The angle between `veca xx vecb and vecb xx veca` is

Options

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

Answer

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 Properties
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 \( \vec{A} \) and \( \vec{B} \), and \( \hat{n} \) is the unit vector perpendicular to the plane containing \( \vec{A} \) and \( \vec{B} \).

### Step 2: Apply the Anti-Symmetry Property
The property of the cross product states that:
\[ \vec{A} \times \vec{B} = -(\vec{B} \times \vec{A}) \]
This means that \( \vec{B} \times \vec{A} \) is equal in magnitude to \( \vec{A} \times \vec{B} \) but points in the opposite direction.

### Step 3: Find the Angle Between the Two Cross Products
Let’s denote:
\[ \vec{C} = \vec{A} \times \vec{B} \]
\[ \vec{D} = \vec{B} \times \vec{A} \]
From the property established in Step 2, we have:
\[ \vec{D} = -\vec{C} \]

### Step 4: Determine the Angle
The angle \( \phi \) between two vectors \( \vec{C} \) and \( \vec{D} \) can be found using the dot product:
\[ \vec{C} \cdot \vec{D} = |\vec{C}||\vec{D}|\cos(\phi) \]
Substituting \( \vec{D} = -\vec{C} \):
\[ \vec{C} \cdot (-\vec{C}) = -|\vec{C}|^2 \]
Thus:
\[ -|\vec{C}|^2 = |\vec{C}||\vec{D}|\cos(\phi) \]
Since \( |\vec{D}| = |\vec{C}| \):
\[ -|\vec{C}|^2 = |\vec{C}|^2 \cos(\phi) \]
Dividing both sides by \( |\vec{C}|^2 \) (assuming \( |\vec{C}| \neq 0 \)):
\[ -1 = \cos(\phi) \]
This implies:
\[ \phi = \pi \]

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

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

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