The named kinds — zero, unit, coinitial, collinear, equal, negative and free

0:00Seven words get defined in one go, and then you are expected to know them. Zero. Unit. Coinitial. Collinear. Equal. Negative. Free. Presented as a list, they look like seven things to memorise, and one of them looks so obvious that nobody checks it. They are not seven of the same kind of thing. Six of them tell you something about the vectors. One tells you only about the drawing.

0:30And there is a single experiment that sorts them, which we are going to run. To run it we need to be careful about what an arrow on a board actually is. It is two things at once. There is where you put the tail. And there is the displacement out of that tail: how far, and which way. A vector is the second of those. The tail is where you happened to start drawing.

0:57That is not a definition anybody hands you up front. It is the thing you have to reverse engineer from the list, and it is easier to just do the experiment. The last of the seven words is the one that licenses everything. Free. A free vector may be picked up and moved anywhere, so long as it is moved parallel to itself. Same length, same direction, new place. It is the same vector afterwards.

1:30Vectors here are always free. Every single one. That sounds like a footnote. It is not. It is permission, and the rest of the material spends it constantly. So here is the experiment. Draw a figure. Ask every one of the seven questions about it. Then pick up every arrow and put it somewhere else. Ask all seven again. And count what changed. Eight arrows on a board. Twenty eight pairs among them.

2:02Before anything moves: three pairs are equal, eleven are collinear, three are a negative pair, six share a length, five are coinitial, five lie along one line, and one pair is drawn one arrow directly on top of the other. Now slide all eight. Each one to a different place. Nothing turned, nothing stretched. Equal: nought pairs changed their answer. Collinear: nought. Negative of: nought. Same length: nought. Coinitial: all five changed. Every single one.

2:39And lying along one line: all five of those changed too. So the sorting is done, and it took no argument. Four of these questions are about the vectors. Coinitial is about the paper. And so, it turns out, is the everyday meaning of the word collinear, which is going to matter more than anything else here. Start with the awkward one. A vector is defined as a quantity with a magnitude and a direction. Both.

3:12Now take an arrow whose two ends are the same point. It goes nowhere. Its magnitude is nought. That is the zero vector. And you will be told two things about it, one immediately after the other. First: it has no definite direction, because its magnitude is nought. Second: it may be given whatever direction happens to suit the argument. Those two sentences do not agree. Neither is withdrawn. Read the first one, and out of forty nine arrows on a grid, forty eight meet the definition of a vector. Read the second, and all forty nine do.

3:54The arrows the two sentences disagree about number exactly one. The zero one. So the clash is real, and it is also as small as a clash can be. The second sentence is the working one, and it earns its keep later: the moment you multiply the zero vector, or take a cross product that comes out zero, you need it to be a vector like any other. There is a second thing worth noticing about the zero vector, and it is a rehearsal for the whole topic.

4:30Take six different points on the board. From each one, draw the segment that goes from that point back to itself. Six drawings. Six different starting points. And one displacement between them, because every one of them goes nowhere. Six pictures, one vector. You will see the zero vector named by two different pairs of letters, as though there were two of them. There are not. Those are two drawings of the same object, and the reason is the free vector permission we have already been given.

5:07Next name. A unit vector is a vector whose magnitude is one. That is the whole definition. The notation is a hat, written over the letter, and it means the unit vector along that vector. And then, having named it, most treatments move straight on without telling you how to build one. The recipe turns up much later. It is worth having now, because it is one line. Take your vector, divide it by its own magnitude. Length one, direction untouched.

5:42On our grid of forty nine arrows, four already have length one: one unit each way along the two axes. Four out of forty nine. But forty eight out of forty nine have a unit vector built for them. Checked: the recipe returns length one at every one of the forty eight, and multiplying back by the magnitude returns the original arrow exactly. The one arrow with no unit vector along it is the zero arrow. You would be dividing by nought. That is the same awkwardness as before, wearing different clothes.

6:22So a unit vector is not something you have to be handed. Nearly every vector has one, and you can build it. Coinitial. Two or more vectors are coinitial when they have the same initial point. They start at the same place. And we have already measured what that is: it is the one name on the list that does not survive being moved. All five coinitial pairs stopped being coinitial the moment the arrows were slid.

6:54Which puts it in an odd position. A moment after being told that these arrows all start at the same point, you are told that any arrow may be put anywhere you like. Both are true. They just are not about the same thing. Coinitial is a question you ask about a picture in front of you. Which of these drawn arrows leave the same corner? That is a real question with a definite answer, and every question you will ever be asked about coinitial vectors is asked about a supplied figure.

7:30It is not a relation between two vectors. It is a relation between two drawings. Once you can slide, any two vectors can be made coinitial, so as a question about vectors it is empty. Now the important one. Collinear. The word is a false friend, and it costs more marks than anything else here. The definition: two or more vectors are collinear if they are parallel to the same line. And then a clause of its own: irrespective of their magnitudes and directions.

8:09Both halves matter and neither one is what the word sounds like. Parallel to the same line. Not lying on the same line. They can be metres apart. Here are four arrows. Three of them are drawn parallel: two out of the same point on the right, and a third one away on a line of its own. The fourth points up and away. By the definition, three of these four are collinear. By the everyday reading of the word, only two are, because only two are actually on one line.

8:43Three against two, on the same figure. That gap is the whole trap. And it is not an accident of this figure. Remember the slide test: lying on one line was one of the two things that failed it. If collinear meant on one line, then moving an arrow could destroy it, and vectors are free to move. The definition has to say parallel. There was never a choice. The second half of that definition does its own damage. Irrespective of magnitudes and directions.

9:20Magnitudes waived: a short arrow and a long one, parallel, are collinear. Directions waived: an arrow and an arrow pointing the exact opposite way, parallel, are collinear. That second one is the sharp edge. A vector is collinear with its own reverse. Checked at all forty eight nonzero arrows on the grid, with no exception to find. People will tell you collinear vectors point the same way. They do not have to. The definition explicitly says so, in a clause most readers skim.

10:01Equal. Two vectors are equal if they have the same magnitude and the same direction, regardless of the positions of their initial points. That last clause looks like tidying up. It is the entire content of the definition. Take it away and see what happens. On our eight arrow figure, the full definition finds three equal pairs. With the position clause struck out, it finds one. And that one pair is exactly the pair that was drawn one arrow on top of the other.

10:37So without the clause, equal stops meaning equal and starts meaning drawn in the same place. It becomes coincidence, which is a fact about the board, and it would have failed the slide test. Every pair the strict reading accepts, the real definition accepts too. It is the two extra pairs that are the point: two arrows in different parts of the board, same length, same direction, genuinely equal. Negative. The negative of a vector is the vector with the same magnitude and the opposite direction.

11:17Draw it as a segment and reverse it: the two letters swap, and you write a minus sign in front. That is one definition. There is a second, given later and never announced as the same thing: the negative is what you get by multiplying the vector by minus one. There it gets a second name, the additive inverse. Two definitions, and nobody tells you they coincide. So check it. Across all forty nine arrows on the grid, reversing the arrow and multiplying by minus one give the same answer every time.

11:57And the second name is earned rather than granted: add any arrow to its own negative and you get the zero vector. All forty nine. The magnitude is untouched, at all forty nine. The line is untouched. Only the sense turns round, and it turns round at every one of the forty eight nonzero arrows. One arrow is its own negative. Exactly one. The zero one, again. Before the worked figures, one more measurement, because it says what the free vector permission is actually spent on.

12:35To add two vectors you put them nose to tail: the second one's tail at the first one's tip. Then the sum runs from the first tail to the second tip. Take our eight arrow figure. Fifty six ordered pairs. How many are already arranged nose to tail, with no moving required? None. Not one of the fifty six. On a square with an arrow on each side, drawn nose to tail round two of its corners, two of twelve are.

13:08So essentially every sum you will ever draw begins by picking a vector up and putting it somewhere else. The triangle law does it. The proof that addition does not care about order does it twice, copying two sides of a parallelogram onto the other two. None of those moves are legal without that one sentence about free vectors. It is the load-bearing idea of the seven, and it is the one that arrives sounding like an afterthought.

13:42Now put it to work. Four arrows again, the ones from before. Three questions, and the answer to each is a different set, which is exactly why three questions get asked. Which are collinear? The first, the second and the fourth. Three of them, parallel to one line, and the fourth is nowhere near the other two. Which are equal? The second and the fourth. Same length, same direction, different places on the board.

14:13Which are coinitial? The first, the second and the third. The three drawn out of the right hand point. Three different answers from three different sets of arrows. Notice that the second arrow appears in all three, and that tells you nothing: these are not competing labels. A vector can be all three at once. Same three questions, asked of a square with an arrow along each side. And everything here depends on reading the arrowheads correctly. The top side points right. The bottom side points left. And both upright sides point downwards.

14:55One warning. When the four sides are labelled, each label carries a little arrow above its letters, and every one of those points rightwards. That mark is notation. It is not the direction of the side. Read those as directions and every answer comes out wrong. Coinitial: the top and the left. They leave the same corner. Equal: the right and the left. Both run down, both four units. They are on opposite sides of the square and they are the same vector.

15:30Collinear but not equal: the top and the bottom. Parallel, same length, opposite senses. Three questions, three different pairs. And the equal pair is not coinitial, while the coinitial pair is not equal, which is the whole reason to keep the ideas apart. Four claims. True or false. One: a vector is collinear with its own negative. True. Because directions are waived. Checked at all forty eight nonzero arrows. Two: two collinear vectors always have equal magnitude. False. Because magnitudes are waived too. One unit east and two units east are collinear and not the same length. The grid holds forty eight counterexamples.

16:24Three: two vectors of equal magnitude are collinear. False, and this one is not even close. One unit east and one unit north are the same length and are not parallel to any common line. Ninety six counterexamples. Four: two collinear vectors of equal magnitude are equal. False. One unit east and one unit west. Collinear, same length, not equal. Twenty four counterexamples. All four turn on the same two words in one definition, and no single counterexample settles more than one of them: the pairs that break the second claim break neither of the others.

17:12The fourth claim deserves one more look, because it is the one that tells you whether the topic has landed. Twenty four counterexamples on that grid. Here is what they are. Every single one of them is an arrow paired with its own negative. Not most. All twenty four, checked. And twenty four is exactly the number of opposite pairs the grid contains, since forty eight nonzero arrows fall into twenty four such pairs.

17:43So the fourth claim fails in precisely one way, and it is the way the first claim already told you about. Collinear plus equal magnitude leaves exactly one freedom unsettled, and that freedom is which way round. Four questions. One idea. If you know that collinear waives the sense, you have answered all four. One last thing, and it is a small piece of arithmetic that nobody points out. Here is a displacement to draw: forty kilometres, thirty degrees west of south. You draw a compass rose, mark the angle from south, draw the arrow, and add a scale bar reading ten kilometres to the unit. The scale bar is the piece people leave off, and it is the piece that makes it an answer rather than a sketch.

18:38The very next one asked for is forty kilometres, thirty degrees east of north. Read as bearings, those are two hundred and ten degrees and thirty degrees. They differ by exactly one hundred and eighty. Half a turn. And both distances are forty. Same magnitude, opposite direction. The second arrow is the negative of the first, and that word was defined on the way past a moment ago. Checked, and not just at those two: at sixteen different bearings, adding half a turn gives the negative every time, and adding a quarter turn never does.

19:21On the drawing, at ten kilometres to the unit, each arrow is four units long, and their two tips are eight units apart. Five things. Six of the seven names are about the vectors. Coinitial is about the drawing, and it is a question you ask of a figure, never of a pair of vectors. Collinear means parallel to a line, not lying on one, and it waives both length and sense. That single sentence answers four true or false claims at once, and a vector is collinear with its own reverse.

20:01Equal waives where each arrow starts, and that clause is the definition, not decoration. Remove it and equal collapses into drawn in the same place. The zero vector is granted a direction by convention, because the alternative sentence breaks everything downstream, and it is the one arrow with no unit vector along it. And free is not a seventh kind sitting beside the other six. It is the permission that makes the other six coherent, and every sum you draw from here on spends it.