If `veca and vecb` are two vectors such that `veca.vecb = 0` and `veca xx vecb = vec0`, then

Options

A. `veca` is parallel to `vecb`
B. `veca` is perpendicular to `vecb`
C. either `veca` or `vecb` is a null vector
D. none of these

Answer

To solve the problem, we need to analyze the conditions given for the vectors \(\vec{a}\) and \(\vec{b}\):

1. **Given Conditions**:
- \(\vec{a} \cdot \vec{b} = 0\)
- \(\vec{a} \times \vec{b} = \vec{0}\)

2. **Understanding the Dot Product**:
The dot product of two vectors is given by:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \]
where \(\theta\) is the angle between the two vectors. Since \(\vec{a} \cdot \vec{b} = 0\), it implies:
\[ |\vec{a}| |\vec{b}| \cos(\theta) = 0 \]
This can happen if either:
- \(|\vec{a}| = 0\) (meaning \(\vec{a}\) is a null vector)
- \(|\vec{b}| = 0\) (meaning \(\vec{b}\) is a null vector)
- \(\cos(\theta) = 0\) (which means \(\theta = \frac{\pi}{2}\) or \(90^\circ\), indicating that the vectors are perpendicular)

3. **Understanding the Cross Product**:
The cross product of two vectors is given by:
\[ \vec{a} \times \vec{b} = |\vec{a}| |\vec{b}| \sin(\theta) \]
Since \(\vec{a} \times \vec{b} = \vec{0}\), it implies:
\[ |\vec{a}| |\vec{b}| \sin(\theta) = 0 \]
This can happen if either:
- \(|\vec{a}| = 0\) (meaning \(\vec{a}\) is a null vector)
- \(|\vec{b}| = 0\) (meaning \(\vec{b}\) is a null vector)
- \(\sin(\theta) = 0\) (which means \(\theta = 0\) or \(\theta = \pi\), indicating that the vectors are parallel)

4. **Combining the Conditions**:
From the dot product, we found that either \(\vec{a}\) or \(\vec{b}\) must be a null vector or they could be perpendicular. From the cross product, we found that either \(\vec{a}\) or \(\vec{b}\) must be a null vector or they could be parallel.

The only scenario that satisfies both conditions (dot product being zero and cross product being zero) simultaneously is when at least one of the vectors is a null vector.

5. **Conclusion**:
Therefore, the correct conclusion is that either \(\vec{a}\) or \(\vec{b}\) is a null vector.

### Final Answer:
Either \(\vec{a}\) or \(\vec{b}\) is a null vector.

---