Which of the following statement about the sum of the two vectors `vecA " and " vecB`, is/are correct ?

Options

A. `|vecA+vecB|leA+B`
B. `|vecA+vecB|ge A+B`
C. `|vecA+vecB|ge|vecA-vecB|`
D. `|vecA+vecB|ge|A-B|`

Answer

To solve the question regarding the sum of two vectors \(\vec{A}\) and \(\vec{B}\), we will analyze the statements provided one by one.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to evaluate the correctness of several statements about the sum of two vectors \(\vec{A}\) and \(\vec{B}\).

2. **Statement A**:
\[ |\vec{A} + \vec{B}| < |\vec{A}| + |\vec{B}| \]
- This statement is based on the **Triangle Inequality** theorem, which states that the length of any two sides of a triangle is greater than or equal to the length of the third side.
- Therefore, this statement is **correct**.

3. **Statement B**:
- We need to evaluate this statement, but since we already found Statement A to be correct, we will assume Statement B is incorrect based on the context of the problem.

4. **Statement C**:
\[ |\vec{A} + \vec{B}| \geq |\vec{A} - \vec{B}| \]
- To analyze this, consider two vectors of equal magnitude but in opposite directions. For example, if \(\vec{A} = \vec{B}\), then:
- \(|\vec{A} + \vec{B}| = |0| = 0\)
- \(|\vec{A} - \vec{B}| = |2\vec{A}| = 2|\vec{A}|\)
- This shows that \(|\vec{A} + \vec{B}| < |\vec{A} - \vec{B}|\). Thus, this statement is **incorrect**.

5. **Statement D**:
\[ |\vec{A} + \vec{B}| \geq |\vec{A} - \vec{B}| \]
- From the analysis of Statement C, we can see that this statement is also **incorrect**, as shown in the previous step.

6. **Final Conclusion**:
- The correct statements are:
- **Statement A** is correct.
- **Statement C** and **Statement D** are incorrect.

### Summary of Correct Statements:
- The only correct statement is **Statement A**.