One plate rotation produces different velocity vectors at different places
Two stations lie on the same ideal rigid plate, yet their velocity arrows have different directions. That is not automatically evidence that the plate is stretching. Motion on a sphere can be described by an angular-velocity vector, and the linear velocity at each point depends on where that point lies relative to the rotation axis.
This guide builds that relationship from a cross product and tests it with original spherical examples. You will change the site without copying its neighbour’s speed, change the reference rotation without inventing a physical acceleration, and investigate why a single station cannot determine every component of an Euler vector.
All positions, angular velocities and uncertainties below are fictional teaching inputs. They are not published plate-model coefficients or a hazard assessment. The existing station-velocity guide explains fitting a trend from a coordinate series; here the separate question is how a rigid-motion model predicts and compares velocities across locations.
Geodesy subject guide · Learning that transfers into life · How X Works hub
Choose your reading route
Route 1 · Chapters 1–2: Define rigid angular motion
Route 2 · Chapters 3–5: Calculate and check local velocity
Route 3 · Chapters 6–8: Change reference and fit geometry
Route 4 · Chapters 9–10: Interpret uncertainty and residuals
Use uncertainty to distinguish a fit from an exact prediction
Route 5 · Chapters 11–12: Transfer and report
Full chapter index · Independent worked case · Questions and practical checks
Chapter index
Define rigid angular motion
Keep angular velocity distinct from a station speed
Define the cross product in compatible axes
Calculate and check local velocity
Work an equatorial site and an axis-aligned limit
Change location without transferring a neighbour’s vector
Convert rotation rates before calculating displacement
Change reference and fit geometry
Translate Cartesian velocity into local directions
Subtract a reference angular velocity consistently
Show what one station can and cannot identify
Interpret uncertainty and residuals
Use uncertainty to distinguish a fit from an exact prediction
Interpret residuals without diagnosing a cause automatically
CHAPTER 1 OF 12 · Define rigid angular motion
1. Keep angular velocity distinct from a station speed
An Euler vector is an angular-velocity vector describing a declared rigid rotation relative to a reference. Its direction identifies the rotation axis and its magnitude the rate. A station velocity is a length-per-time vector at a particular position. The latter follows from the former only after the station geometry and coordinate representation are supplied.
The EarthScope/UNAVCO plate-motion calculator distinguishes the attributed plate motion from the reference motion and supports custom Euler vectors. Those interfaces illustrate why a calculator needs more than one universal plate speed. Its documentation also notes limitations of model uncertainty treatment; an output’s decimal formatting must not be interpreted as an independent accuracy guarantee.
An Euler pole of plate motion is not the instantaneous Earth rotation pole used in pole-tide deformation modelling. Both involve rotational language, but the physical quantities and time scales differ. The pole-tide companion follows Earth-orientation inputs into a deformation model; this article follows an ideal rigid angular velocity into a site’s linear velocity.
Start by identifying the target: predicted rigid motion of a specified point relative to a specified reference under a named model. A velocity value without its reference and position is incomplete. A different arrow on a map can represent a different site, another reference rotation, actual local deformation or a measurement issue; the label alone does not decide which.
CHAPTER 2 OF 12 · Define rigid angular motion
2. Define the cross product in compatible axes
For a rigid spherical teaching model, let r be the geocentric position vector and ω the angular-velocity vector in the same right-handed Cartesian axes. The instantaneous linear velocity is v = ω×r. With ω expressed in radians per year and r in metres, v is in metres per year.
In components, vx = ωy rz−ωz ry, vy = ωz rx−ωx rz, and vz = ωx ry−ωy rx. These equations make the ordering explicit. Reversing the cross product to r×ω reverses every velocity component; the resulting magnitude can remain plausible while the motion direction is wrong.
The cross product is perpendicular to both input vectors. Therefore r·v = 0 under this rigid spherical model: the predicted motion has no geocentric radial component. That is a useful algebraic invariant. It does not prove that a real station has no vertical motion from any other process.
A real plate model and geodetic reference can involve additional definitions beyond this elementary spherical account. Preserve the actual model’s coordinate and velocity conventions. The cross-product examples explain the geometry; they do not establish that every observed station belongs to a perfectly rigid plate or that a global rotation model captures all local deformation.
CHAPTER 3 OF 12 · Calculate and check local velocity
3. Work an equatorial site and an axis-aligned limit
Choose fictional angular velocity ω = (0,0,10⁻⁹) radians per year and station r1 = (6,000,000,0,0) metres. The cross product is v1 = (0,0.006,0) metres per year. In this spherical arrangement at zero latitude and longitude, positive y is local east, so the predicted speed is six millimetres per year eastward.
Check perpendicularity: r1·v1 is zero. The speed also equals |ω|R because the position is perpendicular to the rotation axis. These checks inspect different features of the calculation: a scalar speed relation and a vector-direction relationship. A correct magnitude does not by itself validate the sign of the east component.
Now place a fictional station on the rotation axis at raxis = (0,0,6,000,000). The cross product is zero. A nonzero plate angular velocity therefore does not imply nonzero linear speed at every position. This limit is particularly useful for exposing an implementation that multiplies angular rate by radius without retaining the angle to the axis.
The selected radius and rate are artificial. They teach the relation without importing a real plate assignment or reference frame. For actual stations, use compatible positions, model coefficients and endpoint definitions rather than copying the round equatorial result into an operational epoch-propagation calculation.
CHAPTER 4 OF 12 · Calculate and check local velocity
4. Change location without transferring a neighbour’s vector
Keep the original ω but choose r2 = (3,600,000,4,800,000,0) metres. Its radius is still six million metres. The cross product becomes v2 = (−0.0048,0.0036,0) metres per year. Its speed is √(0.0048²+0.0036²) = 0.006, matching the first equatorial site’s speed but not its Cartesian direction.
The dot product r2·v2 is −17,280+17,280 = zero in the corresponding units. The different Cartesian components follow the site’s different direction around the sphere. Copying v1 to this site would violate the perpendicularity check and would not describe the same rigid rotation.
Choose a third spherical site at geocentric latitude sixty degrees and longitude zero: r3 = (3,000,000,0,3,000,000√3) metres. Its distance from the z rotation axis is three million metres. Its predicted velocity is (0,0.003,0), half the equatorial speed.
The general speed relation is |v| = |ω||r|sinψ, where ψ is the angle between position and rotation axis. A single scalar plate speed hides that geometry. The transferable operation is to evaluate the vector at the new position, not assume that membership in a common rigid plate means identical arrows everywhere.
CHAPTER 5 OF 12 · Calculate and check local velocity
5. Convert rotation rates before calculating displacement
Plate angular rates are often expressed in degrees per million years. Convert the angular unit and the time unit explicitly before using the cross product. One degree per million years is π/(180×10⁶) radians per year, approximately 1.745329252×10⁻⁸. It is not one degree per year divided only by a familiar length scale.
At a fictional equatorial radius of six million metres, that angular rate gives approximately 0.104719755 metres per year, or 104.719755 millimetres per year, when the site is ninety degrees from the axis. Near the axis the speed is smaller according to the sine factor. The example is a conversion, not a typical rate asserted for a particular plate.
A second check reverses the calculation. A six-millimetre-per-year equatorial speed at that radius corresponds to 10⁻⁹ radians per year, approximately 0.05729578 degrees per million years. Keeping units visible in both directions helps catch a missing factor of a thousand between metres and millimetres or a million between years and million years.
Velocity is not a finite coordinate displacement until a time interval and propagation model are supplied. Multiplying a constant velocity by a short interval is a linear approximation to motion; applying a rigid finite rotation follows a different exact geometric relationship. Keep that distinction when extending an instantaneous model into a trajectory.
CHAPTER 6 OF 12 · Change reference and fit geometry
6. Translate Cartesian velocity into local directions
At the spherical site r2, the local radial unit vector is u = (0.6,0.8,0). A corresponding east unit vector is e = (−0.8,0.6,0), and north is n = (0,0,1). These form an orthonormal local basis for the stated equatorial geometry.
Project v2 onto east: e·v2 = (−0.8)(−0.0048)+0.6×0.0036 = 0.006 metres per year. Its north and radial projections are zero. The local result agrees with the first station’s eastward six-millimetre-per-year motion even though their Cartesian component triples differ.
For an oblique Euler vector, local north and east contributions can both be nonzero. Construct the basis at the actual location and project the full vector. Calling Cartesian x east at every site would silently apply the first station’s local geometry to the entire globe.
The spherical radial basis used here is a declared teaching choice. A real product’s local Up may follow ellipsoidal geometry. Use the appropriate coordinate transformation rather than treating the word horizontal as sufficient specification. Different local axes can change component values without changing the underlying physical velocity vector.
CHAPTER 7 OF 12 · Change reference and fit geometry
7. Subtract a reference angular velocity consistently
Suppose ωP describes the attributed rigid plate rotation and ωR the chosen reference rotation, both expressed in compatible axes and units. The relative angular velocity is ωrel = ωP−ωR. The relative station velocity in the corresponding instantaneous account is (ωP−ωR)×r.
Use the original fictional ωP = (0,0,10⁻⁹) and choose ωR = (0,0,0.25×10⁻⁹) radians per year. The relative equatorial velocity at r1 is (0,0.0045,0) metres per year. It differs from six millimetres per year because the reference moves, not because the physical station necessarily accelerated.
If the reference is chosen to rotate with the same ideal plate, ωR = ωP, the predicted relative velocity is zero for every point following that same rigid motion. Zero plate-relative velocity does not imply that the point has zero velocity in every global reference. The qualifier relative to which frame is essential.
Do not subtract scalar speeds when the rotations have different axes. Subtract the angular-velocity vectors first, or subtract compatible full linear-velocity vectors at the same site. Equal speed magnitudes with different directions can leave a nonzero relative vector; a subtraction of magnitudes would erase the directional information.
CHAPTER 8 OF 12 · Change reference and fit geometry
8. Show what one station can and cannot identify
At the fictional site r1 = (R,0,0), the component equations reduce to vx = 0, vy = Rωz and vz = −Rωy. The station constrains two angular-velocity components through its tangential velocity. It does not constrain ωx, the component parallel to its own position vector.
Adding any multiple of r1 to an angular-velocity vector leaves its cross product with r1 unchanged. For example, ω = (0,0,10⁻⁹) and ω′ = (2×10⁻⁹,0,10⁻⁹) predict the same velocity at r1. A single station cannot distinguish them in this idealised model, even with noiseless velocity observations.
Add a second station at rB = (0,R,0). Its equations include vx = −Rωz and vz = Rωx, revealing the previously unobserved component. For synthetic ω = (2,−1,3)×10⁻⁹ radians per year and R = six million metres, the first station velocity is (0,0.018,0.006), while the second is (−0.018,0,0.012).
These two nonparallel positions identify the three angular components in the ideal setup and provide redundant information about ωz. Real network geometry, weighting, covariance and deformation complicate the fit. The lesson is to inspect independent sensitivities rather than assume that a large number of stations automatically supplies an equally informative rotation estimate.
CHAPTER 9 OF 12 · Interpret uncertainty and residuals
9. Use uncertainty to distinguish a fit from an exact prediction
For a restricted example, treat R as exact and estimate ωz from vy/R at the first station. If vy has standard uncertainty one millimetre per year and R = six million metres, the angular standard uncertainty is 0.001/6,000,000, approximately 1.66667×10⁻¹⁰ radians per year.
Predict a compatible equatorial velocity at another position using that same uncertain angular component. Its corresponding standard uncertainty is again one millimetre per year in the appropriate tangential direction. Predictions at different sites share the uncertainty of the fitted Euler vector; they are not independent just because they occupy separate rows in a results table.
If ωz is estimated from both vy at r1 and minus vx at rB, with independent equal one-millimetre-per-year uncertainties, averaging the two equivalent observations reduces its standard uncertainty by √2. This reduction depends on their independence and the rigid-motion model; shared frame errors or local deformation can change the uncertainty account.
The calculator documentation states that it does not incorporate each model’s velocity uncertainty in the displayed calculation. A deterministic output can still be useful for the documented prediction, but its absence of an uncertainty field is not proof of zero uncertainty. A scientific comparison needs an additional supported uncertainty account.
CHAPTER 10 OF 12 · Interpret uncertainty and residuals
10. Interpret residuals without diagnosing a cause automatically
Define a model residual as observed station velocity minus the compatible predicted rigid velocity. A nonzero residual can arise from local deformation, an unsuitable plate assignment, frame mismatch, observation uncertainty or other unmodelled contributions. It is evidence about disagreement under a model, not an automatic identification of a geological cause.
For a fictional test, let the predicted local velocity be east six and north zero millimetres per year. A measured estimate is east seven and north two. The residual is (1,2). Its norm is √5, approximately 2.23607 millimetres per year. That magnitude does not itself establish whether the difference is statistically informative.
Suppose the independent observed and predicted component uncertainties are each one millimetre per year with zero cross-covariance. Each residual component has variance two. The sum of squared standardised residual components is (1²+2²)/2 = 2.5. This is a conditional diagnostic, not a universal threshold or a complete significance assessment for every real network.
A held-out station provides a different test from a station used to fit the Euler vector. Retain which observations determined the model and which evaluated it independently. An excellent fit to the calibration network can coexist with poor predictions elsewhere, particularly near a deforming boundary or under an incorrect plate assignment.
CHAPTER 11 OF 12 · Transfer and report
11. Distinguish a rigid velocity field from finite strain
Two points on a rigidly rotating sphere have velocities that can differ while their separation length remains constant under the exact rotation. Let Δr be their position difference and Δv = ω×Δr their velocity difference. The instantaneous derivative of squared separation is 2Δr·Δv, which vanishes because a vector is perpendicular to its cross product.
Using the original equatorial stations, Δr = r2−r1 = (−2.4×10⁶,4.8×10⁶,0) metres and Δv = (−0.0048,−0.0024,0) metres per year. Their dot product is 11,520−11,520 = zero. The differing velocity arrows do not imply a changing baseline length in this ideal model.
By contrast, a relative velocity with a component parallel to Δr changes separation length to first order. Distinguishing that component from a rigid rotation requires compatible positions and velocities, not simply comparing arrow directions visually. A local deformation analysis also needs an appropriate network geometry and uncertainty treatment.
This calculation does not prove that any real plate is perfectly rigid. It explains what the rigid model predicts and offers a discriminating residual question. The relevant test is whether observations depart from that prediction beyond their supported uncertainty and modelling limitations, rather than whether two arrows happen to look different on a map.
CHAPTER 12 OF 12 · Transfer and report
12. Report the model, reference and residual evidence separately
A reproducible result identifies the plate-motion model, attributed plate, reference rotation, coordinate representation and epoch, rate units, station endpoint, local-axis definition and uncertainty account. If a station lies near a plate boundary or within a deforming region, retain that limitation instead of presenting a rigid interior model as complete local motion.
For the primary fictional example, ω = (0,0,10⁻⁹) radians per year gives six millimetres per year tangential speed at the selected equatorial radius. A reference rotating at one quarter of that rate reduces the relative speed to 4.5. These are geometric teaching results, not measurements of a named tectonic plate.
The existing station-velocity guide handles how coordinate observations are fitted through time. The Helmert companion handles transformations between frame descriptions. Both become relevant before comparing observed velocities with a rigid-motion prediction, but neither operation should be silently substituted for the Euler-vector calculation itself.
The transferable lesson is to keep position and reference attached to angular motion. A plate rotation predicts a velocity field, not one arrow copied around a map. Once that field is defined and checked, residuals become meaningful questions for independent investigation rather than immediate claims that a station, plate or processing system has failed.
Independent worked case and evidence checks
Work a fresh fictional Euler vector ω = (0,2×10⁻⁹,0) radians per year. At rA = (6,000,000,0,0), the predicted velocity is (0,0,−0.012) metres per year. At rB = (0,0,6,000,000), it is (+0.012,0,0). Compute the component cross products independently before comparing the speed magnitudes.
At rC = (0,6,000,000,0), the predicted velocity is zero because the position lies on the rotation axis. These three sites test the direction, magnitude and axis limit of the same rotation. A routine that returns twelve millimetres per year at every location would fail the geometric model despite reproducing two of the speed magnitudes.
Choose a reference vector ωR = (0,0.5×10⁻⁹,0). The relative angular vector is (0,1.5×10⁻⁹,0), and the corresponding nonzero speeds at A and B are nine millimetres per year. Their direction remains as before. The change is a reference subtraction in the declared model, not new evidence that the physical plate slowed.
Now choose a reference with a different axis, ωR = (0,0,10⁻⁹). At rA, the attributed velocity is (0,0,−0.012) and reference velocity (0,0.006,0). Their difference is (0,−0.006,−0.012), with magnitude √0.00018, approximately 0.0134164 metres per year. Subtracting the scalar speeds twelve minus six would incorrectly give six millimetres per year.
As a unit exercise, a rate of half a degree per million years at the same radius gives a maximum tangential speed of approximately 52.3599 millimetres per year. At an angle thirty degrees from the axis, the speed is half that maximum, approximately 26.1799. State the angle to the Euler axis rather than substituting geographic latitude without checking where that axis lies.
For an inverse-geometry check, suppose a noiseless station at rA has observed velocity (0,0.018,0.006) metres per year under a rigid model. It determines ωz = 3×10⁻⁹ and ωy = −10⁻⁹. It does not determine ωx. Write two different ωx values giving the same observation and explain why the unobserved component is parallel to the site position.
Add the earlier rB = (0,6,000,000,0) station with vz = 0.012. It determines ωx = 2×10⁻⁹. If its vx is minus 0.018, it independently agrees with ωz = 3×10⁻⁹. This reconstruction makes the information supplied by each station visible rather than attributing uniqueness to a numerical solver without inspecting the geometry.
Now alter that held-out vx to minus 0.016. The two stations imply different ωz values. With equal independent uncertainty, a least-squares average can provide a compromise, but the discrepancy still needs inspection. A fitted answer does not erase incompatible evidence or establish that the rigid model is the correct physical explanation.
For a residual covariance example, suppose two station predictions share an uncertain reference-rotation contribution. A difference of observed-minus-predicted velocities can cancel part of that shared error under the correct geometry. Treating the two predictions as independent may misstate the uncertainty of their relative motion. Carry the Euler-vector covariance through the site-specific response matrices.
Finally distinguish two tasks in a handover note: estimating each station’s trend from its time series, and fitting or applying a rigid Euler-vector model to those trend estimates. Keep the frames, epochs, uncertainty and calibration/validation roles attached at both stages. Another reader should be able to identify whether a discrepancy entered through temporal estimation, spatial modelling or reference transformation.
One last sensitivity exercise compares two independent estimates of the same angular component. A station with six-million-metre lever arm and tangential uncertainty one millimetre per year gives angular standard uncertainty 1.66667×10⁻¹⁰ radians per year. A second station with three-million-metre lever arm and the same velocity uncertainty gives twice that angular uncertainty. Equal velocity precision does not mean equal information about the rotation.
Under the restricted independent-noise model, inverse-variance weighting gives the first angular estimate four times the weight of the second. Their combined standard uncertainty is the first uncertainty divided by √1.25, approximately 1.490712×10⁻¹⁰ radians per year. Its predicted uncertainty at the six-million-metre lever arm is approximately 0.894427 millimetres per year. The improvement follows the sensitivity and covariance assumptions, not merely the addition of another station name.
Worked comparison
| Quantity or test | What it establishes | What it does not establish alone |
|---|---|---|
| Euler vector | A declared rigid angular motion | One identical linear vector at every site |
| Cross-product prediction | Velocity at a specified compatible position | Absence of local deformation |
| Reference subtraction | Motion relative to a selected rotation | A physical change in the plate’s motion |
| Single-station fit | Two tangential angular sensitivities in the ideal model | All three Euler components |
| Held-out residual | Disagreement with the declared prediction | A unique geological cause |
Questions and practical checks
Should every point on a rigid plate have the same velocity arrow?
No. A common angular velocity produces site-dependent linear vectors.
Does zero plate-relative speed mean no global motion?
No. It may mean the point follows the selected rotating reference.
Can one station determine an entire Euler vector?
Not in the ideal model. A component parallel to its position remains unobserved.
Do different arrows prove plate strain?
No. A rigid rotation can give different arrows while preserving baseline length.
Does a model residual prove local fault movement?
No. Plate assignment, frames, observation uncertainty and other effects must also be investigated.
Continue with the connected subjects
Establish the measured velocity and reference
Estimate station velocity from a coordinate series with temporal uncertainty
Keep coordinate and frame epochs compatible before comparing motion
Compare other geodetic vectors
Distinguish short-term Doppler velocity from a secular station trend
Project vectors and carry shared covariance through changed geometry
Sources and scope
EarthScope/UNAVCO plate-motion calculator: Euler-vector, reference and uncertainty interfaces
EarthScope/UNAVCO geodetic utilities: rigid plate-motion model context
The numerical cases in this guide are original fictional teaching examples, not measurements from a real site. This is an educational explanation, not a field safety instruction, engineering approval or an official certification. For a real project, use qualified practitioners, current product documentation and site-specific evidence. Sources were checked on 31 October 2026.
Connected guides: rotation, reference choices and dynamic loads
Pole-tide deformation: polar-motion inputs, component signs and secular-pole conventions Use this when a changing rotation axis and a chosen reference pole become station-deformation inputs.
Non-tidal ocean loading: bottom pressure, inverted-barometer departures and combined-load accounting Follow this route when a changing ocean mass field must be reconciled with atmospheric loading without double counting.
Helmert transformations: rotation conventions, scale units and inverse checks Use this guide when two coordinate frames disagree about rotation signs, scale units or inverse parameters.