R2 = X2 + Y2, tan φ = Y/X.
(3)
For equilibrium we must have R = 0, which requires X = 0, Y = 0; in words, the sum of the components of the system must be zero for each of two perpendicular directions in the plane.
| Fig. 5. |
A similar procedure applies to a three-dimensional system. Thus if, O being the origin, OH> represent any force P of the system, the planes drawn through H parallel to the co-ordinate planes will enclose with the latter a parallelepiped, and it is evident that OH> is the geometric sum of OA>, AN>, NH>, or OA>, OB>, OC>, in the figure. Hence P is equivalent to three forces Pl, Pm, Pn acting along Ox, Oy, Oz, respectively, where l, m, n, are the “direction-ratios” of OH>. The whole system can be reduced in this way to three forces
X = Σ (Pl), Y = Σ (Pm), Z = Σ (Pn),
(4)
acting along the co-ordinate axes. These can again be combined into a single resultant R acting in the direction (λ, μ, ν), provided
X = Rλ, Y = Rμ, Z = Rν.
(5)