Two forces $P$ and $Q$ act at an angle $\alpha$ and their resultant is $R$. If $Q$ is doubled, the new resultant is perpendicular to $P$. Prove that $Q = R$.

A weight of $20\text{ N}$ is suspended by two strings $OA$ and $OB$ of lengths $3\text{ m}$ and $4\text{ m}$ respectively from two points $A$ and $B$ on the same horizontal line, with $AB = 5\text{ m}$. Find the tensions $T_1$ and $T_2$ in the two strings.

Two forces of magnitudes $8\text{ N}$ and $6\text{ N}$ act at a point at an angle of $60^\circ$ to each other. Calculate: (a) The magnitude of their resultant force $R$. (b) The angle which the resultant makes with the $8\text{ N}$ force.