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/2`
D. `pi/4`

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 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 a 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 \( \vec{A} \times \vec{B} \) will be directed according to the right-hand rule, which gives a direction perpendicular to both \( \vec{A} \) and \( \vec{B} \).
2. Conversely, the vector \( \vec{B} \times \vec{A} \) will point in the opposite direction, since:
\[ \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \]

### Step 3: Analyze the Relationship Between the Two Cross Products
From the above, we can conclude:
\[ \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \]
This means that the vector \( \vec{B} \times \vec{A} \) is in the exact opposite direction of \( \vec{A} \times \vec{B} \).

### Step 4: Calculate the Angle Between the Two Vectors
The angle \( \phi \) between two vectors \( \vec{C} \) and \( \vec{D} \) can be calculated using the dot product:
\[ \cos(\phi) = \frac{\vec{C} \cdot \vec{D}}{|\vec{C}| |\vec{D}|} \]
However, since \( \vec{D} = -\vec{C} \), we have:
\[ \vec{C} \cdot \vec{D} = \vec{C} \cdot (-\vec{C}) = -|\vec{C}|^2 \]
Thus:
\[ \cos(\phi) = \frac{-|\vec{C}|^2}{|\vec{C}| |\vec{D}|} = -1 \]
This implies that:
\[ \phi = 180^\circ \text{ or } \pi \text{ radians} \]

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