Forces `3 O vec A , 5 O vec B ` act along OA and OB. If their resultant passes through C on AB, then

Options

A. C is a mid-point of AB
B. C divides AB in the ratio `2:1`
C. `3 AC = 5 CB`
D. `2 AC = 3 CB`

Answer

To solve the problem of finding the position of point C on line AB where the resultant of the forces \(3 \vec{A}\) and \(5 \vec{B}\) acts, we can follow these steps:

### Step 1: Understand the Forces
We have two forces acting along the lines OA and OB:
- Force \( \vec{F_1} = 3 \vec{A} \)
- Force \( \vec{F_2} = 5 \vec{B} \)

### Step 2: Determine the Resultant Force
The resultant force \( \vec{R} \) can be expressed as:
\[ \vec{R} = \vec{F_1} + \vec{F_2} = 3 \vec{A} + 5 \vec{B} \]

### Step 3: Identify the Position of Point C
Since the resultant passes through point C on line AB, we need to find the position vector of point C. Let’s denote the position vectors of points A and B as \( \vec{A} \) and \( \vec{B} \) respectively.

### Step 4: Use the Section Formula
Point C divides the line segment AB in the ratio of the magnitudes of the forces. Since the forces are \(3\) and \(5\), point C divides AB externally in the ratio \(3:5\).

Using the section formula for external division, the position vector \( \vec{C} \) can be given by:
\[ \vec{C} = \frac{m \vec{B} - n \vec{A}}{m - n} \]
where \(m = 5\) and \(n = 3\).

### Step 5: Substitute the Values
Substituting the values into the formula:
\[ \vec{C} = \frac{5 \vec{B} - 3 \vec{A}}{5 - 3} = \frac{5 \vec{B} - 3 \vec{A}}{2} \]

### Step 6: Final Expression
Thus, the position vector of point C is:
\[ \vec{C} = \frac{5 \vec{B} - 3 \vec{A}}{2} \]

### Conclusion
This gives us the position of point C on line AB where the resultant of the forces \(3 \vec{A}\) and \(5 \vec{B}\) acts.

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