Solution:

Let $3 \theta$ be the angle between the forces P and Q. It is given that the resultant R of P and Q divides the angle between them in the ratio 1: 2. This means that the resultant makes an angle $ \theta$ with the direction of P and angle $2 \theta$ with the the direction of Q.

Therefore, $ P=\frac{R \sin 2 \theta}{\sin 3 \theta}$ and $\theta=\frac{R \sin \theta}{\sin 3 \theta} \\ $

$ \Rightarrow \frac{P}{Q}=\frac{\sin 2 \theta}{\sin \theta}=2 \cos \theta \\ $

Also, $ Q=\frac{R \sin \theta}{\sin 3 \theta} \Rightarrow Q=\frac{R}{3-4 \sin ^{2} \theta} \\ $

$ \Rightarrow \frac{R}{Q}=3-4 \sin ^{2} \theta \Rightarrow \frac{R}{Q}=-1+4 \cos ^{2} \theta \Rightarrow \frac{R}{Q}+1=(2 \cos \theta)^{2} \\ $

From, (i) and (ii), we get,

$ \left(\frac{P}{Q}\right)^{2}=\frac{R}{Q}+1 \Rightarrow \frac{R}{Q}=\frac{P^{2}-Q^{2}}{Q^{2}} \Rightarrow R=\frac{P^{2}-Q^{2}}{Q} \\ $