Let the pair of vector `veca,vecb` and `vecc,veccd` each determine a plane. Then the planes are parallel if

Options

A. `(vecaxxvecc)xx(vecbxxvecd)=vec0`
B. `(vecaxxvecc).(vecbxxvecd)=vec0`
C. `(vecaxxvecc)xx(veccxxvecd)=vec0`
D. `(vecaxxvecc).(veccxxvecd)=vec0`

Explanation

To determine the condition under which two planes defined by the pairs of vectors \(\vec{a}, \vec{b}\) and \(\vec{c}, \vec{d}\) are parallel, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Planes**:
- The vectors \(\vec{a}\) and \(\vec{b}\) define the first plane.
- The vectors \(\vec{c}\) and \(\vec{d}\) define the second plane.

2. **Finding Normal Vectors**:
- The normal vector to the plane defined by vectors \(\vec{a}\) and \(\vec{b}\) can be found using the cross product:
\[ \vec{N_1} = \vec{a} \times \vec{b} \]
- Similarly, the normal vector to the plane defined by vectors \(\vec{c}\) and \(\vec{d}\) is:
\[ \vec{N_2} = \vec{c} \times \vec{d} \]

3. **Condition for Parallel Planes**:
- For two planes to be parallel, their normal vectors must be parallel. This occurs if the angle between the two normal vectors is either \(0^\circ\) or \(180^\circ\).
- Mathematically, this means:
\[ \vec{N_1} \times \vec{N_2} = \vec{0} \]
- This condition can also be expressed in terms of the magnitudes and the sine of the angle between the normal vectors:
\[ |\vec{N_1}| |\vec{N_2}| \sin \theta = 0 \]
- Here, \(\theta\) is the angle between the normal vectors \(\vec{N_1}\) and \(\vec{N_2}\).

4. **Conclusion**:
- Therefore, for the planes defined by the vectors \(\vec{a}, \vec{b}\) and \(\vec{c}, \vec{d}\) to be parallel, the condition is:
\[ \vec{a} \times \vec{b} \text{ is parallel to } \vec{c} \times \vec{d} \]
- This can be simplified to:
\[ \vec{a} \times \vec{b} = k (\vec{c} \times \vec{d}) \text{ for some scalar } k \]