If `vecAxxvecB=vecC` then find out the correct one :-

Options

A. `vecA.vecB=0`
B. `vecA.vecCne0`
C. `vecB.vecCne0`
D. `(vecA+vecB).vecC=0`

Answer

To solve the problem, we need to analyze the given information about the vectors A, B, and C, where \( \vec{A} \times \vec{B} = \vec{C} \).

### Step-by-Step Solution:

1. **Understanding the Cross Product**:
The cross product \( \vec{A} \times \vec{B} \) results in a vector \( \vec{C} \) that is perpendicular to both \( \vec{A} \) and \( \vec{B} \). Therefore, we can conclude:
\[ \vec{C} \perp \vec{A} \quad \text{and} \quad \vec{C} \perp \vec{B} \]

2. **Dot Product of Perpendicular Vectors**:
The dot product of two perpendicular vectors is zero. Hence, we can write:
\[ \vec{A} \cdot \vec{C} = 0 \quad \text{and} \quad \vec{B} \cdot \vec{C} = 0 \]

3. **Analyzing the Options**:
- **Option A**: \( \vec{A} \cdot \vec{B} = 0 \)
- This option states that \( \vec{A} \) and \( \vec{B} \) are perpendicular. However, we cannot conclude this from the information given. The only thing we know is that \( \vec{C} \) is perpendicular to both \( \vec{A} \) and \( \vec{B} \). Therefore, this option is **incorrect**.

- **Option B**: \( \vec{C} \cdot \vec{A} = 0 \)
- This option is true since we established that \( \vec{C} \) is perpendicular to \( \vec{A} \).

- **Option C**: \( \vec{C} \cdot \vec{B} = 0 \)
- This option is also true since \( \vec{C} \) is perpendicular to \( \vec{B} \).

- **Option D**: \( \vec{A} + \vec{B} \cdot \vec{C} = 0 \)
- This can be rewritten as \( \vec{A} \cdot \vec{C} + \vec{B} \cdot \vec{C} = 0 \). Since both \( \vec{A} \cdot \vec{C} = 0 \) and \( \vec{B} \cdot \vec{C} = 0 \), their sum is also zero. Thus, this option is **correct**.

4. **Conclusion**:
- The correct options based on the analysis are \( \vec{C} \cdot \vec{A} = 0 \), \( \vec{C} \cdot \vec{B} = 0 \), and \( \vec{A} + \vec{B} \cdot \vec{C} = 0 \).
- Therefore, the correct answer is **Option D**.