Forces proportional to `AB , BC` and `2 CA` act along the slides of a triangle `ABC` in magnitude and direction by

Options

A. CA
B. AC
C. BC
D. CB

Explanation

To solve the problem, we will analyze the forces acting along the sides of triangle ABC and apply the triangle law of vector addition.

### Step-by-Step Solution:

1. **Identify the Forces**:
We have forces acting along the sides of triangle ABC. The forces are proportional to the lengths of the sides:
- Force along AB is proportional to \( AB \)
- Force along BC is proportional to \( BC \)
- Force along CA is proportional to \( 2CA \)

2. **Express the Forces as Vectors**:
We can denote the forces as vectors:
- Let \( \vec{F_{AB}} \) be the force along AB.
- Let \( \vec{F_{BC}} \) be the force along BC.
- Let \( \vec{F_{CA}} \) be the force along CA.

According to the problem:
\[ \vec{F_{AB}} \propto \vec{AB}, \quad \vec{F_{BC}} \propto \vec{BC}, \quad \vec{F_{CA}} \propto 2\vec{CA} \]

3. **Apply the Triangle Law of Vector Addition**:
According to the triangle law of vector addition, the sum of the vectors around a closed triangle is zero:
\[ \vec{F_{AB}} + \vec{F_{BC}} + \vec{F_{CA}} = 0 \]

4. **Rearranging the Equation**:
From the equation above, we can rearrange it to express one of the forces in terms of the others:
\[ \vec{F_{AB}} + \vec{F_{BC}} = -\vec{F_{CA}} \]

5. **Substituting the Given Proportions**:
Given that \( \vec{F_{CA}} \) is proportional to \( 2\vec{CA} \), we can write:
\[ \vec{F_{AB}} + \vec{F_{BC}} = -2\vec{CA} \]

6. **Final Expression**:
Rearranging gives us:
\[ \vec{F_{AB}} + \vec{F_{BC}} + 2\vec{CA} = 0 \]

7. **Conclusion**:
This indicates that the forces are balanced according to the triangle law of vector addition. The correct option based on our analysis is option A.