Why some measurements need a direction attached and others do not

0:00How tall are you? One number, one unit, and the question is closed. A metre seventy. There is nothing left to add. Now here is a footballer with the ball at her feet, and the question is how she should strike the pass. Say she should strike it at eight metres a second and you have not answered. Eight metres a second where? The answer is not incomplete because you were vague. It is incomplete because a number is genuinely not enough.

0:32That gap is the whole of this topic. Some measurements are finished by a number. Some are not finished until a direction is attached as well. What almost every course does next is hand you two lists of names to memorise. We are going to do something better than that: we are going to build the second kind of quantity, in order, and watch the classification fall out of the construction.

1:00Start with the lists anyway, because there is something hiding in them. On one side, the measurements a single number settles: time, work, money, resistance, volume, area, temperature, density, mass, voltage, length, distance and speed. Thirteen of them. On the other, the ones that are not settled until a direction comes too: weight, momentum, force, velocity, acceleration, the strength of an electric field, and displacement. Seven. The first kind we will call scalars. The second kind, vectors.

1:40Twenty names. And a student reading them naturally asks: what have the seven got in common that the thirteen have not? Stare at the words and nothing comes. So instead of staring, measure. Give every one of the twenty names its physical dimension, and ask which of them carry the same one. Five pairs do. And here is what makes them worth finding: three of those five straddle the divide. Same dimension, opposite sides.

2:13They stand over exactly two dimensions. A length, and a length each second. Speed against velocity, which is the pair everybody quotes. And then displacement, which is awkwardly paired twice, because the length dimension carries two of the scalar names and not one. Distance against displacement, and length against displacement. The other two pairs do not straddle anything. Distance and length are both scalars; weight and force are both vectors. So sharing a dimension is not what the split is about.

2:52Which means it cannot be the units. Whatever separates these two columns, no unit will ever show it to you. Here is the construction. It has two steps and they happen in an order, and the order is the part nobody tells you. Step one. Draw a line in space. Not a segment, not a ray. A line, running off in both directions forever. Now give it a direction, by choosing an arrowhead. And the thing to notice is how few choices you have. Not many. Two.

3:28That is worth checking rather than asserting. Take a line whose direction arrow is three units long, and scan every multiple of it on a grid of seventy three, looking for the ones exactly one unit long. Two come back. A third of the way along, each way. Ask the same scan for an arrow of length two and it finds two others. Ask it for one of length nought and it finds none at all, because the one arrow with no length points nowhere and is not a direction.

4:03And the two it finds are each other turned round. Same line, opposite ways, and adding them gives nothing. So one line admits two directed lines. Now, are those two different objects, or is that just bookkeeping? Different. But not in the way you might guess. Take eight points along the line built one way, and check each against the line built the other way. All eight are on both. Slide any of them a hundredth off the line and it is on neither.

4:38One collection of points. Two directions. The drawing changes in exactly one place — where the arrowhead sits — and that one place is the whole difference. Nothing else about the line moved at all. Step one is done. We have a direction. Now watch carefully, because here is the step that gets skipped. How long is this thing? It has no length. Not an unknown length, not a length we have not measured. None.

5:10Take a routine that finds the two farthest apart points it can, and turn it loose on the line. Ask it to beat ten. It does. A hundred. A thousand. Ten thousand. A million. Five bounds, and it beats all five, and it would beat any bound you named, because there is always more line. An arrowhead on an unbounded line fixes a direction and supplies no magnitude whatever. If you think an arrow carries a length, this is where that belief comes apart.

5:47Step two. Cut it down. Mark two points, and throw away everything outside them. Now run the same routine again. The very same routine, on the very same line, with the very same arrowhead. Beat ten? It cannot. A hundred? It cannot. It beats none of the five bounds it beat a moment ago. What it gives instead is a number. Three. The farthest apart two points of this segment can be is three, and out of a hundred and thirty six pairs it compared, exactly one pair is that far apart — the two ends.

6:27The arrow did not change between those two runs. The cut did. And to make sure it is the cut being read and not the line, cut the same directed line at a different second point. The number becomes six. That is where a magnitude comes from. Not from the arrowhead. From the cutting. Now the definition, and it will read differently than it would have five minutes ago. A vector is a directed line segment: a quantity that has both a magnitude and a direction.

7:03Read cold, that is a shopping list. Two ingredients, apparently interchangeable, and no picture of why an arrow is the right drawing for it. Read as the end of what we just built, it is the two steps in order. The direction came first, from choosing one of two arrowheads on a whole line. The magnitude came second, from cutting that line down. That ordering is why the drawing is an arrow rather than a number or a dot. The shaft is the segment. The head is the choice.

7:40The two points where we cut get names. The end the arrow leaves from is the initial point. The end it lands on is the terminal point. Call them A and B, and the arrow from one to the other is written as the two letters with an arrow across them, and read aloud as vector A B. Or give the whole thing a single letter with an arrow over it, and read that as vector a.

8:08The distance between the two ends is the magnitude. Some people call it the length. Same thing. And it is written three ways: the two letters inside vertical bars, the single letter inside vertical bars, or just the plain letter with nothing over it at all. That last one is the one that catches people. Strip the arrow off a vector and what is left is its magnitude — a bare number.

8:38Now to the thing the definition hides by listing its two demands side by side. They are not symmetric. Not remotely. Here is how you can see it. Take twelve arrows. Run one operation on all of them, and count what moved. Turn every arrow round. Direction moved at all eleven places where there was a direction to move — the twelfth arrow is the zero one, which has no direction to turn. Magnitude moved at none of the twelve.

9:12Now stretch every arrow by three. Magnitude moved at all eleven. Direction moved at none. Two operations. Each moves exactly one of the two halves and leaves the other untouched. The same routine, both times, so the difference cannot be coming from the routine. And to be sure the routine can see anything at all, show it an operation that does nothing. It reports nothing moved. Add a hundredth to every magnitude afterwards and it catches all twelve, the zero arrow included, because nought and a hundredth are different numbers.

9:55So both halves move. But only one of them can move to anywhere it likes. Turning an arrow round is always available. Any direction can be reversed, at any time, and you land on another perfectly good direction. Now try to do the corresponding thing to the magnitude. Try to drive a length below nought. Double every arrow backwards — scale it by minus two — and measure. Not one of the twelve comes back negative. The magnitude of minus twice an arrow is twice its magnitude, not minus it.

10:34The reason is in the arithmetic. A magnitude is a square root of a sum of squares. Squares are never negative, and the root refuses a negative input outright rather than returning one. Which is why writing that the magnitude of a vector is less than nought has no meaning. Not a rule to memorise. There is no operation anywhere that produces one. Write the sign in front of the length instead — treat it as though it carried the minus — and eleven of the twelve go below nought at once. That is the cost of the mistake, counted.

11:17Now the classification, which we have earned rather than assumed. Six measurements. Five seconds. A thousand cubic centimetres. Ten newtons. Thirty kilometres an hour. Ten grams a cubic centimetre. Twenty metres a second due north. The verdicts are: time, scalar. Volume, scalar. Force, vector. Speed, scalar. Density, scalar. Velocity, vector. And there is exactly one rule doing all six. Name the quantity, then look the name up. Not: does it look directional. Not: how many numbers does it feel like. Name the quantity. That step is the entire job, and everything after it is a lookup.

12:07The reason to insist on that is the shortcut, which is what students actually carry away: if a direction is written down, it is a vector; if not, it is a scalar. Run both rules over twelve items — those six, and six more. Ten kilograms, two metres north west, forty degrees, forty watts, a charge measured in coulombs, and twenty metres a second, every second. The shortcut answers every time and is wrong twice.

12:40It is wrong at ten newtons, because no direction is written and force is a vector anyway. And it is wrong at twenty metres a second every second, for exactly the same reason: that is an acceleration, and acceleration is a vector, bearing or no bearing. Those two are not exceptions to be memorised. They are the same item twice, and they are what tells you the shortcut is a different rule that happens to agree most of the time — nine right against seven, over the items where both can speak.

13:18The test is never whether somebody wrote a direction down. It is which quantity the measurement names. Now something honest, which most treatments skate over. Run our rule over those twelve and it is right at every item it can speak on, and wrong at none. But it refuses three times. An angle. A power. A charge. Three of the twelve name a quantity that is on neither list, and a rule that looks names up on lists has nothing to say about a name that is not on one.

13:55You still get the right answers — all three are scalars — but not from the lists. From knowing one more fact about each quantity. That is worth saying out loud, because it tells you what the lists actually are. They are not a definition. They are twenty worked examples of a decision that has to be made quantity by quantity, and your job is to be able to make it for the twenty first.

14:25Back to the two items that look identical and are not: thirty kilometres an hour, and twenty metres a second due north. Is it the units? Convert. Thirty kilometres an hour is twenty five thirds of a metre a second. Twenty metres a second is seventy two kilometres an hour. Same kind of number, and neither is even larger than the other once written the same way. Is it the number itself? Take eight values. Every one of them reads as a speed, and every one of them is the magnitude of some arrow. All eight on both sides.

15:05What neither side will take is a value below nought. Three of those are refused by both — and that is the same refusal the magnitude made earlier, not a second rule. So the number cannot separate them and the unit cannot separate them. What separates them is two words. And notice what those two words do. Strike them out and you do not merely lose a direction. Twenty metres a second due north stops being a velocity and becomes a speed. Two metres north west stops being a displacement and becomes a distance. The phrase renames the quantity, and then the lists decide.

15:50Which brings us to the pair that causes the most trouble of all, and the one the two lists put on opposite sides without ever explaining. Walk four kilometres out and four kilometres back. How far did you walk? Eight. How far are you from where you started? Nothing at all. Same two legs, two completely different questions. Distance adds up each leg on its own. Displacement adds the legs as arrows first, and takes one length at the end.

16:24Twelve walks were measured. Displacement falls short at eight of them and equals the distance at four, and there is no walk anywhere where it is the larger. It cannot be. And the four where they agree are not a coincidence. They are exactly the walks whose legs all point the same way. Four of each, and the same four. Here is the near miss, and it is the one you would say. That the legs all lie along one line. That collects six walks — and two of them are walks where displacement is strictly less. Four out and four back lies along one line. So does two out and one back.

17:06Along one line is not enough. The walker must not turn back. And when the readings do differ, they differ properly: the narrowest margin over all eight is about half a unit, and the widest is a walk of twelve that ends where it began. Four things. The split is built, not classified. A line takes one of two arrowheads, and that is the direction, carrying no length at all. Cutting it down between two points is what supplies the magnitude. The definition is those two steps, in that order.

17:46The two halves are not interchangeable. One operation moves the direction and leaves the magnitude alone; another moves the magnitude and leaves the direction alone; and no operation anywhere drives a length below nought. To sort a measurement, name the quantity first. Looking for a written bearing is a different rule that agrees most of the time and fails at force and at acceleration. And distance and displacement are not two ways of measuring one thing. They agree only when the walk never turns back.

18:26One question we have left open on purpose. Every arrow we drew sat somewhere — it had a starting point. Does a vector belong at the place it is drawn, or can you pick it up and carry it elsewhere? That answer changes what you are allowed to do with these, and it is not obvious. The word itself is Latin. Vector: a carrier, one who carries. It came into mathematics through the work of William Rowan Hamilton, born in eighteen oh five, died in eighteen sixty five — a man who spent the better part of a decade trying to multiply arrows in three dimensions.

19:09The carrying, it turns out, was the easy part.