
B. `pi`
C. `PI//4`
D. `PI//2`

Answer

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

### Step 1: Understand 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 \( \vec{B} \).

### Step 2: Determine the Direction of the Cross Products
1. The vector \( \vec{A} \times \vec{B} \) is directed out of the plane formed by \( \vec{A} \) and \( \vec{B} \).
2. The vector \( \vec{B} \times \vec{A} \) is directed into the plane formed by \( \vec{A} \) and \( \vec{B} \).

### Step 3: Analyze the Relationship Between the Two Vectors
From the properties of the cross product, we know:
\[ \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \]
This indicates that \( \vec{B} \times \vec{A} \) is in the opposite direction to \( \vec{A} \times \vec{B} \).

### Step 4: Calculate the Angle Between the Two Vectors
Since \( \vec{A} \times \vec{B} \) points out of the plane and \( \vec{B} \times \vec{A} \) points into the plane, the angle \( \phi \) between these two vectors is:
\[ \phi = 180^\circ = \pi \text{ radians} \]

### Conclusion
The angle between the vectors \( \vec{A} \times \vec{B} \) and \( \vec{B} \times \vec{A} \) is \( \pi \) radians.

### Final Answer
The correct option is \( \pi \).

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