For vectors `vec(A) and vec(B)` making an angle `theta` which one of the following relation is correct?

Options

A. `vec(A)xxvec(B)=vec(B)xxvec(A)`
B. `vec(A)xxvec(B)=ABsintheta`
C. `vec(A)xxvec(B)=ABcostheta`
D. `vec(A)xxvec(B)=-vec(B)xxvec(A)`

Answer

To solve the problem regarding the relationship between two vectors \(\vec{A}\) and \(\vec{B}\) making an angle \(\theta\), we will analyze the options provided and determine which one is correct.

### Step-by-Step Solution:

1. **Understanding Vector 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 \(\hat{n}\) is the unit vector perpendicular to the plane formed by \(\vec{A}\) and \(\vec{B}\), and \(\theta\) is the angle between the two vectors.

2. **Analyzing the Options**:
- **Option 1**: \(\vec{A} \times \vec{B} = \vec{B} \times \vec{A}\)
- This is incorrect because the cross product is anti-commutative. Therefore, \(\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})\).

- **Option 2**: \(\vec{A} \times \vec{B} = AB \sin(\theta)\)
- This is partially correct as it gives the magnitude of the cross product, but it does not account for direction. The correct expression should include the direction as well, hence this option is also incorrect.

- **Option 3**: \(\vec{A} \times \vec{B} = AB \cos(\theta)\)
- This is incorrect because the cross product involves \(\sin(\theta)\), not \(\cos(\theta)\).

- **Option 4**: \(\vec{A} \times \vec{B} = -(\vec{B} \times \vec{A})\)
- This is correct. It reflects the anti-commutative property of the cross product, where the direction of \(\vec{A} \times \vec{B}\) is opposite to that of \(\vec{B} \times \vec{A}\).

3. **Conclusion**:
The correct relation among the options given is:
\[ \vec{A} \times \vec{B} = -(\vec{B} \times \vec{A}) \]