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 problem, we need to analyze the statements that can be derived from the condition that two vectors A and B are equal (A = B).

### Step-by-Step Solution:

1. **Understanding Vector Equality**:
- Two vectors A and B are equal if and only if their magnitudes and directions are the same.
- Mathematically, this can be stated as:
\[ A = B \implies |A| = |B| \text{ and } \text{direction of } A = \text{direction of } B \]

2. **Analyzing the Statements**:
- We need to evaluate the following statements to find out which one is incorrect:
1. \( |A| = |B| \)
2. \( A + B = 2A \)
3. \( A \cdot B = |A| |B| \)
4. \( A + B \neq 0 \)

3. **Evaluating Each Statement**:
- **Statement 1**: \( |A| = |B| \)
- This is correct since A and B are equal.
- **Statement 2**: \( A + B = 2A \)
- Since \( A = B \), we can substitute B with A:
\[ A + B = A + A = 2A \]
- This statement is correct.
- **Statement 3**: \( A \cdot B = |A| |B| \)
- The dot product of two equal vectors A and B can be expressed as:
\[ A \cdot B = |A| |B| \cos(\theta) \]
where \( \theta \) is the angle between A and B. Since A and B are equal, \( \theta = 0 \) and \( \cos(0) = 1 \):
\[ A \cdot B = |A| |B| \implies A \cdot A = |A|^2 \]
- This statement is also correct.
- **Statement 4**: \( A + B \neq 0 \)
- Since \( A = B \), we have:
\[ A + B = A + A = 2A \]
- If A is a non-zero vector, then \( 2A \neq 0 \). However, if A is the zero vector, then \( A + B = 0 \). Therefore, this statement is not universally true.

4. **Conclusion**:
- The statement that is not correct is:
\[ A + B \neq 0 \]
- Thus, the incorrect statement is **Statement 4**.