Resultant magnitude
Magnitude of the vector sum, not the sum of magnitudes.
Physics · Forces & momentum
Resultant force calculator
Add two, three or four forces in a plane. Enter each magnitude and counterclockwise angle from +x; inspect the resultant, its direction and signed components.
Created v1.0
By Thibaut Grzelak, Mechanical Analysis Engineer
02
Results
Direction from +x
Signed angle from −180° to 180°. Indeterminate at zero resultant (hidden).
Horizontal component
Positive toward +x.
Vertical component
Positive toward +y.
Relation and numerical substitution
R = √(ΣF_x² + ΣF_y²)—
Concurrent forces or a translational force balance in two dimensions. All angles share the same Cartesian reference.
01Formulas and symbols
Formulas used
ΣF_x
ΣF_x = F1 × cos(θ1) + F2 × cos(θ2)Positive toward +x.ΣF_y
ΣF_y = F1 × sin(θ1) + F2 × sin(θ2)Positive toward +y.ΣF_x
ΣF_x = F1 × cos(θ1) + F2 × cos(θ2) + F3 × cos(θ3)Positive toward +x.ΣF_y
ΣF_y = F1 × sin(θ1) + F2 × sin(θ2) + F3 × sin(θ3)Positive toward +y.ΣF_x
ΣF_x = F1 × cos(θ1) + F2 × cos(θ2) + F3 × cos(θ3) + F4 × cos(θ4)Positive toward +x.ΣF_y
ΣF_y = F1 × sin(θ1) + F2 × sin(θ2) + F3 × sin(θ3) + F4 × sin(θ4)Positive toward +y.R
R = √(ΣF_x² + ΣF_y²)Magnitude of the vector sum, not the sum of magnitudes.θ_R
θ_R = atan2(ΣF_y, ΣF_x)Signed angle from −180° to 180°. Indeterminate at zero resultant (hidden).02Assumptions and limits
Scope of validity
- Concurrent forces or a translational force balance in two dimensions. All angles share the same Cartesian reference.
03Validation example
Reference numerical case
- 30 N at 0° plus 40 N at 90° gives ΣFx = 30 N, ΣFy = 40 N, R = 50 N and θR = 53.1301°.
04References
Technical references
- University Physics 1, §2.2 Vector components — OpenStax
FAQ
Can I enter negative force magnitudes?
Use a nonnegative magnitude and reverse the direction by 180°. Components may be negative.
Does R = 0 imply complete static equilibrium?
Only translational force balance. Rotational equilibrium also requires the sum of moments to vanish.
05Model and conventions
Add two, three or four forces in a plane. Enter each magnitude and counterclockwise angle from +x; inspect the resultant, its direction and signed components.
06Limits of this model
This does not calculate moments or the line of action of nonconcurrent loads. Tiny components at floating-point round-off scale are treated as zero.
07Common mistakes
Do not add magnitudes unless all forces share a direction. A zero resultant has no defined angle: its direction is deliberately omitted.