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.

State and prove Lami's Theorem for three coplanar forces acting at a point in static equilibrium. Hence, if a mass of $10\text{ kg}$ is suspended by two strings of lengths $3\text{ m}$ and $4\text{ m}$ from two points on a horizontal ceiling $5\text{ m}$ apart, find the tension in each string (taking $g = 9.8\text{ m/s}^2$).