Stretching by a scalar, and dividing a vector by its own length

0:00This topic behaves differently from the ones around it, and knowing that in advance will save you twenty minutes of hunting for something that is not there. You are given four statements about multiplying a vector by a number, and none of the four is proved. Then there is one piece of algebra, three lines long, and that one piece of algebra produces the object almost everything afterwards runs on.

0:29Four things asserted. One thing derived. If you go at this the way you went at the last topic, looking for the argument behind each claim, you will be frustrated the whole way through, because behind those four claims there is no argument and there cannot be one. So let us separate the two halves properly, and then measure the half that can be measured. The four statements are these. Multiply a vector by a number and you get a vector. It lies along the same line as the one you started with. It points the same way if the number is positive, and the opposite way if the number is negative. And its length is the size of the number times the original length.

1:19That is not a theorem. That is what the operation means. You are being told what the symbol does, and there is nothing underneath it to uncover. Compare it with vector addition, where two different constructions were given for one operation and the work was showing they agree. Here there is one construction, and it is a definition. But there is still something worth doing, and it is not nothing.

1:49A statement you cannot prove can still be checked, in one particular sense: you can ask whether the thing checking it is capable of returning no. So take a grid of a hundred and twenty five arrows and eleven numbers, and pair them up every way. One thousand three hundred and seventy five pairs. Of those, one thousand two hundred and forty have a number that is not nought and an arrow that is not the zero arrow.

2:20Is the stretched copy along the original line? At all one thousand two hundred and forty. Every time. And here is the part that makes that count mean anything. Run the same routine on a rule that applies the number to only one part of the arrow, and it finds three hundred and seventy six. Run it on a rule that stretches and then shifts the answer sideways, and it finds thirty six.

2:49So the routine can say no. It said yes to stretching because stretching does it, not because it says yes to everything. Same for direction. Of those one thousand two hundred and forty, six hundred and twenty have a positive number and six hundred and twenty a negative one. Every one of the six hundred and twenty positives leaves the direction alone. Every one of the six hundred and twenty negatives turns it exactly round. And not one negative number leaves a direction where it was.

3:27Now the fourth claim, and the one where the notation is doing real work. The length of the stretched vector is written as the size of the number, in modulus bars, times the length of the original. Bars round the vector, which you expect. And bars round the number too. Those second bars look like decoration. They are not. Read the rule with the bars and it holds at all one thousand three hundred and seventy five pairs. Read it without them, and it holds at seven hundred and fifty five and fails at six hundred and twenty.

4:07And those six hundred and twenty are exactly the pairs where the number is below nought and the arrow is not the zero arrow. Every single one of them. What goes wrong there is not a small error. At every one of those six hundred and twenty, the barless reading returns a length below nought. A negative length. That is not a slightly wrong answer; it is not an answer at all, because a length is a distance, and a distance is never negative.

4:41The real magnitude, over all one thousand three hundred and seventy five, is never once below nought. The bars are what keep it that way. Draw the whole of the third claim at once. One arrow, and five multiples of it: the arrow itself, half of it, twice it, minus half of it, and minus twice it. The first three point one way. The last two point the other way.

5:09All five lie along one line, and between them they carry three different lengths: half, one, and twice. The two negatives repeat two of those three lengths, pointing backwards. Now do something to that picture, because it shows you how much a single sign is carrying. Drop one minus sign. Just the fourth one, so minus a half becomes a half, and leave everything else alone. The picture had three arrows pointing one way and two the other. Now it has four one way and one the other.

5:46The two pictures differ at exactly one of the five arrows, and the loss shows up as a repetition: five different arrows become four, because one of the labels now appears twice. A sign in front of a number is not an ornament on the number. It is the direction of the arrow, and it is the easiest thing in this topic to lose while reading quickly. Take the number to be minus one, and look at what comes out.

6:17Same length, opposite direction. This gets called the negative of the vector, and then a second name, the additive inverse, and a note that a vector and this thing add to the zero vector in either order. You have met this object already. When vectors were first drawn, the negative of a vector was built geometrically: take the segment and read its two letters the other way round, from the head back to the tail.

6:48So there are two definitions of one thing, arriving a long way apart, and nothing anywhere says they are the same. Which means it is worth checking. Over all one hundred and twenty five arrows of the grid, multiplying by minus one and swapping the letters produce the same arrow. All one hundred and twenty five times. It keeps the magnitude at all one hundred and twenty five, turns the direction at all one hundred and twenty four that have a direction, and adds to the original to give the zero arrow, in either order, at all one hundred and twenty five.

7:30And it is minus one specifically, not merely a negative number. Minus two turns all one hundred and twenty four round in exactly the same way. But it keeps the magnitude at one arrow out of the hundred and twenty five, and that arrow is the zero one. Reversal is about the sign. Length is about the size. Minus one is the only number that does the first without doing the second.

8:01One line, easy to skip, and it closes a case you might not have noticed was open. Any number times the zero vector is the zero vector. Try all eleven numbers. All eleven send the zero arrow to the zero arrow, so eleven different scalings produce exactly one result between them. And at none of the eleven does the answer come out with a direction. That is the whole content. Scaling can stretch a direction, shrink it, or turn it round. It cannot manufacture one out of nothing.

8:39Which matters in about five minutes, when we choose a number designed to give a vector length one, and have to exclude this arrow by hand. Here is the first claim turned into a tool. Every scaled copy is along the original line. Read that backwards and you get something useful: two vectors are along one line exactly when one of them is a number times the other, with the number not nought and neither vector the zero one.

9:12Collinearity, which sounds geometric, is now one equation in one unknown number. And notice what that lets you do. Asked to show two vectors are collinear, you do not draw anything. You find the number. So let us not evaluate a formula. Let us search. Take every ordered pair of arrows from the grid, fifteen thousand six hundred and twenty five of them, and for each pair look through a net of fractions for one that carries the first arrow to the second.

9:46The geometry finds three hundred and fifty two pairs lying along one line. The search finds a number for three hundred and fifty three. They disagree at exactly one pair. And that pair is the zero arrow with itself. Among pairs where neither arrow is the zero one, they never disagree at all, over all fifteen thousand six hundred and twenty five. Which tells you precisely what the two exclusions in the criterion are for, and that neither one is decoration.

10:23Here is an item you will meet in some form: four statements about two collinear vectors, and the question is which of them are incorrect. Incorrect, plural. That is a hint, and it gets missed. The four. One: one vector is a number times the other. Two: each is the other, or the other turned round. Three: their parts are not in proportion. Four: they point one way and differ only in length.

10:54Decide each of them at every collinear pair on the grid. Three hundred and fifty two pairs. The first holds at all three hundred and fifty two. The second holds at two hundred and forty eight, so it fails at a hundred and four. Being collinear does not mean being equal or reversed. A stretch is not a reversal. The third holds at nought of them. It fails at all three hundred and fifty two, because parts in proportion is exactly what collinear means.

11:29The fourth holds at a hundred and seventy six and fails at a hundred and seventy six. Exactly half. And the half it fails at is the half where the number is negative and one of the two is turned round. Three of the four are wrong. Find one, stop, and you lose the two marks that were sitting there. Three rules arrive together under one heading, and the heading calls all three distributive laws.

12:01The first: two numbers added, then one scaling, equals two scalings added. The second: one scaling inside another equals a single scaling by the product of the two numbers. The third: two vectors added, then one scaling, equals two scalings added. All three are true. Tried nine hundred and seventy two, nine hundred and seventy two, and four thousand three hundred and seventy four times respectively, stretching breaks none of them.

12:34And each of those three tests can fail on its own, which is worth showing, because three tests that all pass are only interesting if each of them could have failed separately. Stretch by the square of the number instead, and the first rule breaks six hundred and fifty times, and the other two never. Stretch by twice the number, and the second breaks six hundred and fifty times, and the other two never.

13:02Stretch and then shift the answer sideways, and all three break. Three separate tests, three separate ways of failing. So all three passing for stretching means three separate things. Now read the three again, not for whether they are true, but for what is actually written in them. The first has an addition in it, and the thing being added is two numbers. The third has an addition in it, and the thing being added is two vectors.

13:34The second has no addition in it at all. Count the additions in each rule, by kind, off the way each one is written. One number addition and no vector addition. No additions of either kind. No number addition and one vector addition. So of three rules filed under the word distributive, exactly one distributes over an addition of vectors, and exactly one contains no addition whatsoever. The middle rule is not a distributive law. Nothing can distribute over an operation that is not present. What it says is that nesting two scalings is the same as one scaling by the product, which is associativity, filed under a name that does not describe it.

14:27The heading is a collective label. It fits one of the three exactly, one of them loosely, and the middle one not at all. That is forty seconds of sorting, and it is the difference between three rules memorised and three rules understood. And one more thing about those three rules, which is why the four claims have to be stated separately rather than derived from the laws. The three rules do not pin the operation down.

14:58Here is a rule that obeys all three. Stretch the arrow by the number, and then throw the third part away, flattening everything onto one plane. It breaks none of the three. Not once in nine hundred and seventy two, nine hundred and seventy two, or four thousand three hundred and seventy four tries. And it is not stretching. It differs from stretching at ninety of a hundred and sixty two pairs. It lies along the original line at only two hundred and forty of the one thousand two hundred and forty live pairs. And it obeys the magnitude rule at exactly the same two hundred and forty.

15:43So the three laws are not a definition. The four claims are the definition, and the laws are consequences you are allowed to use. Now the three lines this whole topic exists for. You are allowed to choose the number. So choose it to be one over the length of the vector. Feed that into the magnitude rule. The length of the result is the size of one over the length, times the length. And that is one.

16:15So a vector divided by its own length is a vector of length one pointing the way the original points. It gets a hat, and the hat turns up in almost everything that follows. That is it. Three lines, one substitution, and it is the only thing derived here. Check it over the grid. Of the hundred and twenty five arrows, a hundred and twenty four are not the zero one. Divide each by its own length, and all one hundred and twenty four come out with length exactly one.

16:50And the test is capable of saying no. Divide by the square of the length instead, which is the commonest slip, and you get length one at six of the hundred and twenty four. Those six are exactly the six that already had length one, where a number and its square are the same number. The choice comes with a condition, written twice in one line: once as an inequality and once in words.

17:20The vector must not be the zero vector. This is thirty seconds and it is the part everybody omits. The reason is arithmetic, not taste. The number you chose was one over the length. The zero vector has length exactly nought, and one over nought is not a number. In the checking here, that is not an error message. The routine is asked for the unit vector of the zero arrow and returns nothing at all, which is the honest answer. One arrow out of the hundred and twenty five is refused, and it is that one.

18:00And now the line about the zero vector from a moment ago earns its place. If you could scale the zero vector into something with a direction, you could get round the condition. You cannot, because every number sends it straight back to itself. Carry the condition. The first time a question hands you a sum that cancels, you will divide by nothing without it. Once you can make length one, you can make any length.

18:31Want a vector of length seven pointing the way a given vector points? Normalise first, then stretch by seven. Over the hundred and twenty four nonzero arrows of the grid, that gives length exactly seven at all one hundred and twenty four. The tempting shortcut is to skip the first step and just multiply the original by seven. That gives length seven at six of the hundred and twenty four.

19:02The same six. The ones that were already length one, where the shortcut and the real thing happen to coincide. Two steps, in that order. Normalise, then scale. The first step is what makes the second step mean what you wanted it to mean. Now a harder one, and the order is again the whole lesson. You are given two vectors and asked for the vector of length one along their sum.

19:31Add first. Then normalise the sum. The wrong move, and it is a natural one, is to normalise each of them and add the results. Take every pair from a smaller grid where both arrows and their sum are nonzero. Six hundred and fifty pairs. Add then normalise, and you get length one at all six hundred and fifty. Of course you do. That is what normalising does. Normalise then add, and you get length one at forty eight of the six hundred and fifty.

20:07And the two orders give the same answer at forty eight pairs. Which sounds like the same forty eight, and it is, but not for the reason you would guess. Those forty eight are exactly the pairs where the two arrows and their sum all have the same length. That is a coincidence of lengths at particular pairs, not a rule about order. So there is no version of the wrong order that works in general. Add, then normalise.

20:39One last item, because it catches people who have done everything else right. You are given a vector and asked for the number that turns it into a vector of length one. Suppose the vector has length the root of three. Then the size of your number, times the root of three, must be one, so the size of the number is one over the root of three. The size. Which leaves two numbers, not one.

21:07Plus one over the root of three, and minus one over the root of three. Both give a vector of length one. They point opposite ways, and a vector of length one pointing backwards is still a vector of length one. The question asked for a multiplier, not for a direction. Both signs are answers. And the number is worth a glance, because it is not a fraction. Try eighty one fractions running from minus three and a third to three and a third in twelfths, and not one of them works. Three times the square of the answer is one, and no fraction does that.

21:51Six things. Multiplying a vector by a number is defined, not derived. Four claims, no proofs, and none of them missing. The one derivation is the unit vector. The bars go round the number as well as round the vector, because without them the rule returns a negative length at every pair with a negative number. Minus one gives you the negative of a vector, which is the same object that was built geometrically much earlier, arriving again under a different name.

22:28Collinear means one is a number times the other, with the number not nought and neither vector the zero one, and both of those exclusions are load-bearing. Of the three rules gathered under the word distributive, one distributes over an addition of vectors, one over an addition of numbers, and one contains no addition at all. And the unit vector is a vector over its own length, legal for every vector but one, and the order of operations is normalise first, then do whatever else you were going to do.

23:06Almost everything after this is written in hats. This is where they come from.