The angle between `(bar(A) xx bar(B)) and (bar(B) xx bar(A))` is (in radiant)

Options

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

Answer

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

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

### Step 2: Apply the Anti-parallel Property
We know from the properties of the cross product that:
\[ \bar{B} \times \bar{A} = -(\bar{A} \times \bar{B}) \]
This means that the vector \( \bar{B} \times \bar{A} \) is in the opposite direction to \( \bar{A} \times \bar{B} \).

### Step 3: Determine the Angle Between the Two Vectors
Since \( \bar{A} \times \bar{B} \) and \( \bar{B} \times \bar{A} \) are in opposite directions, they are considered anti-parallel. The angle \( \phi \) between two anti-parallel vectors is:
\[ \phi = \pi \text{ radians} \quad (180^\circ) \]

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

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