Figure 1. Parallelogram of forces. Two forces act simultaneously on an object in different directions. The resultant force, and the direction of any subsequent movement, is defined by the diagonal of a parallelogram whose sides are drawn parallel to the applied forces with lengths proportional to their magnitudes. This is an application of vector diagrams.
A parallelogram is a quadrilateral (four-sided figure) whose opposite sides are parallel, and whose opposite angles, therefore, are equal. The diagonals of a parallelogram bisect each other. A parallelogram of base b and height h has an area:
A = bh = ab sinA = ab sinB.
The height of a parallelogram is h = a sinA = a sinB. The sides a, b, c, and d and diagonals p and q of a parallelogram satisfy the equality:
p2 + q2 = a2 + b2 + c2 + d2
Special cases of a parallelogram are a rhombus, which has sides of equal length, a rectangle, which has two sets of parallel sides that are perpendicular to each other, and a square, which meets the conditions of both a rectangle and a rhombus.
Parallelogram of forces
If a particle is under the action of two forces, which are represented in magnitude and direction by the two sides of a parallelogram drawn from a point, the resultant of the two forces is represented by the diagonal of the parallelogram drawn from that point (see Figure 1).
Parallelogram of velocities
If a body has two component velocities, represented in magnitude and direction by two adjacent sides of a parallelogram drawn from a point, the resultant velocity of the body is represented by the diagonal of the parallelogram drawn from that point. The parallelogram of velocities, like the parallelogram of forces, is a particular case of the parallelogram of vectors.
A familiar application of it is to find the resultant velocity of an object, such as a boat, that is subject to a current acting in a different direction in the one in which it is trying to move.