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**Addition of Forces.** The concept of force resultant can be applied to a concurrent force system which is written in the Cartesian vector form. If P is the resultant of F1, F2, F3, â€¦ hence

**P** =Î£**F**_{i}

= (*F*_{1x} + *F*_{2x} + â€¦)**i** + (*F*_{1y} + *F*_{2y} + â€¦)**j** + (*F*_{1z} + *F*_{2z} + â€¦)**k**

= Î£*F*_{ix}**i** + Î£*F*_{iy}**j** + Î£*F*_{iz}**k** **(2.5)**

whereÎ£*F*_{ix}, Î£*F*_{iy}, and Î£*F*_{iz} are the sums of the magnitudes of the forces in the corresponding direction.

**2.8 RESOLUTION OF FORCES**

Any force can be *resolved* into its components. There are a number of methods that can be used to resolve a force. The method use depends on the problem at hand. The different methods are described below.

**2.8.1. Parallelogram Law**

A force acting at any point can be resolved into components that act in two desired directions through the parallelogram law. Force **F** in Figure 2.12(a), for example, can be replaced by two components acting in directions 1 and 2. The resolution is implemented by drawing a parallelogram with **F** as the diagonal and its two non-parallel sides along directions 1 and 2,

Note that resolution is the reverse process of adding two forces into their resultant. Hence the resolution involves six quantities, i.e. the magnitude and direction of **F** and of the two components, where two of them can be determined when the other four are known.

**Rectangular components.** When force **F** is resolved into perpendicular directions, we obtain the rectangular components. This is shown in Figure 2.13 for a 2D case, where the respective perpendicular directions are represented by the* x*-axis and the *y*-axis. In this case, we get

*F*_{x} = *F* cos *Î¸*

*F*_{y} =* F* sin *Î¸*

where *Î¸* = tan^{-1} (*F*_{y}/*F*_{x})

**Addition Of Forces.** Forces can be added by using their components, normally the rectangular components in the Cartesian directions. Consider forces **F**1 and **F**2 that are concurrent at *O* being added using the force polygon, Figure 2.14(a). The resultant **P** is

**P** = **F**1 + **F**2

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