Three forces P,Q and R are acting at a point in the plane. The angle between P & Q and Q & R are `150^(@) & 120^(@)` respectively, then for equilibrium, forces P,Q & R are in the ratio

Options

A. `1:2:3`
B. `1:2:sqrt(3)`
C. `3:2:1`
D. `sqrt(3):2:1`

Explanation

To solve the problem of finding the ratio of the forces P, Q, and R acting at a point in the plane, we can use the conditions for equilibrium. The forces are given to be at angles of 150° between P and Q, and 120° between Q and R.

### Step-by-Step Solution:

1. **Understand the Forces and Angles**:
- We have three forces: P, Q, and R.
- The angle between P and Q is 150°.
- The angle between Q and R is 120°.

2. **Apply the Law of Sines for Equilibrium**:
- For the system to be in equilibrium, the following relationship must hold:
\[ \frac{P}{\sin(120°)} = \frac{Q}{\sin(90°)} = \frac{R}{\sin(150°)} \]

3. **Calculate the Sine Values**:
- \(\sin(90°) = 1\)
- \(\sin(120°) = \sin(180° - 60°) = \sin(60°) = \frac{\sqrt{3}}{2}\)
- \(\sin(150°) = \sin(180° - 30°) = \sin(30°) = \frac{1}{2}\)

4. **Set Up the Ratios**:
- From the equilibrium condition, we can write:
\[ \frac{P}{\frac{\sqrt{3}}{2}} = \frac{Q}{1} = \frac{R}{\frac{1}{2}} \]

5. **Express P, Q, and R in Terms of a Common Variable**:
- Let \(k\) be the common ratio:
- \(P = k \cdot \frac{\sqrt{3}}{2}\)
- \(Q = k\)
- \(R = k \cdot \frac{1}{2}\)

6. **Find the Ratio of P, Q, and R**:
- Now we can express the ratio of the forces:
\[ P : Q : R = \left(k \cdot \frac{\sqrt{3}}{2}\right) : k : \left(k \cdot \frac{1}{2}\right) \]
- Dividing each term by \(k\):
\[ P : Q : R = \frac{\sqrt{3}}{2} : 1 : \frac{1}{2} \]

7. **Simplify the Ratio**:
- To eliminate the fractions, multiply each term by 2:
\[ P : Q : R = \sqrt{3} : 2 : 1 \]

### Final Result:
The ratio of the forces P, Q, and R is:
\[ \sqrt{3} : 2 : 1 \]