Given that `veca.vecb=0 and veca xx vecb=0`. What can you conclude about the vectors `veca and vecb`.

Options

A. `veca bot vecb`
B. `veca||vecb`
C. `veca=0`
D. Either `|veca|=0 or |vecb|=0`

Answer

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

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

2. **Understanding the Dot Product**:
The dot product \( \vec{a} \cdot \vec{b} \) is defined as:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta \]
where \( \theta \) is the angle between the two vectors. Since \( \vec{a} \cdot \vec{b} = 0 \), we can conclude that:
\[ |\vec{a}| |\vec{b}| \cos \theta = 0 \]
This implies that either:
- \( |\vec{a}| = 0 \) (i.e., \( \vec{a} \) is the zero vector)
- \( |\vec{b}| = 0 \) (i.e., \( \vec{b} \) is the zero vector)
- \( \cos \theta = 0 \) (i.e., \( \theta = 90^\circ \), meaning \( \vec{a} \) and \( \vec{b} \) are perpendicular)

3. **Understanding the Cross Product**:
The cross product \( \vec{a} \times \vec{b} \) is defined as:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta \]
Since \( \vec{a} \times \vec{b} = 0 \), we have:
\[ |\vec{a}| |\vec{b}| \sin \theta = 0 \]
This implies that either:
- \( |\vec{a}| = 0 \) (i.e., \( \vec{a} \) is the zero vector)
- \( |\vec{b}| = 0 \) (i.e., \( \vec{b} \) is the zero vector)
- \( \sin \theta = 0 \) (i.e., \( \theta = 0^\circ \) or \( \theta = 180^\circ \), meaning \( \vec{a} \) and \( \vec{b} \) are parallel)

4. **Conclusion**:
From the conditions derived from both the dot product and the cross product, we have:
- From \( \vec{a} \cdot \vec{b} = 0 \): \( \vec{a} \) and \( \vec{b} \) are either perpendicular, or one of them is the zero vector.
- From \( \vec{a} \times \vec{b} = 0 \): \( \vec{a} \) and \( \vec{b} \) are either parallel, or one of them is the zero vector.

The only way for both conditions to hold true simultaneously is if at least one of the vectors is the zero vector. Therefore, we can conclude:
\[ \text{Either } \vec{a} = \vec{0} \text{ or } \vec{b} = \vec{0} \]