Resolve three dimensional force systems with confidence. Track components, moments, equilibrants, and direction angles instantly. Build accurate mechanical insight from every entered force vector.
Use coordinate direction angles in degrees. Enter point coordinates for each force to calculate moments about the origin. Leave unused force magnitudes at 0.
These sample values match the default entries already loaded in the form.
| Force | Magnitude (N) | Alpha (°) | Beta (°) | Gamma (°) | x (m) | y (m) | z (m) |
|---|---|---|---|---|---|---|---|
| F1 | 120 | 60 | 75 | 45 | 1.20 | 0.30 | 0.00 |
| F2 | 80 | 125 | 60 | 70 | 0.00 | 1.10 | 0.50 |
| F3 | 65 | 80 | 135 | 55 | -0.80 | 0.40 | 0.90 |
| F4 | 45 | 110 | 95 | 35 | 0.60 | -0.20 | 1.30 |
1) Direction ratios from entered angles: l = cos(alpha), m = cos(beta), n = cos(gamma)
2) Valid unit direction: u = (l, m, n) / sqrt(l² + m² + n²)
3) Force components: Fx = Fux, Fy = Fuy, Fz = Fuz
4) Resultant force: R = ΣF = (ΣFx)i + (ΣFy)j + (ΣFz)k
5) Resultant magnitude: |R| = sqrt((ΣFx)² + (ΣFy)² + (ΣFz)²)
6) Resultant direction angles: alphaR = cos⁻¹(ΣFx / |R|), betaR = cos⁻¹(ΣFy / |R|), gammaR = cos⁻¹(ΣFz / |R|)
7) Moment of each force about origin: M = r × F
8) Resultant moment: Mo = Σ(r × F)
9) Equilibrant: E = -R
10) Pitch of the wrench: p = (R · Mo) / |R|²
11) Nearest central axis point: r0 = (R × Mo) / |R|²
They are forces whose lines of action do not lie in one plane. Such systems require full three dimensional resolution into x, y, and z components.
Three coordinate direction angles define how each force is oriented in space relative to the x, y, and z axes. That lets the calculator resolve 3D force components correctly.
Real direction cosines must satisfy cos²alpha + cos²beta + cos²gamma = 1. If your entries miss that condition slightly, the tool normalizes them so the direction remains physically valid.
The resultant is the net force vector from all entered forces. The equilibrant has the same magnitude but opposite direction, so it balances the system.
Coordinates define the position vector of each applied force. The calculator uses that vector with the cross product to find each moment about the origin.
Use consistent units throughout. Magnitude is typically in newtons, coordinates in meters, and resulting moments then appear in newton meters.
Yes. Just leave unused rows at zero magnitude. Those rows become inactive and do not affect the final resultant or moment calculations.
Pitch shows the moment component that remains parallel to the resultant force. It is useful when describing the equivalent wrench of a spatial force system.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.