parallelogram of forces
A parallelogram, two adjacent sides of which are vectors having the same initial point and representing two forces acting simultaneously upon this point, such that the resultant of the forces is given by the diagonal of the parallelogram that passes through this point.
parallelogram of forces: vector diagram for force resolution
The parallelogram of forces is a graphical method for finding the resultant of two forces acting at a single point. You draw the two force vectors as adjacent sides of a parallelogram, starting from the same point, and the diagonal of the parallelogram gives both the magnitude and direction of the resultant force. This is not abstract geometry: it reflects a physical truth about how forces combine in space.
In practice, you construct the diagram by drawing the first force vector to scale, then from its endpoint, drawing the second force vector, also to scale and at the correct angle between them. From the starting point, you complete the parallelogram by drawing lines parallel to each vector. The diagonal connecting your starting point to the opposite corner is your resultant. You measure this diagonal's length against your scale to get the resultant's magnitude, and measure its angle to get its direction relative to your reference axes.
When and why it matters
This method matters wherever forces meet at a point: a load hanging from two cables at an angle, a rope under tension on a pulley, structural members meeting at a joint. It becomes essential when the angle between forces is not 90 degrees, because simple addition of force magnitudes fails; the geometry of the angle determines whether forces reinforce or partially cancel each other. For cables meeting at a shallow angle, the tensions required to support a load are much higher than the load itself.
The parallelogram construction assumes forces are concurrent, meaning they act along lines that meet at one point. It also assumes the forces are in equilibrium or you are seeking their combined effect on a body small enough to treat as a point mass. For distributed loads or couples (rotational forces), you need different methods.
Modern engineers often solve this problem algebraically using vector components, but the parallelogram remains valuable for quick estimates on site, for teaching force behavior, and for checking calculations. The principle also extends to three forces (parallelepipeds) and more, though graphical methods become impractical and vector math takes over. The name itself dates from the geometric construction: the figure is literally a four-sided polygon with opposite sides parallel.