If `vec(A) xx vec(B) = vec(B) xx vec(A)`, then the angle between `A to B` is

Options

A. `pi//2`
B. `pi//3`
C. `pi`
D. `pi//4`

Answer

To solve the problem, we need to analyze the given equation:

\[ \vec{A} \times \vec{B} = \vec{B} \times \vec{A} \]

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

2. **Properties of Cross Product**:
It is known that:
\[ \vec{A} \times \vec{B} = -(\vec{B} \times \vec{A}) \]
This means that the cross product is anti-commutative. Therefore, for the equation \(\vec{A} \times \vec{B} = \vec{B} \times \vec{A}\) to hold true, it must be that:
\[ \vec{A} \times \vec{B} = -(\vec{A} \times \vec{B}) \]
This implies:
\[ 2(\vec{A} \times \vec{B}) = 0 \]
Hence, \(\vec{A} \times \vec{B} = 0\).

3. **Condition for Zero Cross Product**:
The cross product of two vectors is zero when:
\[ |\vec{A}| |\vec{B}| \sin(\theta) = 0 \]
This can occur if either \(|\vec{A}| = 0\), \(|\vec{B}| = 0\), or \(\sin(\theta) = 0\). The sine function is zero when:
\[ \theta = 0 \text{ or } \theta = \pi \]
This means that the vectors are either in the same direction or in opposite directions.

4. **Conclusion**:
Since the problem asks for the angle between \(\vec{A}\) and \(\vec{B}\), and we have established that the only angle that satisfies the condition \(\vec{A} \times \vec{B} = \vec{B} \times \vec{A}\) is when \(\theta = \pi\) (180 degrees), we conclude that:

\[ \text{The angle between } \vec{A} \text{ and } \vec{B} \text{ is } \pi \text{ radians.} \]