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# Addition and Subtraction of Cartesian Vectors Mechanical Engineering Notes | EduRev

## Mechanical Engineering : Addition and Subtraction of Cartesian Vectors Mechanical Engineering Notes | EduRev

The document Addition and Subtraction of Cartesian Vectors Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Engineering Mechanics - Notes, Videos, MCQs & PPTs.
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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£Fi

=  (F1x + F2x + â€¦)i + (F1y + F2y + â€¦)j + (F1z + F2z + â€¦)k

= Î£Fixi + Î£Fiyj + Î£Fizk          (2.5)

whereÎ£Fix, Î£Fiy, and Î£Fiz 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

Fx = F cos Î¸

Fy = F sin Î¸

where  Î¸ = tan-1 (Fy/Fx)

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 + F

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