Measure plate motion in a shared frame

Compare two motions before naming their relationship

Two markers can both move east while becoming farther apart, closer together or maintaining their separation. Their individual arrows are not enough: compare their velocities in the same reference frame. A reference frame is the coordinate system relative to which positions and motions are reported.

The investigation uses an invented local example, not a live map or GPS feed. Plate A lies west of a straight north-south boundary, and plate B lies east. Positive east points to the right; positive north points upward. The across-boundary control supplies B’s eastward velocity relative to A, and the along-boundary control supplies B’s northward velocity relative to A. Negative settings reverse those directions.

Subtract matching components

In the shared frame, A always moves 2 cm/year east and 1 cm/year north. At the default controls, B moves 5 cm/year east and 5 cm/year north. Imagine watching from a marker moving with A: subtract A’s motion from B’s motion.

The relative east component is 5 − 2 = 3 cm/year. The relative north component is 5 − 1 = 4 cm/year. Written as a pair, B relative to A is (3, 4) cm/year. The order is always east, then north. An ordered pair is a compact way of keeping those two perpendicular directions separate.

More generally, let a be the across-boundary control and b the along-boundary control, both in cm/year. The browser constructs A = (2, 1) and B = (2 + a, 1 + b). Thus B − A = (a, b). Read this as “subtract the first plate’s east and north velocities from the second plate’s corresponding velocities.” These formulas specify the teaching model rather than estimating velocities from measurements.

Draw the relative vector

On squared paper draw perpendicular east and north axes. Choose one square to represent 1 cm/year and label that scale. From the origin move three squares east and four north. Draw the arrow from the origin to the endpoint. Its horizontal and vertical parts reproduce the controls; its diagonal length represents relative speed.

For perpendicular components, the Pythagorean relation gives speed = √(a² + b²). At (3, 4), this is √(9 + 16) = 5 cm/year. Adding 3 + 4 would count two perpendicular segments, not the straight arrow. Speed is nonnegative, whereas a component can be negative to record direction.

Change a to −3 while keeping b = 4. Predict the arrow’s direction and length before checking. The arrow now points northwest, but its speed remains 5 cm/year because squaring removes the component’s sign. Equal speeds therefore need not mean equal velocities.

Zero relative motion does not freeze the frame

Set a = 0 and b = 0. Both plates then have shared-frame velocity (2, 1) cm/year, so their relative velocity is zero. They move together in this constructed frame. Saying “B does not move relative to A” is more precise than saying “neither plate moves.”

As a paper experiment, add (7, −2) cm/year to both default shared-frame velocities. A becomes (9, −1) and B becomes (12, 3). Subtraction still gives (3, 4). A common change of translational frame changes the individual velocities but cancels out of their difference. This exercise keeps the axes parallel; it does not explore rotating coordinate systems or spherical plate geometry.

Connect the arithmetic to evidence

Scientists use repeated satellite positioning to estimate present crustal velocities and geological records to infer longer-term average motions. Those estimates have timescales, reference frames and uncertainties; the browser’s exact slider values are supplied assumptions. USGS explanation of measuring plate motion.

Create an evidence ledger with separate rows for supplied quantities, calculated quantities and missing observations. Put A’s velocity and the control settings in the first row, B minus A in the second, and an actual station time series in the third. This prevents a calculated arrow from being mislabeled as a new measurement.

Transfer to a different pair

In another shared east-north frame, A moves (−1, 2) cm/year and B moves (2, −2) cm/year. Calculate B relative to A and its speed before revealing the answer. Does the north component point north?

Check the plate-motion transfer

B minus A is (2 − (−1), −2 − 2) = (3, −4) cm/year. Its speed is 5 cm/year. The negative north component means southward motion relative to A. Neither adding the two velocities nor dropping the signs would preserve the correct relationship.