If `vec(A)` is perpendicular to `vec(B)`, then

Options

A. `vec(A)xxvec(B)=0`
B. `vec(A).[vec(A)+vec(B)]=A^(2)`
C. `vec(A).vec(B)=AB`
D. `vec(A).[vec(A)+vec(B)]=A^(2)+AB`

Explanation

To solve the problem, we need to analyze the implications of the statement "If vector A is perpendicular to vector B."

### Step-by-Step Solution:

1. **Understanding Perpendicular Vectors**:
- When two vectors A and B are perpendicular, it means that the angle θ between them is 90 degrees.

2. **Using the Dot Product**:
- The dot product of two vectors A and B is given by:
\[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos(\theta) \]
- Since θ = 90 degrees, we have:
\[ \cos(90^\circ) = 0 \]
- Therefore,
\[ \vec{A} \cdot \vec{B} = 0 \]

3. **Analyzing the Options**:
- **Option 1**: \( \vec{A} \times \vec{B} = 0 \)
- This is incorrect because the cross product of two perpendicular vectors is not zero; it is equal to the product of their magnitudes and the sine of the angle between them.
- **Option 2**: \( \vec{A} \cdot \vec{A} + \vec{B} = |\vec{A}|^2 \)
- This option is incorrect as it does not hold true in general.
- **Option 3**: \( \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \)
- This is incorrect as we established that \( \vec{A} \cdot \vec{B} = 0 \).
- **Option 4**: \( \vec{A} + \vec{B} = |\vec{A}|^2 + |\vec{B}| \)
- This is also incorrect.

4. **Conclusion**:
- The only correct statement derived from the condition that vector A is perpendicular to vector B is:
\[ \vec{A} \cdot \vec{B} = 0 \]
- Therefore, none of the provided options are correct based on the standard definitions of vector operations.

### Final Answer:
None of the options are correct based on the condition that vector A is perpendicular to vector B.