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 $Q = R$.

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.

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 $g = 9.8\text{ m/s}^2$).