Four forces acting on a particle keep it in equilibrium, then :-

Options

A. the force must be coplanar.
B. the force cannot be coplanar
C. the forces may or may not be coplanar.
D. if three of these forces are coplanar, so must be the fourth.

Answer

To solve the question regarding the four forces acting on a particle in equilibrium, we need to analyze the conditions under which equilibrium is maintained.

### Step-by-Step Solution:

1. **Understanding Equilibrium**:
- A particle is in equilibrium when the net force acting on it is zero. This means that all the forces acting on the particle must balance each other out.

2. **Analyzing the Forces**:
- Let’s denote the four forces acting on the particle as \( F_1, F_2, F_3, \) and \( F_4 \).
- For the particle to be in equilibrium, the vector sum of these forces must be zero:
\[ F_1 + F_2 + F_3 + F_4 = 0 \]

3. **Co-planarity of Forces**:
- The first option suggests that the forces must be coplanar. However, we can have forces that are not coplanar and still maintain equilibrium.
- For instance, if two forces are acting in a plane and two others are acting perpendicular to that plane, they can still balance each other out.

4. **Case Analysis**:
- **Case 1**: All four forces are coplanar. If they are equal in magnitude and opposite in direction, they will cancel each other out, leading to equilibrium.
- **Case 2**: Two forces are in the plane, and two are acting perpendicular to that plane. If the perpendicular forces also balance each other, the particle can still be in equilibrium.

5. **Conclusion on Co-planarity**:
- From the above analysis, we conclude that the forces may or may not be coplanar. Therefore, the first option is incorrect.

6. **Evaluating Other Options**:
- **Option B** states that forces cannot be coplanar, which is also incorrect since they can be coplanar.
- **Option C** states that the forces may be coplanar or may not be coplanar, which is correct.
- **Option D** states that if three of the forces are coplanar, then the fourth must also be coplanar. This is true because if three forces are in a plane and the fourth force is not, it cannot balance the other three forces.

7. **Final Answer**:
- The correct options are C and D.