Parallel forces are added as scalars or algebraic numbers. The resultant of parallel forces is the algebraic sum of these forces and has their direction. Determining the resultant involves determining its position, that is, the distance from any force in the system whose position is known.
We will use the following algorithm to solve the problem of reducing a parallel force system:
1) Choosing a parallel force system vector \(\overline{e}\)
2) Determining the coefficients \(\overline{a_i}\) (measures of each force relative to the system vector):
\(a_i=|F_i|\) if the direction of \(\overline{F}\) is consistent with the direction of \(\overline{e}\)
\(a_i=-|F_i|\) if the direction of \(\overline{F}\) is opposite to the direction of \(\overline{e}\).
3) Sum of the parallel force system \(\overline{S}=\sum_{i=1}^n a_i\cdot \overline{e}\)
4) Determining the lead vectors of the force attachment points \(\overline{r_i}\)
5) Calculating the vector that determines the position of the center of the parallel force system \(\overline{OO*}=\frac{\sum a_i\cdot \overline{r_i}}{\sum a_i}\)