Composition and Resolution of Vectors

The process of finding a single vector which will have the same effect as a set of vectors acting on a body is known as composition of vectors. The resolution of vectors is exactly the opposite process of composition i.e., it is the process of finding two or more vectors which will have the same effect as that of a single vector acting on the body.

Parallelogram Law of Vectors

The parallelogram law of vectors enables us to determine the single vector called resultant vector which can replace the two vectors acting at a point with the same effect as that of the two vectors. This law was formulated based on experimental results on a body subjected to two forces. This law can be applied not only to the forces but to any two vectors like velocities, acceleration, momentum etc.

This law states that ‘if two forces (vectors) acting simultaneously on a body at a point are represented in magnitude and directions by the two adjacent sides of a parallelogram, their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes thorough the point of intersection of the two sides representing the forces (vectors)’.

Fig.1 Representation of Parallelogram Law of Vectors

In the Fig. 1, the force F1 = 4 units and the force F2 = 3 units are acting on a body at a point A. To get the resultant of these forces, according to this law, construct the parallelogram ABCD such that AB is equal to 4 units to the linear scale and AC is equal to 3 units. Then according to this law, the diagonal AD represents the resultant in magnitude and direction. Thus the resultant of the forces F1 and F2 is equal to the units corresponding to AD in the direction α to F1.

Triangle Law of Vectors

Referring to Fig. 1. (b), it can be observed that the resultant AD may be obtained by constructing the triangle ABD. Line AB is drawn to represent F1 and BD to represent F2. Then AD should represent the resultant of F1 and F2.

Thus we have derived the triangle law of forces from the fundamental law of parallelogram. The Triangle Law of Forces (vectors) may be stated as ‘if two forces (vectors) acting on a body are represented one after another by the sides of a triangle, their resultant is represented by the closing side of the triangle taken from the first point to the last point.’

Polygon Law of Forces (Vectors)

If more than two forces (vectors) are acting on a body, two forces (vectors) at a line can be combined by the triangle law and finally resultant of all forces (vectors) acting on the body may be obtained.

Fig. 2 Representation of Polygon Law of Vectors

A system of four concurrent forces acting on a body are shown in Fig. 2. AB represents F1 and BC represent F2. Hence according to triangle law of forces AC represents the resultant of F1and F2, say R1. If CD is drawn to represent F3, then from the triangle law of forces AD represents the resultant of R1 and F3. In other words, AD represents the resultant of F1, F2 and F3. Let it be called as R2.

Similarly, the logic can be extended to conclude that AE represents the resultant of F1, F2, F3 and F4. The resultant R is represented by the closing line of the polygon ABCDE in the direction from A to E. Thus we have derived the polygon law of the forces (vectors) and it may be stated as ‘if a number of concurrent forces (vectors) acting simultaneously on a body are represented in magnitude and direction by the sides of a polygon, taken in an order, then the resultant is represented in magnitude and direction by the closing side of the polygon, taken from the first point to the last point’.

Graphical (Vector) Method for the Resultant Force

It is another name for finding out the magnitude and direction of the resultant force by the polygon law of forces. It is done as discussed below.

1) Construction of space diagram (position diagram)

It means the construction of a diagram showing the various forces (or loads) along with their magnitude and lines of action.

2) Use of Bow’s notations

All the forces in the space diagram are named by using the Bow’s notations. It is a convenient method in which every force (or load) is named by two capital letters, placed on its either side in the space diagram.

Fig. 3 Denoting a Force by Bow’s Notation

3) Construction of vector diagram (force diagram)

It means the construction of a diagram starting from a convenient point and then go on adding all the forces by vector addition one by one (keeping in view the directions of the forces) to some suitable scale. Now the closing side of the polygon, taken in opposite order, will give the magnitude of the resultant force (to the scale) and its direction.