Mechanics

Force is a vector quantity, it has both a magnitude and a direction. The SI unit of force is the Newton (N). The force vector changes if either, or both, the magnitude and direction change. In understanding the balance of forces acting on a system like a bike, it is frequently convenient to look at the forces acting on the bike in a vertical, longitudinal and transverse direction. This can be done by resolving individual forces into their components in these three directions (selected to be a Cartesian coordinate system as shown in the top diagram).

For the force vector, A, shown, the components in the x,y,z directions shown are also vectors described by a magnitude, A_x,A_y,A_z, and a direction, i, j, k. This gives: A = A_xi + A_yj + A_zk , where bold characters denote vector quantities and regular characters magnitudes. The quantities i, j, k are the Unit vectors in the x, y, z directions and have a magnitude of 1. The components in a given direction can then be added to give the total force acting in that direction (lower diagram).

If a body is in a state of rest or moving with a constant velocity, the sum of the force components acting on it in each coordinate direction must be zero. SA_xi = 0, SA_yj = 0, SA_zk = 0. The S symbol is used to represent a sum over all of the components in a given coordinate direction.

From: Hibbler, "Statics and Mechanics of Materials," Macmillan (1993)