If A=B , then which of the following is not correct ?

Options

A. `hatA=hatB`
B. `|A|=|B|`
C. `AhatB=BhatA`
D. `A+B=hatA+hatB`

Answer

To solve the question, we need to analyze the statements given in the context of vector equality. The question states that if \( A = B \), we need to identify which of the following statements is not correct.

### Step-by-Step Solution:

1. **Understanding Vector Equality**:
- Given \( A = B \), it implies that both the magnitudes and directions of vectors \( A \) and \( B \) are equal.
- Mathematically, this can be expressed as:
\[ |A| = |B| \quad \text{and} \quad \text{Direction of } A = \text{Direction of } B \]

2. **Analyzing the Statements**:
- **Statement 1**: \( A \cap = B \cap \)
- Here, \( A \cap \) and \( B \cap \) represent the unit vectors in the direction of \( A \) and \( B \) respectively. Since the directions are the same and magnitudes are equal, this statement is correct.

- **Statement 2**: \( |A| = |B| \)
- This statement is directly derived from the given condition \( A = B \). Hence, this statement is also correct.

- **Statement 3**: \( A \cdot B \cap = B \cdot A \cap \)
- This statement can be interpreted as the scalar product of vector \( A \) with the unit vector of \( B \) being equal to the scalar product of vector \( B \) with the unit vector of \( A \). Since both vectors are equal, this statement holds true as well.

- **Statement 4**: \( A + B = A \cap + B \cap \)
- This statement suggests that the sum of the vectors \( A \) and \( B \) is equal to the sum of their unit vectors. However, this is incorrect because the left side represents the vector addition of \( A \) and \( B \) (which has a certain magnitude), while the right side represents the addition of unit vectors (which always has a magnitude of 1). Therefore, this statement is not correct.

3. **Conclusion**:
- The statement that is not correct is:
\[ A + B \neq A \cap + B \cap \]

### Final Answer:
The statement that is not correct is \( A + B = A \cap + B \cap \).