mark the correct statement .

Options

A. `|veca + vecb | ge veca | + | vecb|`
B. `|veca + vecb|le |veca | vecb|`
C. `|veca - vecb|ge |veca | + vecb|`
D. All of the above

Answer

To solve the problem of determining the correct statement regarding the vectors A and B, we will analyze the given options based on vector addition and properties.

### Step-by-step Solution:

1. **Understanding Vector Addition**:
- We have two vectors, A and B. For example, let vector A = 3i (along the x-axis) and vector B = 4j (along the y-axis).
- The resultant vector A + B can be found using the Pythagorean theorem since these vectors are perpendicular to each other.

2. **Calculating the Magnitude of A + B**:
- The magnitude of the resultant vector A + B can be calculated as:
\[ |A + B| = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. **Analyzing the Statements**:
- **Option A**: \( |A + B| \geq |A| + |B| \)
- Here, \( |A| = 3 \) and \( |B| = 4 \), so \( |A| + |B| = 3 + 4 = 7 \).
- Since \( 5 < 7 \), this statement is **not correct**.

- **Option B**: \( |A + B| \leq |A| + |B| \)
- We already found \( |A + B| = 5 \) and \( |A| + |B| = 7 \).
- Since \( 5 \leq 7 \), this statement is **correct**.

- **Option C**: \( |A - B| \geq |A| - |B| \)
- To analyze this, we need to consider \( |A - B| \).
- The magnitude of \( A - B \) can be calculated as:
\[ |A - B| = |3i - 4j| = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- Here, \( |A| - |B| = 3 - 4 = -1 \).
- Since \( 5 \geq -1 \) holds true, but this inequality does not provide a definitive conclusion about the relationship, we cannot conclude that this statement is correct without additional context.

- **Option D**: \( |A - B| \leq |A| - |B| \)
- From the previous calculation, we found \( |A - B| = 5 \) and \( |A| - |B| = -1 \).
- Since \( 5 \leq -1 \) is **not correct**, this statement is also **not correct**.

4. **Conclusion**:
- The only statement that holds true based on our analysis is **Option B**: \( |A + B| \leq |A| + |B| \).

### Final Answer:
The correct statement is **Option B**.

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