Particles of masses `m_(1)` and `m_(2)` are at a fixed distance apart. If the gravitational field strength at `m_(1)` and `m_(2)` are `vecI_(1)` and `vecI_(2)` respectively. Then

Options

A. `m_(1)vecI_(1)+m_(2)vecI_(2)=0`
B. `m_(1)vecI_(2)+m_(2)vecI_(1)=0`
C. `m_(1)vecI_(1)-m_(2)vecI_(2)=0`
D. `m_(1)vecI_(2)-m_(2)vecI_(1)=0`

Explanation

To solve the problem, we need to analyze the gravitational field strengths at the positions of two masses, \( m_1 \) and \( m_2 \), which are at a fixed distance apart. The gravitational field strength at \( m_1 \) is denoted as \( \vec{I}_1 \) and at \( m_2 \) as \( \vec{I}_2 \).

### Step-by-Step Solution:

1. **Understanding Gravitational Field Strength**:
The gravitational field strength \( \vec{I} \) at a mass due to another mass is given by the formula:
\[ \vec{I} = \frac{G \cdot m}{d^2} \]
where \( G \) is the gravitational constant, \( m \) is the mass creating the field, and \( d \) is the distance from the mass to the point where the field is being measured.

2. **Gravitational Field Strength at \( m_1 \)**:
The gravitational field strength at mass \( m_1 \) due to mass \( m_2 \) is:
\[ \vec{I}_1 = \frac{G \cdot m_2}{d^2} \]
This field strength acts towards \( m_2 \).

3. **Gravitational Field Strength at \( m_2 \)**:
Similarly, the gravitational field strength at mass \( m_2 \) due to mass \( m_1 \) is:
\[ \vec{I}_2 = \frac{G \cdot m_1}{d^2} \]
This field strength acts towards \( m_1 \).

4. **Direction of Gravitational Forces**:
The gravitational field strength \( \vec{I}_1 \) at \( m_1 \) acts towards \( m_2 \), and \( \vec{I}_2 \) at \( m_2 \) acts towards \( m_1 \). Therefore, if we consider the direction of \( \vec{I}_1 \) as positive, then \( \vec{I}_2 \) will be negative when considering the force balance.

5. **Applying Newton's Third Law**:
According to Newton's third law, the forces exerted by the masses on each other are equal in magnitude and opposite in direction. Thus, we can express the gravitational forces as:
\[ F_1 = m_1 \cdot \vec{I}_1 \quad \text{and} \quad F_2 = m_2 \cdot (-\vec{I}_2) \]

6. **Setting Up the Equation**:
Since the forces are equal and opposite, we can write:
\[ m_1 \cdot \vec{I}_1 + m_2 \cdot (-\vec{I}_2) = 0 \]
Rearranging gives:
\[ m_1 \cdot \vec{I}_1 - m_2 \cdot \vec{I}_2 = 0 \]

7. **Conclusion**:
Thus, we have derived the relationship:
\[ m_1 \cdot \vec{I}_1 = m_2 \cdot \vec{I}_2 \]
This corresponds to the option:
\[ m_1 \cdot \vec{I}_1 - m_2 \cdot \vec{I}_2 = 0 \]

### Final Answer:
The correct answer is:
\[ m_1 \cdot \vec{I}_1 - m_2 \cdot \vec{I}_2 = 0 \]