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: Calculate \( \vec{B} \times \vec{A} \)
Using the property of the cross product, we know:
\[ \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \]
This means that \( \vec{B} \times \vec{A} \) is equal to the negative of \( \vec{A} \times \vec{B} \).

### Step 3: Determine the Relationship Between the Two Vectors
Since \( \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \), the two vectors \( \vec{A} \times \vec{B} \) and \( \vec{B} \times \vec{A} \) are antiparallel. Antiparallel vectors point in opposite directions.

### Step 4: Find the Angle Between Antiparallel Vectors
The angle between two antiparallel vectors is \( \pi \) radians (or 180 degrees). This is because they are in opposite directions.

### Conclusion
Thus, the angle between the vectors \( \vec{A} \times \vec{B} \) and \( \vec{B} \times \vec{A} \) is:
\[ \pi \text{ radians} \]

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

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