Vector Calculator for Balancing and Vibration
Unbalance is a vector. It has a size and an angular location, and the moment you do anything useful with it — add a trial weight, subtract a previous run, work out what correction to apply — you are doing vector arithmetic, not ordinary arithmetic.
This calculator adds and subtracts two vectors. Enter each one in polar form (magnitude and angle) or in rectangular form (X and Y). Leave the other pair blank and it will be filled in for you.
Vector Calculator
Vector 1
Vector 2
Angles are measured counter-clockwise from the X axis. Units cancel, so any consistent unit works.
On this page
Why balancing is vector arithmetic
A balancing machine reports unbalance as a magnitude and an angle — say 180 g·mm at 47°. Two such readings cannot be combined by adding the magnitudes. 100 at 0° plus 100 at 180° is zero, not 200. Everything in the correction process depends on respecting that.
The two operations that matter:
Addition combines unbalances that exist together — the contribution of two components on a shared assembly, or an existing unbalance plus a weight you are about to add.
Subtraction isolates a change. This is the operation behind trial weight work: measure the rotor, add a known trial weight, measure again, and the vector difference between the two readings is the effect of the trial weight alone.
Converting between polar and rectangular form
Vectors are read in polar form and calculated in rectangular form. The conversion both ways:
X = Magnitude × cos(Angle) Y = Magnitude × sin(Angle) Magnitude = √(X² + Y²) Angle = atan2(Y, X)
Addition and subtraction are done on the X and Y components separately, then converted back. Doing it any other way produces a wrong answer that looks plausible, which is the most expensive kind.
Worked example: trial weight
A rotor reads 120 g·mm at 30° on the initial run. A trial weight is added and the rotor is run again, now reading 200 g·mm at 75°.
The effect of the trial weight is the vector difference between the second reading and the first:
Run 2 − Run 1 = (200 ∠ 75°) − (120 ∠ 30°)
Resolving each into components, subtracting, and converting back gives the trial weight’s effect vector. That vector is what lets you scale and rotate the trial weight into the correction the rotor actually needs. The ratio of the original unbalance to the trial effect gives the size; the angle between them gives the rotation.
Enter those numbers in the calculator above and you will get both the sum and the difference, along with the rectangular components of each input.
Angle conventions, and the mistake they cause
This calculator measures angles counter-clockwise from the X axis, which is the standard mathematical convention.
Balancing machines do not universally follow it. Some report clockwise. Some reference a keyway, a timing mark, or an encoder zero rather than a coordinate axis. Some report the location of the heavy spot; others report where to add or remove material, which is 180° opposite.
None of that changes the arithmetic, but all of it changes the answer you act on. Before applying a correction, confirm three things about your machine: which direction positive angles run, where zero is, and whether the reported angle is the heavy spot or the correction location. A 180° error doubles the unbalance instead of removing it, and it is the single most common way a correction makes a rotor worse.
Frequently asked questions
Can I enter one vector in polar form and the other in X and Y?
Yes. Each vector is resolved independently. Fill in whichever pair you have for each one.
What if I enter both pairs for the same vector?
Magnitude and angle take precedence, and the X and Y fields are overwritten with the resolved values.
Why does the result show a negative angle?
Angles are returned in the range −180° to +180°. A result of −53° is the same direction as 307°.
Does this work for units other than unbalance?
Yes. The arithmetic is unit-agnostic, so it works equally for vibration amplitude and phase, force, or displacement, as long as both vectors use the same unit.
Does it handle more than two vectors?
Two at a time. For more, combine the first two, then use that result as Vector 1 for the next.
Related resources
ISO 21940-11 balance tolerance · Process capability calculator · GD&T symbols reference · Balancing machines
