Mark the correct statement.

Options

A. `|vec(a)+vec(b)|ge|vec(a)|+|vec(b)|`
B. `|vec(a)+vec(b)|le|vec(a)|+|vec(b)|`
C. `|vec(a)-vec(b)|le|vec(a)|+|vec(b)|`
D. All of the above

Explanation

To solve the problem of determining the correct statement regarding the inequalities involving vectors A and B, we can follow these steps:

### Step 1: Understand the Magnitude of the Sum of Two Vectors
We start with the expression for the magnitude of the sum of two vectors A and B:
\[ |\mathbf{A} + \mathbf{B}| = \sqrt{|\mathbf{A}|^2 + |\mathbf{B}|^2 + 2|\mathbf{A}||\mathbf{B}|\cos\theta} \]
where \(\theta\) is the angle between vectors A and B.

### Step 2: Analyze the Cosine Function
The cosine function varies between -1 and 1. Therefore:
- When \(\theta = 0^\circ\) (vectors in the same direction), \(\cos\theta = 1\).
- When \(\theta = 180^\circ\) (vectors in opposite directions), \(\cos\theta = -1\).

### Step 3: Establish the Maximum and Minimum Values
Using the cosine values:
1. **Maximum Value**:
\[ |\mathbf{A} + \mathbf{B}| \text{ is maximum when } \theta = 0^\circ: \]
\[ |\mathbf{A} + \mathbf{B}| \leq |\mathbf{A}| + |\mathbf{B}| \]

2. **Minimum Value**:
\[ |\mathbf{A} + \mathbf{B}| \text{ is minimum when } \theta = 180^\circ: \]
\[ |\mathbf{A} + \mathbf{B}| \geq ||\mathbf{A}| - |\mathbf{B}|| \]

### Step 4: Formulate the Inequalities
From the above analysis, we can conclude:
\[ ||\mathbf{A}| - |\mathbf{B}|| \leq |\mathbf{A} + \mathbf{B}| \leq |\mathbf{A}| + |\mathbf{B}| \]

### Step 5: Compare with Given Statements
Now, we compare the derived inequalities with the statements provided in the question:
1. \( |\mathbf{A} + \mathbf{B}| \geq |\mathbf{A}| + |\mathbf{B}| \) (False)
2. \( |\mathbf{A} + \mathbf{B}| \leq |\mathbf{A}| + |\mathbf{B}| \) (True)
3. \( |\mathbf{A} - \mathbf{B}| \leq |\mathbf{A}| + |\mathbf{B}| \) (True)

### Conclusion
The correct statements from the options provided are:
- Statement 2: \( |\mathbf{A} + \mathbf{B}| \leq |\mathbf{A}| + |\mathbf{B}| \) is true.
- Statement 3: \( |\mathbf{A} - \mathbf{B}| \leq |\mathbf{A}| + |\mathbf{B}| \) is also true.

Thus, the correct options are 2 and 3.

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