Parallelogram Law of Forces — when two forces acting simultaneously on a particle are represented (in magnitude and direction) by two adjacent sides of a parallelogram, their resultant is represented by:

Introduction / Context:
The Parallelogram Law of Forces is a foundational result in engineering mechanics for adding two concurrent forces. It allows engineers to replace two forces by a single equivalent resultant that acts through the same point with a direction and magnitude obtained geometrically. This law underpins graphical statics, vector addition, and equilibrium checks for pin-jointed structures and particles.

Given Data / Assumptions:

• Two forces act simultaneously on a single particle (concurrent at one point).
• Each force is represented by a directed segment equal in magnitude and direction to the real force.
• The two segments are drawn as adjacent sides of a parallelogram.

Concept / Approach:

Construct a parallelogram with the two force vectors as adjacent sides. The diagonal drawn from the common tail (the point of concurrency) gives the vector sum of the two forces. This diagonal's length represents the magnitude of the resultant, and its orientation gives the resultant's direction. Analytically, this matches vector addition F_R = F_1 + F_2, including the cosine rule for magnitude and the law of sines for direction if needed.

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Resolve the two forces into perpendicular components and add components algebraically; the combined vector equals the geometrical diagonal. Computed magnitude via the cosine rule agrees with the diagonal length.

Why Other Options Are Wrong:

Longer/shorter side are not generally aligned with the resultant. The non-intersecting diagonal does not start from the concurrency point, so it cannot represent the resultant of the two applied forces. The average of diagonals lacks physical meaning for vector sum.

Common Pitfalls (misconceptions, mistakes):

Drawing the diagonal from the wrong corner; forgetting that the law applies only to concurrent forces; mixing up magnitude scaling when using non-uniform drawing scales.

Final Answer:

the diagonal of the parallelogram which passes through the point of intersection of the forces