Let `vecC=vecA+vecB`

Options

A. `|vecC|` is always greater than `|vecA|`
B. It is possible to have `|vecC|lt|vecA| and |vecC|lt|vecB|`
C. C is always equal to A+B
D. C is never equal to A+B

Answer

To solve the problem where \(\vec{C} = \vec{A} + \vec{B}\), we need to analyze the given options regarding the magnitudes of the vectors involved. Let's go through the steps systematically.

### Step 1: Understanding the Vectors
We have:
\[ \vec{C} = \vec{A} + \vec{B} \]
This means that the vector \(\vec{C}\) is the resultant of the vectors \(\vec{A}\) and \(\vec{B}\).

### Step 2: Analyzing the Magnitudes
We need to evaluate the following options regarding the magnitudes:
1. \(|\vec{C}| > |\vec{A}|\)
2. \(|\vec{C}| < |\vec{A}|\) and \(|\vec{C}| < |\vec{B}|\) is possible.
3. \(|\vec{C}| = |\vec{A} + \vec{B}|\)
4. \(|\vec{C}| \neq |\vec{A} + \vec{B}|\)

### Step 3: Testing with Examples
Let's consider specific examples to evaluate the options.

#### Example 1:
Let \(\vec{A} = 3 \hat{i}\) and \(\vec{B} = -\hat{i}\).
- Magnitude of \(\vec{A}\):
\[ |\vec{A}| = 3 \]
- Magnitude of \(\vec{B}\):
\[ |\vec{B}| = 1 \]
- Now, calculate \(\vec{C}\):
\[ \vec{C} = \vec{A} + \vec{B} = 3\hat{i} - \hat{i} = 2\hat{i} \]
- Magnitude of \(\vec{C}\):
\[ |\vec{C}| = 2 \]

From this example:
- \(|\vec{C}| = 2 < |\vec{A}| = 3\), hence option 1 is incorrect.
- \(|\vec{C}| < |\vec{A}|\) is true, and \(|\vec{C}| < |\vec{B}|\) is not true, so option 2 is partially true but requires more analysis.

#### Example 2:
Let \(\vec{A} = 3 \hat{i}\) and \(\vec{B} = \hat{j}\).
- Magnitude of \(\vec{A}\):
\[ |\vec{A}| = 3 \]
- Magnitude of \(\vec{B}\):
\[ |\vec{B}| = 1 \]
- Now, calculate \(\vec{C}\):
\[ \vec{C} = \vec{A} + \vec{B} = 3\hat{i} + \hat{j} \]
- Magnitude of \(\vec{C}\):
\[ |\vec{C}| = \sqrt{3^2 + 1^2} = \sqrt{10} \]
- Here, \(|\vec{C}| \neq |\vec{A}| + |\vec{B}|\) since \(3 + 1 = 4\) and \(\sqrt{10} < 4\).

### Step 4: Evaluating Options
- **Option 1**: Incorrect, as shown in Example 1.
- **Option 2**: Possible, as shown in Example 1 and Example 2.
- **Option 3**: Incorrect, as shown in Example 2.
- **Option 4**: Incorrect, as shown in Example 2.

### Conclusion
The only correct option is:
**Option 2**: It is possible to have \(|\vec{C}| < |\vec{A}|\) and \(|\vec{C}| < |\vec{B}|\).