# The Law of Parallelogram

THE LAW OF PARALLELOGRAM

This law states that, If two forces acting at a point be represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by lite diagonal of the parallelogram passing through that point

Given OA = P force and OB = Q force as per Fig. 1.5 (a).

We construct parallelogram OBCA as per Fig. 1.5 (b). Let us drop perpendicular CD on extension of OA.

Thus,               AD= Q cos α,CD=Q sin α

∆OCD is a right angled triangle,

:.                                  OC2 =OD2 +CD2

Or                                R2 = (P + Q cos α)2 + (Q sin α)2

= P2 + Q2 cos2 a + 2PQ cos a+ Q2 sin2 α

= p2 + Q2 (cos2 a+ sin2 a)+ 2PQ cos α.

or

R2 = P2 + Q2 + 2PQ cos α                   [since cos2 a+ sin2 a= 1]

or

R= √p2 +Q2 +2PQ cos α

Equation (1.1) gives the magnitude of the resultant force R.

Direction of R :

Let θ be the angle of R with P

tan θ = CD/OD =Q sin a/ P+Q cos α

θ  = tan -1 (Q sin α /P+ Q cos α)