The parallelogram of forces is a method for solving (or visualizing) the results of applying two forces to an object. When more than two forces are involved, the geometry is no longer a parallelogram, but the same principles apply to a polygon of forces. The resultant force due to the application of a number of forces can be found geometrically by drawing arrows for each force. The parallelogram of forces is a graphical manifestation of the addition of vectors.
Newton's proof
Preliminary: the parallelogram of velocity Suppose a particle moves at a uniform rate along a line from A to B (Figure 2) in a given time (say, one second), while in the same time, the line AB moves uniformly from its position at AB to a position at DC, remaining parallel to its original orientation throughout. Accounting for both motions, the particle traces the line AC. Because a displacement in a given time is a measure of velocity, the length of AB is a measure of the particle's velocity along AB, the length of AD is a measure of the line's velocity along AD, and the length of AC is a measure of the particle's velocity along AC. The particle's motion is the same as if it had moved with a single velocity along AC.
Newton's proof of the parallelogram of force Suppose two forces act on a particle at the origin (the "tails" of the vectors) of Figure 1. Let the lengths of the vectors F1 and F2 represent the velocities the two forces could produce in the particle by acting for a given time, and let the direction of each represent the direction in which they act. Each force acts independently and will produce its particular velocity whether the other force acts or not. At the end of the given time, the particle has both velocities. By the above proof, they are equivalent to a single velocity, Fnet. By Newton's second law, this vector is also a measure of the force which would produce that velocity, thus the two forces are equivalent to a single force.
Bernoulli's proof for perpendicular vectors We model forces as Euclidean vectors or members of R 2 {\displaystyle \mathbb {R} ^{2}} . Our first assumption is that the resultant of two forces is in fact another force, so that for any two forces F , G ∈ R 2 {\displaystyle \mathbf {F} ,\mathbf {G} \in \mathbb {R} ^{2}} there is another force F ⊕ G ∈ R 2 {\displaystyle \mathbf {F} \oplus \mathbf {G} \in \mathbb {R} ^{2}} . Our final assumption is that the resultant of two forces doesn't change when rotated. If R : R 2 → R 2 {\displaystyle R:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} is any rotation (any orthogonal map for the usual vector space structure of R 2 {\displaystyle \mathbb {R} ^{2}} with det R = 1 {\displaystyle \det R=1} ), then for all forces F , G ∈ R 2 {\displaystyle \mathbf {F} ,\mathbf {G} \in \mathbb {R} ^{2}}
R ( F ⊕ G ) = R ( F ) ⊕ R ( G ) {\displaystyle R\left(\mathbf {F} \oplus \mathbf {G} \right)=R\left(\mathbf {F} \right)\oplus R\left(\mathbf {G} \right)}
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