Resultant Force Calculator
Calculate resultant force magnitude and direction from multiple force vectors using component addition and Pythagorean theorem.
What Is Resultant Force?
Force is a vector quantity with both magnitude and direction. When two or three forces act on a body at the same point, their combined effect is a single resultant force found by vector addition. This is the net force that determines acceleration through Newton's second law.
Vector Addition of Forces
Resolve each force into horizontal and vertical components:
$$F_x = \sum F_i \cos\theta_i, \quad F_y = \sum F_i \sin\theta_i$$The resultant magnitude and direction are:
$$R = \sqrt{F_x^2 + F_y^2}, \quad \theta_R = \arctan\left(\frac{F_y}{F_x}\right)$$Angles are measured from the positive x axis, counterclockwise. Use \(\mathrm{atan2}(F_y, F_x)\) to get the correct quadrant.
Worked Example
Two perpendicular 100 N forces at 0° and 90° give \(F_x = 100\,\mathrm{N}\), \(F_y = 100\,\mathrm{N}\), and \(R = 100\sqrt{2} \approx 141.4\,\mathrm{N}\) at 45°. Adding a third force changes both magnitude and direction of the net vector.
Related tools: Vector Addition Calculator, Force Calculator, and Magnitude of Acceleration Calculator.
Frequently Asked Questions
How many forces can this calculator combine?
You can add two or three force vectors. Each force needs a magnitude in newtons and an angle in degrees from the positive x axis.
Which angle convention is used?
Angles are measured counterclockwise from the +x axis. 0° points right, 90° points up, 180° points left, and 270° points down.
What if forces act at different points?
This tool finds the vector sum of forces. If forces act at different locations, you may also need to compute net torque for rotational effects.
Can the resultant be zero?
Yes. If vector components cancel, the net force is zero and the object is in translational equilibrium (no acceleration from those forces).
How is resultant force related to acceleration?
Newton's second law states \(\vec{F}_{net} = m\vec{a}\). The resultant force divided by mass gives the acceleration vector in the same direction.
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