1. **Stating the problem:** We want to prove the equation of the resultant force when two forces act at a point using the parallelogram of forces method. 2. **Concept:** When two forces $\vec{F_1}$ and $\vec{F_2}$ act at a point, their resultant $\vec{R}$ can be found by constructing a parallelogram where the two forces are adjacent sides. The diagonal of the parallelogram from the point of application represents the resultant force. 3. **Formula:** If $F_1$ and $F_2$ are magnitudes of the forces and $\theta$ is the angle between them, the magnitude of the resultant $R$ is given by the law of cosines: $$ R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos \theta} $$ 4. **Proof steps:** - Consider two forces $\vec{F_1}$ and $\vec{F_2}$ acting at a point with angle $\theta$ between them. - Place $\vec{F_1}$ along the x-axis for simplicity. - The components of $\vec{F_2}$ are: - $F_2 \cos \theta$ along x-axis - $F_2 \sin \theta$ along y-axis - The resultant vector $\vec{R} = \vec{F_1} + \vec{F_2}$ has components: - $R_x = F_1 + F_2 \cos \theta$ - $R_y = F_2 \sin \theta$ - The magnitude of $\vec{R}$ is: $$ R = \sqrt{R_x^2 + R_y^2} = \sqrt{(F_1 + F_2 \cos \theta)^2 + (F_2 \sin \theta)^2} $$ - Expanding and simplifying: $$ R = \sqrt{F_1^2 + 2 F_1 F_2 \cos \theta + F_2^2 \cos^2 \theta + F_2^2 \sin^2 \theta} $$ - Using the identity $\cos^2 \theta + \sin^2 \theta = 1$: $$ R = \sqrt{F_1^2 + 2 F_1 F_2 \cos \theta + F_2^2} $$ 5. **Conclusion:** This matches the law of cosines formula for the resultant force magnitude, proving the equation using the parallelogram of forces. This method visually and mathematically confirms how two forces combine to form a resultant force.