The formula this page uses
R = (ΣFₓ, ΣF_y), |R| = √(ΣFₓ² + ΣF_y²), θ = atan2(ΣF_y, ΣFₓ)
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Resultant Vector Calculator
Solve resultant vector problems with clear steps, notation, and a final check.
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Calculate without using AI.
Evaluate the governing formula locally in your browser. Define each known quantity once, then change values to test another case instantly.
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Vectors
Resultant Vector Calculator explained
The short version
- The resultant is the single vector that does the same job as two or more vectors together.
- Add components, not magnitudes: 60 and 80 give 140 only when they point the same way.
- Break each vector into x and y parts, add the parts, then rebuild the length and the angle.
What each part means
SymbolWhat it means
Fₓ — Horizontal componentF · cos θ for each vector, in the same force or speed unit.
F_y — Vertical componentF · sin θ for each vector.
|R| — Resultant magnitudeThe strength of the single equivalent vector.
θ — Resultant directionMeasured anticlockwise from the positive x-axis unless the problem sets another reference.
Show your work: a full example
- Two forces on one point60 N at 30° and 80 N at 120°
- Components of the first(60 cos 30°, 60 sin 30°) = (51.96, 30.00)
- Components of the second(80 cos 120°, 80 sin 120°) = (−40.00, 69.28)
- Add the x parts and the y parts(51.96 − 40.00, 30.00 + 69.28) = (11.96, 99.28)
- Magnitude of the resultant√(11.96² + 99.28²) = √(143.0 + 9856.5) ≈ 100.0 N
- Directionθ = atan2(99.28, 11.96) = 83.13° from the positive x-axis
- Cross-checkthe two forces are 90° apart, so √(60² + 80²) = √10000 = 100 N exactly
A second, different case
- A different case: a boat crossing a riverthe boat heads north at 8 km/h, the current pushes east at 6 km/h
- Write the componentsboat (0, 8) and current (6, 0)
- Add(6, 8)
- Magnitude√(36 + 64) = √100 = 10 km/h over the ground
- Directionthe angle east of north is arctan(6/8) = 36.87°, so the boat drifts well downstream
- Now aim upstream to land straight acrosssin α = 6/8 = 0.75, so α = 48.59° upstream of north
- The cost of aiming upstreamthe useful northward speed drops to √(8² − 6²) = √28 ≈ 5.29 km/h
Copy-ready example
60 N at 30° and 80 N at 120°
Two forces on one point
Questions about the Resultant Vector Calculator
Why can I not just add the two magnitudes?
Because part of each vector pulls in a direction the other does not. Only when they are exactly parallel does every bit of both add up, which is the 140 N row in the table. At 180° they fight and leave just 20 N.
What is the equilibrant?
The vector that cancels the resultant, so it has the same magnitude and the opposite direction. For the worked example it is 100 N at 83.13° + 180° = 263.13°. Adding it to the two original forces leaves the point in balance.
How do bearings differ from the angles here?
A bearing is measured clockwise from north, while these angles are anticlockwise from east. The boat's resultant is 36.87° east of north, which is bearing 036.9°, but 53.13° in the standard convention.
Can three vectors have a zero resultant?
Yes, and that is exactly what equilibrium means. If the components in x sum to zero and the components in y sum to zero, the object does not accelerate, no matter how large the individual forces are.