The angle between vectors `(vec(M)xx vec(N))` and `(vec(N)xx vec(M))` is

Options

A. `0^(@)`
B. `60^(@)`
C. `90^(@)`
D. `180^(@)`

Answer

To find the angle between the vectors \( \vec{M} \times \vec{N} \) and \( \vec{N} \times \vec{M} \), we can follow these steps:

### Step-by-Step Solution:

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

2. **Calculating \( \vec{M} \times \vec{N} \)**:
When we compute \( \vec{M} \times \vec{N} \), the resulting vector is perpendicular to both \( \vec{M} \) and \( \vec{N} \).

3. **Calculating \( \vec{N} \times \vec{M} \)**:
The cross product \( \vec{N} \times \vec{M} \) can be rewritten using the property of cross products:
\[ \vec{N} \times \vec{M} = -(\vec{M} \times \vec{N}) \]
This means that \( \vec{N} \times \vec{M} \) is in the opposite direction to \( \vec{M} \times \vec{N} \).

4. **Finding the Angle Between the Two Vectors**:
Since \( \vec{N} \times \vec{M} \) is in the opposite direction of \( \vec{M} \times \vec{N} \), the angle between these two vectors is:
\[ \theta = 180^\circ \]

5. **Conclusion**:
Therefore, the angle between the vectors \( \vec{M} \times \vec{N} \) and \( \vec{N} \times \vec{M} \) is \( 180^\circ \).

### Final Answer:
The angle between the vectors \( \vec{M} \times \vec{N} \) and \( \vec{N} \times \vec{M} \) is \( 180^\circ \).