Addition & Subtraction of Cartesian Vectors | Engineering Mechanics - Civil Engineering (CE) PDF Download

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, … hence

P =ΣF_i

=  (F_1x + F_2x + …)i + (F_1y + F_2y + …)j + (F_1z + F_2z + …)k

= ΣF_ixi + ΣF_iyj + ΣF_izk          (2.5)

whereΣF_ix, ΣF_iy, and ΣF_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 θ

 F_y = F sin θ

where  θ = 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 F1 and F2 that are concurrent at O being added using the force polygon, Figure 2.14(a).  The resultant P is

P = F1 + F2