Position, Displacement, and Reference Frames
MECH.NEWT.POS.0001
Position, Displacement, and Reference Frames
▶ Video lesson
§ The physical picture
Stand at a fixed lamppost in a park and watch a drone hover. To pin down the drone completely you must say two things: how far it is from the lamppost, and in which direction. Draw that as an arrow — tail at the lamppost, head at the drone — and you have the position vector . The lamppost is playing the role of the origin: the agreed reference point that all measurements hang from.
Lay a coordinate grid over the park, axes crossing at the lamppost, and the arrow casts two "shadows": one along the -axis of length , one along the -axis of length . Those shadows are the components of the position vector, and they are just the coordinates of the drone. The single arrow and the pair of numbers carry identical information — the arrow is the geometric costume, the pair of numbers the algebraic one, and the first skill of this topic is changing costumes quickly.
Now let a dog loose in the park. It sprints in loops, doubles back, investigates three trees, and finally flops down 20 m from where it started. Its GPS collar logs 300 m of running — that is distance, a scalar that only ever grows. Its displacement is the single straight arrow from the release point to the flop point: 20 m, northeast. A crow flying directly between those two points would trace exactly that arrow. Displacement is the crow's-eye summary of any journey, however chaotic — and if the dog trots back to its owner, the arrow shrinks to the zero vector while the odometer keeps its 300 m. The two measures answer different questions, and mechanics needs both: the kinematic machinery (velocity, the of every equation you will meet) runs on displacement, while path-dependent costs — fuel burned, work done against friction — run on distance.
One subtlety deserves respect from day one: the origin is your choice. Put it at the launch pad, the classroom door, the centre of the Earth — physics does not care, as long as you commit to one origin and stick with it for the whole problem. Two mission controllers tracking the same drone from different vans write down different position vectors, and both are right. What they always agree on is the arrow between any two things they both watch — because shifting the origin shifts every position vector by the same amount, and differences shrug that shift off. Measure the same trip from two different origins and every position vector changes, but the subtraction wipes the bookkeeping out: everyone, everywhere, gets the same displacement arrow.
The full package — an origin, a set of (usually orthogonal) axes, and a clock, all rigidly attached to some physical body — is a reference frame. The park frame, the train frame, the Earth-centred frame of GPS: each is a complete measuring apparatus, and the same event gets different numbers in each. That dependence is not a defect; it is the central organizing fact of kinematics. The craft, built below, is telling apart the quantities that are private to a frame (positions, and — once frames move relative to each other — displacements too) from the quantities every rigid frame agrees on (the distance between two particles at the same instant).
§ The math of "where"
Everything in this topic is a conversation between an arrow's two costumes: the component triple and the magnitude-plus-direction description. With axes chosen, the position of a point is written
where: — position vector from origin to the particle; — signed coordinates (negative means the other side of the origin); — unit vectors, arrows of length exactly 1 along the axes, which do nothing but carry direction. Read (1) as walking instructions, not abstract algebra: go metres along the -direction, then along , then up, and the arrow points to wherever you stopped. For motion in a plane, drop the last instruction. How long is the arrow? The components are the legs of a right triangle (in 3D, two nested right triangles) whose hypotenuse is the arrow itself — the drone 40 m east and 30 m north is not 70 m from the lamppost, because the straight line cuts the corner. Pythagoras does the cutting:
where: — the particle's distance from the origin, always positive or zero. The squares kill the signs, which is why a magnitude can never be negative: a point at is exactly as far from the origin as one at . Notice also what (2) throws away: knowing the distance is "5 km away", a whole circle of possibilities. The missing half of the answer lives in the angle: in the plane, measured anticlockwise from the -axis, plus 180° whenever , because arctan alone cannot tell a point from its diametric opposite — the worked examples spring this trap on a real radar fix. Now do the first genuine piece of vector algebra of mechanics. If a particle moves from to , the arrow that gets you from the head of to the head of is the displacement, tail to head — — so the definition writes itself:
where: — displacement (the Greek always reads "final minus initial"); — the endpoint positions measured from one origin. Read the right-hand side as an instruction: locate the finish, locate the start, subtract. Nothing about the route between them survives; path-independence is built into the definition at birth. Mind the order: final minus initial, always; swapping the two describes the return trip, . And each component settles its own account — a 2D displacement is two 1D displacements filed jointly.
Once distance has no such shortcut — it must honour the route, because the route is exactly what it measures. Chain the instants: in each short interval the particle covers of ground, and the odometer adds those up:
where: — path length (distance travelled), a scalar that can only grow while anything moves; — the velocity previewed for the topic ahead. The inequality is the headline: the straight arrow can never beat the odometer, and Derivation B below gives it a two-line reason. Equality holds only for straight, never-reversing motion — the crow's route. Two objects at once bring the last idea: the position of relative to — the arrow a sailor on ship would draw to ship — is another difference of positions from a common origin:
where: — positions from a common origin; — the arrow from to , origin-proof for exactly the same reason displacement is. And if a whole second frame (a train, a drifting boat) carries its origin along a trajectory as seen from frame , the tail-to-tail triangle relates the two descriptions of any particle:
where: — position in frame ; — position of the same particle in frame ; — where 's origin sits in . Everything changes with time the frames are in relative motion, and now even displacements disagree between frames — the walker at 1 m/s in the train's books and 1.86 km in the ground's. That is the doorway to Galilean transformations, opened properly two topics on; for now it is enough to see that (6) is just (5) wearing a time-dependence. One organizing table before the derivations — every equation and the one job it does:
| equation | job | reach for it when |
|---|---|---|
| , eq. (1) | coordinates → arrow | a problem hands you grid coordinates |
| , eq. (2) | components → distance from origin | the question is "how far?" (add the quadrant-checked angle for "which way?") |
| , eq. (3) | two positions → net change | anything asks "how far from the start?" |
| , eq. (4) | route → odometer reading | fuel, friction, path-dependent costs |
| , eq. (5) | two objects → relative position | "where is B as seen from A?" |
| , eq. (6) | frame → frame | translating between two observers' books |
§ Reference frames: what changes, what survives
A frame is more than an origin. Two observers may also tilt their axes relative to each other — the surveyor aligns with north, the architect with the street grid — and, more drastically, one frame may move relative to the other. The most general relation between rigid, non-moving frames is a rotation plus a translation: , where is a rotation matrix and a constant shift. Components change under rotation even though the arrow itself does not; and they are just the drone's coordinates. Do all these bookkeepings agree on? The answer — the separation between two particles — is the deepest result of this article, and it earns a real proof.
One more frame remains: the moving one, eq. (6) with genuinely changing. A passenger walking forward at 1 m/s in a train doing 30 m/s has position ; in one second her displacement is 1 m in the train's books and 31 m in the ground's. Displacement is origin-proof but not motion-proof. Yet even here the invariant survives: two passengers sitting 10 m apart in both frames at any instant, because the frames' relative motion shifts both passengers' positions identically at each instant. That is the frame that makes a problem trivial — the train frame for the passengers, the ground frame for the signal engineer — and telling them apart is genuinely professional skill in mechanics, and the systematic rules for converting velocities between such frames wait in the Galilean-transformation article that follows.
§ Limiting cases
| Case | Result | Meaning |
|---|---|---|
| one dimension | the vector collapses to a single signed number; sign carries direction | |
| straight trip, no reversal | the only case where arrow and odometer agree | |
| round trip (finish = start) | zero displacement, any amount of distance | |
| origin shift by constant | unchanged | positions are private, differences are public |
| rigid rotation of axes | components of change, unchanged | separations are the frame-proof observable |
| frame moving relative to yours | ; displacements now differ between frames | origin-proof is not motion-proof — see Derivation C |
§ Worked examples
1 · The drone on the pad (confidence builder)
A drone lifts off from its launch pad — take the pad as origin, east as , north as . It ends up hovering 40 m east and 30 m north. (a) Write its position vector. (b) How far is it from the pad? (c) In what direction?
Step 1. Components are read straight off: , , so .
Step 2. Magnitude by (2): — the 3-4-5 triangle scaled by 10.
Step 3. Direction: both components positive, first quadrant, no correction needed. north of east.
Step 4. Sanity check by projecting back: and — returns the original components exactly.
2 · The ship on the radar (exam level)
A harbour master takes the harbour master's tower as origin, with east and north. A ship shows up at coordinates ; some minutes later at . Find (a) the magnitude and direction of the ship's initial position vector, and (b) its displacement between the two fixes.
Step 1. Initial components: (west of the tower), (north of it) — second quadrant. Magnitude: (a tidy 5-12-13 triangle scaled by 5).
Step 2. Naive calculator angle: . That points east-southeast — a ship to the west.
Step 3. Quadrant correction (, so add ): anticlockwise from east — north of west.
Step 4. Displacement by (3), final minus initial, component by component: , . So : the ship moved east and slightly south, toward the harbour mouth.
Step 5. Magnitude: (a 7-24-25 triangle). Note that the displacement is nowhere near either position vector — net change and location are different animals, and only the subtraction connects them.
3 · One passenger, two frames (exam level)
A train moves due east at a steady 30 m/s. A passenger walks toward the front of the train at 1.0 m/s (relative to the train) for 60 s. Find her displacement in the train frame and in the ground frame. Then check that the separation between her and a friend seated 10 m behind her is the same in both frames at the end of the walk.
Step 1. Train frame: she walks a straight line at 1.0 m/s for 60 s, so toward the front. (No reversals, so here distance equals displacement magnitude.)
Step 2. Ground frame: use eq. (6) in reverse, . The train's origin advances while she walks 60 m within it, so east.
Step 3. The two answers differ by exactly — the accumulated shift of the moving origin. This is Derivation C made flesh: displacement is origin-proof under a fixed shift, but a moving origin shifts the two endpoints by different amounts, and the difference no longer cancels.
Step 4. Separation check: at the end of the walk the friend (who stayed seated) is 10 m behind her in the train frame; at that same instant, both positions have picked up the identical offset , so the separation is — the same in both books, exactly as Derivation C demands.
Step 5. Sanity check with speeds: in the ground frame she covers 1860 m in 60 s, an average of — the train's 30 plus her 1. Velocity addition, met here for the first time, is just the time derivative of eq. (6).
§ Where this lives today
Every time your phone locks onto GPS it solves a position-vector problem far grander than any homework: the receiver measures its distance to four or more satellites and computes its position vector in a frame whose origin is the centre of the Earth, then converts to several thousand kilometres, updated every second, accurate to a few metres. Converting that fix to your map grid is eq. (6) plus a rotation: a change of reference frame, performed billions of times a day, with Derivation C silently guaranteeing that the map gets distorted in the process. The delivery-app rider's "1.2 km away" is the crow flies (a displacement magnitude) while the route says 2.1 km of actual streets. Robotics leans on the whole trio at once: a warehouse robot's wheel odometry integrates eq. (4) (odometry), a running vector sum of little legs maintains its displacement from the dock — dead reckoning, eq. (3) iterated — and periodic fixes re-anchor its position in the warehouse frame, because odometry drifts and the vector sum is only as good as its last small error. And when a drone's "return to home" button is pressed, it does not retrace its wandering outbound path: it flies its displacement vector, reversed — the straightest possible statement of everything this article just taught.
§ Check yourself
§ Going further
Texts. Morin, Introduction to Classical Mechanics, Ch. 1-2 · Taylor, Classical Mechanics, Ch. 1 · Goldstein, Poole & Safko, 3rd ed., Ch. 1 and Ch. 4 (orthogonal transformations) · Halliday, Resnick & Walker, Fundamentals of Physics, Ch. 2-4 as an on-ramp.
Provenance. Definitions of position, displacement, distance, and reference frame follow Morin Ch. 1-2, Taylor Ch. 1, and Landau-Lifshitz §1. A verifier should recompute: (i) the telescoping cancellation showing displacement is path- and origin-independent; (ii) the triangle-inequality bound ; (iii) the invariance using ; and (iv) the worked arithmetic (e.g. ). No empirical constants are quoted beyond illustrative example speeds/distances.
Builds on: Vectors in Kinematics and Coordinate Systems. Leads to: Velocity and Acceleration · Relative Motion and Galilean Transformations · Newton's First Law and Inertial Frames.
§ The key results
| quantity | formula | remember it as |
|---|---|---|
| position | walking instructions from the origin | |
| how far from origin | Pythagoras on the shadows | |
| direction (2D) | quadrant check first, calculator second | |
| displacement | final minus initial, always in that order | |
| distance | the odometer — never beaten by the crow | |
| relative position | arrow from A to B; origin-proof | |
| frame change | subtract where the other origin went |
§ Invariance scoreboard
- Shift the origin by constant — positions change; displacements, relative positions, separations all survive
- Rotate the axes — components change; lengths and separations survive ()
- Let the frame move — even displacements change; separations at one instant still survive
- Round trip — always; distance keeps every metre
§ Anchor numbers
- Drone at : — the 3-4-5 triangle scaled by 10
- Ship at : — calculator says ; forces
- Hike 5 km north + 3 km east: distance 8 km, displacement
- Train walker: 60 m (train frame) vs 1860 m (ground frame) — origin-proof is not motion-proof
§ How to use it (60 seconds)
Step 1. Fix origin, axes, positive direction; keep them for the whole problem. Step 2. Positions: components → magnitude by (2), direction by quadrant-checked arctan. Step 3. Displacement: subtract final minus initial, component by component; sketch as a check. Step 4. Sanity: distance, and any frame-independent claim should survive shifting the origin.
§ Traps
- Distance is NOT the magnitude of displacement — a 400 m lap has
- arctan answers only between and : no quadrant check, no credit
- is the return trip — order matters
- Bare coordinates without a declared origin and frame are meaningless
- "Origin-proof" holds for constant shifts only — a moving frame changes displacements
⟨ student | Fermi ⟩ = understanding