The angle between Vectors `(vec(A)xxvec(B))` and `(vec(B)xxvec(A))` is

Options

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

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 the unit vector perpendicular to the plane formed by \(\vec{A}\) and \(\vec{B}\).

### Step 2: Determine the Direction of the Cross Products
- The vector \(\vec{A} \times \vec{B}\) is directed according to the right-hand rule and is perpendicular to the plane formed by \(\vec{A}\) and \(\vec{B}\).
- The vector \(\vec{B} \times \vec{A}\) is also directed according to the right-hand rule, but since the order of the vectors is reversed, it will point in the opposite direction. Thus, we can express it as:
\[ \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \]

### Step 3: Find the Angle Between the Two Vectors
The angle \(\phi\) between two vectors \(\vec{X}\) and \(\vec{Y}\) can be found using the formula:
\[ \cos(\phi) = \frac{\vec{X} \cdot \vec{Y}}{|\vec{X}| |\vec{Y}|} \]
In our case, we have:
\[ \vec{X} = \vec{A} \times \vec{B} \quad \text{and} \quad \vec{Y} = \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \]
Thus, we can say:
\[ \vec{Y} = -\vec{X} \]

### Step 4: Calculate the Angle
Since \(\vec{Y}\) is the negative of \(\vec{X}\), the angle between them is:
\[ \phi = \pi \text{ radians} \quad \text{(or 180 degrees)} \]

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

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