The angle between vectors `(AxxB)and(BxxA)` is

Options

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

Explanation

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

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

### Step 2: Determine the Direction of Each Cross Product
Using the right-hand rule:
- For \( \mathbf{A} \times \mathbf{B} \), point your fingers in the direction of \( \mathbf{A} \) and curl them towards \( \mathbf{B} \). Your thumb will point in the direction of \( \mathbf{A} \times \mathbf{B} \).
- For \( \mathbf{B} \times \mathbf{A} \), point your fingers in the direction of \( \mathbf{B} \) and curl them towards \( \mathbf{A} \). Your thumb will point in the direction of \( \mathbf{B} \times \mathbf{A} \).

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

### Step 4: Calculate the Angle Between the Two Vectors
Since \( \mathbf{B} \times \mathbf{A} \) is in the opposite direction to \( \mathbf{A} \times \mathbf{B} \), the angle \( \phi \) between them is:
\[ \phi = 180^\circ \text{ (or } \pi \text{ radians)} \]

### Conclusion
Thus, the angle between the vectors \( \mathbf{A} \times \mathbf{B} \) and \( \mathbf{B} \times \mathbf{A} \) is \( 180^\circ \).

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