Example Data Table
| Input mode |
Vector 1 |
Vector 2 |
Vector 3 |
Expected resultant |
| 2D polar |
40 N at 20° |
55 N at 135° |
30 N at 270° |
22.609 N at 93.304° |
| 3D components |
Fx 18, Fy 12, Fz 5 |
Fx -7, Fy 20, Fz 10 |
Fx 4, Fy -6, Fz 8 |
Rx 15, Ry 26, Rz 23 |
| 3D polar |
60 at 30°, 10° elevation |
45 at 120°, -5° elevation |
35 at 250°, 20° elevation |
Use spatial direction output |
Formula Used
For 2D polar vectors:
Fx = F cos θ
Fy = F sin θ
Fz = 0
For 3D polar vectors with elevation from the x-y plane:
Fx = F cos φ cos θ
Fy = F cos φ sin θ
Fz = F sin φ
For component entry:
Rx = F1x + F2x + F3x
Ry = F1y + F2y + F3y
Rz = F1z + F2z + F3z
Resultant magnitude R = √(Rx² + Ry² + Rz²)
Azimuth θR = atan2(Ry, Rx)
Elevation φR = atan2(Rz, √(Rx² + Ry²))
Direction angles are α = cos⁻¹(Rx/R), β = cos⁻¹(Ry/R), and γ = cos⁻¹(Rz/R).
The equilibrant is E = -R.
How to Use This Calculator
- Select the input mode that matches your force data.
- Use polar mode when each force has magnitude and angle.
- Use component mode when x, y, and z values are known.
- Enter all three vectors with the same force unit.
- Use positive and negative signs to show direction.
- Choose degrees or radians for angle inputs.
- Press Calculate Result to view output above the form.
- Use Download CSV for spreadsheet records.
- Use Download PDF after calculation for a report file.
Resultant Force in Three Vectors
A resultant force replaces several forces with one force. It has the same overall effect on the body. In many physics problems, three forces act at once. Each force may pull in a different direction. The calculator resolves them into x, y, and z parts. It then adds the matching parts. This method is safer than drawing alone. It also works for precise laboratory data.
Why Components Matter
A vector has size and direction. The size is the magnitude. The direction is described by angles or components. Component form separates the vector into perpendicular axes. These axes do not interfere with each other. So the x parts can be added directly. The y parts can also be added directly. The z parts are included for spatial force systems. After summing parts, the final magnitude comes from the square root formula.
Advanced Direction Reading
The resultant direction tells where the combined force points. The azimuth shows the angle in the horizontal plane. The elevation shows how far the vector rises above that plane. Direction cosines give angles from the positive x, y, and z axes. These values help in statics, mechanics, robotics, and field measurement work. The equilibrant is also useful. It has the same magnitude as the resultant. It points in the opposite direction. That force would balance the system.
Practical Uses
Use this tool when checking free body diagrams. It can verify crane loads, cable pulls, wind forces, or applied lab forces. You can enter data as polar values, two dimensional components, or three dimensional components. Keep units consistent across all three vectors. Newtons are common. Pounds force also work when every input uses the same unit. A correct sign is important. A negative component changes direction. A wrong angle can change the final answer greatly.
Good Calculation Habits
Review the coordinate convention before entering values. Use counterclockwise angles from the positive x axis unless your problem states otherwise. For 3D polar input, use elevation from the x-y plane. Check the component table after solving. Large opposing components often cancel. Small rounding changes may appear. Increase decimal places for more detailed reports. Export the CSV for spreadsheets. Use the report button for saved worksheets.
FAQs
What is a resultant force?
A resultant force is one force that replaces several forces. It gives the same combined effect. It has magnitude, direction, and components.
Can this calculator handle three dimensional vectors?
Yes. Choose 3D components or 3D polar mode. The calculator then includes the z component, elevation angle, and direction angles.
What angle convention is used?
The 2D angle and azimuth are measured from the positive x axis. Positive angles rotate counterclockwise in the x-y plane.
What is elevation angle?
Elevation is measured from the x-y plane. A positive value points upward. A negative value points downward.
What is the equilibrant force?
The equilibrant is equal in magnitude to the resultant. It points in the opposite direction. It would balance the three forces.
Can I use pounds force instead of newtons?
Yes. Enter lb, lbf, N, or another label. Keep all three input forces in the same unit for correct output.
Why are component signs important?
Signs show direction along each axis. A negative x component points left. A negative y component points downward in standard diagrams.
Why does the result show direction angles?
Direction angles describe the resultant relative to the positive x, y, and z axes. They are useful in 3D statics and mechanics.