The Physics
Opus in profectus

Distance & Displacement

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I've got to assume that everybody reading this has an idea of what distance is. It's one of those innate concepts that doesn't seem to require explanation. Nevertheless I've come up with a preliminary definition that I think is rather good. Distance is a measure of the interval between two locations. (This is not the final definition.) The "distance" is the answer to the question, "How far is it from this to that or between this and that?"

How far is it?
how far is it possible answer standard answer
Earth to sun 1 astronomical unit 1.5 × 1011 m
66th to 86th Street in NYC 1 mile 1.6 × 103   m
heel to toe on a man's foot 1 foot 3.0 × 10−1 m

You get the idea. The odd thing is that sometimes we state distances as times.

How far is it?
how far is it possible answer standard answer
International Space Station 90 minutes per orbit 40,000,000 m
Chicago to Milwaukee 90 minutes by train 00,150,000 m
Central Park to Battery Park 90 minutes on foot 00,010,000 m

They're all ninety minutes, but nobody but a fool would say they were they were all the same distance. What's being described in these examples is not distance, but time. In casual conversation, it's often all right to state distances this way, but in most of physics this is unacceptable.

That being said, let me deconstruct the definition of distance I just gave you. Every year in class, I do the same moronic demonstration where I start at one side of the lecture table and walk to the other side and then ask "How far have I gone?" Look at the diagram below and then answer the question.

There are two ways to answer this question. On the one hand, there's the sum of the smaller motions that I made: two meters east, two meters south, two meters west; resulting in a total walk of six meters. On the other hand, the end point of my walk is two meters to the south of my starting point. So which answer is correct? Well, both. The question is ambiguous and depends on whether the questioner meant to ask for the distance or displacement.

Let's clarify by defining each of these words more precisely. Distance is a scalar measure of the interval between two locations measured along the actual path connecting them. Displacement is a vector measure of the interval between two locations measured along the shortest path connecting them.

How far does the Earth travel in one year? In terms of distance, quite far (the circumference of the Earth's orbit is nearly one trillion meters), but in terms of displacement, not far at all (zero, actually). At the end of a year's time the Earth is right back where it started from. It hasn't gone anywhere.

Your humble author occasionally rides his bicycle from Manhattan to New Jersey in search of discount そば (soba) and さけ (sake) at a large Japanese grocery store on the other side of the Hudson River. Getting there is a three step process.

  1. Follow the Hudson River 8.2 km upriver.
  2. Cross using the George Washington Bridge (1.8 km between anchorages).
  3. Reverse direction and head downriver for 4.5 km.

The distance traveled is a reasonable 14 km, but the resultant displacement is a mere 2.7 km north. The end of this journey is actually visible from the start. Maybe I should buy a canoe.

Distance and displacement are different quantities, but they are related. If you take the first example of the walk around the desk, it should be apparent that sometimes the distance is the same as the magnitude of the displacement. This is the case for any of the one meter segments but is not always the case for groups of segments. As I trace my steps completely around the desk the distance and displacement of my journey soon begin to diverge. The distance traveled increases uniformly, but the displacement fluctuates a bit and then returns to zero.

This artificial example shows that distance and displacement have the same size only when we consider small intervals. Since the displacement is measured along the shortest path between two points, its magnitude is always less than or equal to the distance.

How small is small? The answer to this question is, "It depends". There is no hard and fast rule that can be used to distinguish large from small. DNA is a large molecule, but you still can't see it without the aid of a microscope. Compact cars are small, but you couldn't fit one in your pocket. What is small in one context may be large in another. Calculus has developed a more formal way of dealing with the notion of smallness and that is through the use of limits. In the language of calculus the magnitude of displacement approaches distance as distance approaches zero. In symbols, that statement looks like this.

Δs → 0  ⇒  Δs → |Δr|


What would be a good symbol for distance? Hmm, I don't know. How about d? Well, that's a fine symbol for us Anglophones, but what about the rest of the planet? (Actually, distance in French is spelled the same as it is in English, but pronounced differently, so there may be a reason to choose d after all.) In the current era, English is the dominant language of science, which means that many of our symbols are based on the English words used to describe the associated concept. Distance does not fall into this category. Still, if you want to use d to represent distance, how could I stop you?

All right then, how about x? Distance is a simple concept and x is a simple variable. Why not pair them up? Many textbooks do this, but this one will not. The variable x should be reserved for one-dimensional motion along a defined x-axis (or the x-component of a more complex motion). Still, if you want to use x to represent distance, how could I stop you?

English is currently the dominant language of science, but this has not always been the case nor is there any reason to believe that it will stay this way forever. Latin was preeminent for a long time, but it is little used today. Still, there are thousands of technical and not so technical words in the English language that have Latin roots.

Imagine some object traveling along an arbitrary path on top of an infinite two-dimensional grid. Place an observer anywhere in space — on or off the path, it doesn't matter. Make their position the origin of the grid. Draw an arrow from this observer to the moving object at any moment. This is our position vector. It's a vector because it has a magnitude (a size) and a direction. This vector points away from the observer much like the spokes of a wheel point away from its hub.

The Latin word for spoke is radius. For this reason, we will use r0 (r nought) for the initial position vector, r for the position vector any time after that, and Δr (delta r) for the change in position — the displacement.

Keep imagining our imaginary object traveling along an arbitrary path, but this time ignore the coordinate system. Think about the path the same way you think about traveling on a highway. There is no x or y coordinate on a highway (and certainly no z). No up, down, left, or right. No north, south, east, or west. There is only forward. Coordinates are for sailors or pilots. Locations on highways are indicated with mileposts. How far down the road have you gone. How much distance have you covered?

The Latin word for distance is spatium. It's also the source of the English word space. For this reason, we will use s0 (s nought) for the initial position on a path, s for the position on the path any time after that, and Δs (delta s) for the space traversed going from the one position to another — the distance.

If you think Latin deserves its reputation as a "dead tongue" then I can't force you to use these symbols, but I should warn you that their use is quite common. Old habits die hard. Use of spatium goes back to the first book on kinematics as we know it — Dialogues Concerning Two New Sciences (1638) by Galileo Galilei.

In uno stesso moto equabile, lo spazio percorso in un tempo più lungo è maggiore dello spazio percorso in un tempo più breve.   In the case of one and the same uniform motion, the distance traversed during a longer interval of time is greater than the distance traversed during a shorter interval of time.
Galileo Galilei, 1638   Galileo Galilei, 1638

OK, that was actually Italian. Galileo wrote to the people of the Mediterranean boot in his regional dialect, but the rest of Europe would have most likely read a Latin translation.

Spatium transactum tempore longiori in eodem motu aequabili maius esse spatio transacto tempore breviori.   In the case of one and the same uniform motion, the distance traversed during a longer interval of time is greater than the distance traversed during a shorter interval of time.
Galilaeus Galilaei, 1638   Galileo Galilei, 1638


The SI unit of distance and displacement is the meter [m]. A meter is a little bit longer than the distance between the tip of the nose to the end of the farthest finger on the outstretched hand of a typical adult male. Originally defined as one ten millionth of the distance from the equator to the north pole as measured through Paris (so that the Earth's circumference would be 40 million meters); then the length of a precisely cut metal bar kept in a vault outside of Paris; then a certain number of wavelengths of a particular type of light. The meter is now defined in terms of the speed of light. One meter is the distance light (or any other electromagnetic radiation of any wavelength) travels through a vacuum after 1/299,792,458 of a second.

Multiples (like km for road distances) and divisions (like cm for paper sizes) are also commonly used in science.

There are also several natural units that are used in astronomy and space science.


Let's change how we observe the world and see how it affects distance and displacement. A symmetric operation is a change that results in no change. Quantities that are not affected by a change are said to show a symmetry. The opposite of symmetry is asymmetry and the opposite of symmetric is asymmetric.

First, the location of the observer does not matter. Place the origin wherever it's convenient (or wherever it's inconvenient). It won't matter. Distance and displacement are not affected by a translation of the origin. There is no special place when it comes to measuring distance and displacement. All locations in the universe are equivalent. Space is homogeneous.

Second, the orientation of the axes is irrelevant. Point them in any direction you want (or don't want). Just keep the x-axis perpendicular to the y-axis. (This you must not change.) Distance and displacement are not affected by a rotation of the axes. There is no special direction when it comes to measuring distance or displacement. All directions are equivalent. Space is isotropic.

Third, and most difficult to state in words, the chirality or handedness of the coordinate system is also irrelevant. Frequently, the x-axis points to the right and the y-axis points up (that is, toward the top of a page, blackboard, whiteboard, computer display, etc.). If we add a third z-axis, in what direction should it point: in or out (that is, into or out of the page, blackboard, etc.)? If you chose out, then you've made a right-handed coordinate system. If you chose in, then it's a left-handed coordinate system.

The two possible coordinate systems are like hands because they are mirror images of one another. No amount of rotation will ever allow you to line up all the parts of your left hand onto all the parts of your right hand. Align the fingers and thumbs of both hands and your palms will face in opposite directions. Align your palms and fingers and your thumbs will point in opposite directions. The Greek word for hand is χερι (kheri), so this property of hands and coordinate systems (and organic molecules and magnetic interactions) is called chirality. It is equivalent to a reflection in a mirror. A right-handed coordinate system is right-handed when viewed directly but left-handed when viewed in a mirror — when viewed through the looking glass, to use a literary reference.

Distance and displacement are not affected by a reflection of the coordinate system. This is not true for all physical quantities, however. The ones that don't work the same when viewed in a mirror are called pseudovectors. Some examples of pseudovetors are torque, angular momentum or spin, and magnetic field. The direction of a pseudovector is always related to a hand rule of some sort (like the one used in vector multiplication). But as we have just discussed and as everyone knows, right hands become left hands and left hands become right hands when viewed in a mirror. Wrong hand means wrong direction. Space appears to know the difference between left and right.