Suppose that two objects A and B are moving with velocities `vecv_(A)` and `vecv_(B)` (each with respect to some common frame of reference). Let `vecv_(AB)` represent the velocity of A with respect to B. Then

Options

A. `vecv_(AB) + vecv_(BA) = 0`
B. `vecv_(AB) - vecv_(BA) = 0`
C. `vecv_(AB) = vecv_(A) + vecv_(B)`
D. `|vecv_(AB)| ne |vecv_(BA)|`

Explanation

To solve the problem, we need to analyze the relationship between the velocities of two objects A and B, and their relative velocities.

### Step-by-Step Solution:

1. **Define the Velocities**:
Let \(\vec{v}_A\) be the velocity of object A and \(\vec{v}_B\) be the velocity of object B, both with respect to a common frame of reference.

2. **Relative Velocity Definition**:
The velocity of A with respect to B, denoted as \(\vec{v}_{AB}\), can be defined as:
\[ \vec{v}_{AB} = \vec{v}_A - \vec{v}_B \]

3. **Velocity of B with respect to A**:
Similarly, the velocity of B with respect to A, denoted as \(\vec{v}_{BA}\), can be defined as:
\[ \vec{v}_{BA} = \vec{v}_B - \vec{v}_A \]

4. **Relate \(\vec{v}_{AB}\) and \(\vec{v}_{BA}\)**:
We can express \(\vec{v}_{BA}\) in terms of \(\vec{v}_{AB}\):
\[ \vec{v}_{BA} = -(\vec{v}_{AB}) \]

5. **Magnitude Relationship**:
From the above relationship, we see that:
\[ |\vec{v}_{BA}| = |\vec{v}_{AB}| \]
This means the magnitudes of the velocities of A with respect to B and B with respect to A are equal.

6. **Evaluate Options**:
- **Option 1**: \(\vec{v}_{AB} + \vec{v}_{BA} = 0\) is correct because \(\vec{v}_{BA} = -\vec{v}_{AB}\).
- **Option 2**: \(\vec{v}_{AB} - \vec{v}_{BA} = 0\) is incorrect because \(\vec{v}_{AB} - (-\vec{v}_{AB}) = 2\vec{v}_{AB} \neq 0\).
- **Option 3**: \(\vec{v}_{AB} = \vec{v}_A + \vec{v}_B\) is incorrect as shown in the definitions.
- **Option 4**: The magnitudes are equal, so this option is incorrect as it states they are not equal.

7. **Conclusion**:
The only correct statement is that the velocity of A with respect to B plus the velocity of B with respect to A equals zero.

### Final Answer:
The correct option is:
\[ \vec{v}_{AB} + \vec{v}_{BA} = 0 \]