This PoSWW is a group effort, on one of the sub-topics in VCAA’s current curriculum:
parallel and perpendicular vectors
There is minimal guidance in the Study Design on what the sub-topic is intended to cover and, in particular, no definitions of “parallel vector” and “perpendicular vector” are provided. Our specific concern is the role of the zero vector.*
Below are definitions and implied definitions from three current Year 12 VCE Specialist Mathematics textbooks. We have attempted to be as fair as possible, for each text selecting the clearest definitions/descriptions we could locate.
*) One might be inclined to argue that this is a minor concern. For those thus inclined, we’ll address the argument soon with a new WitCH.
CAMBRIDGE
NELSON
JACARANDA
UPDATE: (21/02/23)
When I use a word, it means just what I choose it to mean – neither more nor less. Of course that might be different from what I chose it to mean two minutes ago.*
CAMBRIDGE
JACARANDA
NELSON
*) To be fair, Jacaranda refers to a vector resolute “parallel to” or “perpendicular to” a given vector; this is not incorrect language (although Jacaranda’s presentation has other issues).
Ouch! OK… so Cambridge SM3&4 on page 150 specifies that any set of vectors that contains the zero vector is a linearly dependent set. Also, any set of vectors that contains two or more parallel vectors is a linearly dependent set. Is it not just easier (don’t worry, rhetorical question) to say the zero vector is parallel to all vectors?
I do see the issue with the cross product of two parallel vectors and admit I failed to consider that situation. Not sure it invalidates my claim though (but I also admit needing to do a lot more thinking on the matter!)
I’ll have a (new) WitCH on independence stuff tomorrow.
> Is it not just easier (don’t worry, rhetorical question) to say the zero vector is parallel to all vectors?
The problem is that we would like to say that being parallel is transitive; i.e. that if u is parallel to v, and v is parallel to w, then u is parallel to w. But if the zero vector is parallel to all vectors, then *all* vectors must be parallel to each other, which is clearly far worse than having a clunky description.
From what’s given here, I don’t think the Cambridge text has made any errors so far, except for maybe the statement about “same direction or opposite directions” (which one can argue is not supposed to be a definition but merely an intuition (though one can also argue that such an intention should be made clear)). Certainly any set of vectors that contains the zero vector is linearly dependent since any nonzero multiple of the zero vector yields the zero vector; and by their own definition of two parallel vectors, it’s certainly true that any set of vectors containing two or more parallel vectors is a linearly dependent set. At least the Cambridge text is consistent on this (unless I’m missing something).
Suppose you have two non-zero vectors, a and b. Can you always split b into components parallel and perpendicular to a?
Ideally you “should”, but there’s no way to have both this and transitivity of parallel-ness. (One way to fix this might be to reword this as “you can always split b into components that are scalar multiples of vectors parallel and perpendicular to a”, though of course this is somewhat clumsy wording.)
Let me know if you have a preferred solution in mind that I’ve missed.
I think it’s obvious for at least these contexts that you permit the zero vector to be parallel and perpendicular to everything. Then you can split into components without clumsy wording, then two vectors being dependent is identical to them being parallel, then the vectors perpendicular to a given vector forms a subspace/line/plane, and so on.
But in any case, it is simply insane to have VCAA include this stuff in VCE without telling anyone what they mean.
I’ve spent the better part of an hour pondering what I thought I knew about a zero vector and have come to the following stumbling blocks:
1. I say THE zero vector, not the zero vector in 2D or similar, so my language implies that there is only one zero vector, which is the same for all space. This seems to be unusual. I’m not sure how big an issue it is and whether I need to change my language usage.
2. Either the zero vector has no direction (which would make it a scalar?) or it has a direction that cannot be defined. I’m not sure either of these is a correct statement, but I’m more comfortable with the latter.
2a. If the zero vector has a direction, then we can talk about vectors which are parallel and/or perpendicular to the zero vector as these are all geometric ideas. Maybe it is easier to say the zero vector has no direction…
…but then it is not a vector. Bugger!
3. If we take the geometric definition of a scalar product or vector product, can one of the vectors be the zero vector? How do you take the sine or cosine of an angle which cannot be defined? Does that mean these products cannot be defined?
4. I’m getting the sense that textbook writers may have decided this is all too difficult to deal with, and it is easier to just say “non-zero vector
” in examples and exercises. Maybe VCAA will do the same.
5. But the zero vector (unless we say it cannot exist, but that has other consequences…) is a vector and does obey some rules of vector arithmetic, so… I’m not sure.
Thanks, RF. Brief replies:
1) “The” zero vector is in reference to a particular vector world.
2) I don’t see that you can talk about the zero vector having a “direction”.
2)(a) The point is, at some stage you have to switch from geometric ideas/motivation/definition to algebraic. When you do, it is much easier to manage the zero vector.
3) Stick to the algebraic definitions of the products. Only a screwball like the Mathologer would define these geometrically.
4) Of course it’s “too difficult”, and it’s fair enough to duck the question to the extent you can. But the textbook writers shouldn’t write crap, and they shouldn’t write things that will then come back to bite them. (Which leads into tomorrow’s WitCH …)
5) The zero vector is a vector because, as you are suggesting, and as I’m demanding, vectors are now algebraic things.
Hi RF. Just on 3, yes you can do it. But it isn’t good IMO for teaching purposes, not at school nor at university. I think it’s only point is as a mental exercise.
One approach is to define the dot product in terms of the angle. Then, say that the zero vector is the limit of vectors
where 
as 
. The choice of 
doesn’t matter (if we end up doing something strange where the choice of 
actually matters, we declare our computation to be not allowed.)
Then
and we see that the result 
as 
.
Anyway.
This is one of the problems I have with how vectors are usually taught: at school I was taught that a scalar was a magnitude, and a vector was a magnitude with a direction, and I’m pretty sure this is how they are commonly taught.
Of course, this doesn’t make much sense, because -1 is not a magnitude, yet is a scalar, and you can’t really say that the zero vector has a direction.
I think that textbooks should define scalars and vectors as they actually are: the elements of a field and the elements of a vector space, respectively. One might say that this is too complicated for high school students, but I say not, because (a) the scalar and vector axioms need not actually be tested, and (b) you don’t really need to know much about them (in high school) other than the fact that they exist.
Thanks, A, but no. Your concern is valid, but your suggested solution is absurd.
What you point out is a very general and well-known issue; sometimes it is addressed well, sometimes poorly and sometimes not at all. But you simply have to accept that many, probably all, mathematical concepts, including vectors, will be introduced to school kids in simpler or vaguer or more visual or specialised forms. Yes, this inevitably creates issues, because you then want to have kids adjust their thinking to the more sophisticated concepts, and they may not want to loosen the hold on what they understand: you want to teach fraction multiplication and they still want to think in terms of pies. But there’s no magic fix, and simply throwing kids in the deep end would be fun to watch, but would result in many watery corpses.
Okay, but there’s surely got to be a better way to introduce scalars and vectors other than that “a scalar is a magnitude, and a vector is a magnitude with a direction.” Maybe as column vectors? I don’t think it would be too difficult to connect that with a visual representation, showing that each element of the column vector corresponds to a different dimension, or something along those lines.
“and they may not want to loosen the hold on what they understand:”
I suppose that’s the issue I have with imprecise and vague definitions. My experience at school was that teachers would teach imprecise definitions, incorrect or strange methods, and arbitrary restrictions as dogma, treat them as dogma, and teach students to treat them as dogma.
Yes, of course you have a point. I didn’t say that this stuff was typically done well. I’ll reply to you later, with my thoughts on “vectors”. The general issue you raise is worth a post: now on “the list”.
Hi again, A. Sorry to be slow to get back to you.
The more I’ve thought about how to answer you, the more my answer is turning into an essay, a whole post. I think that essay/post would be well worth the effort, but it’ll take effort and I’d rather it not be buried in a comment. So, i’ll make my answer (kind of) brief, and hopefully it won’t be too much a wait before I write more on it.
The first time kids will run into “vectors” is typically around Year 9 or Year 10, when they consider forces and velocities and the like, what are referred to as “vector quantities”. It seems to me fine and natural to refer to such physical quantities as having “magnitude and “direction”. (Yeah, a zero velocity would have no “direction”, but who cares?) There is essentially no vector algebra done at this stage, except maybe 1-D stuff, noting the positive/negative direction of motion, and the like. In particular, scalar multiplication plays little or no role, and I see no use for the term “scalar” here.
Around Year 11, students will get onto considering “vectors” as “arrows” and then in terms of coordinates. So, already, we have three, successively more abstract notions of “vector”. That’s even before considering the troublesome zero vector, much less fields and higher dimensions and independence and so on.
But kids are not going to happily move from physical vectors to coordinate vectors, if that move goes by way of pure mathematics. Coordinates are hard enough and abstract enough as it is. Yes, in the end, the abstraction clarifies everything, but not in the beginning. Kids have to be led into abstraction, not thrown into it.
So what do you do? First of all, I think you just have to accept that kids, because of the subject content and the need for intuition, must begin with the concrete and then be moved, gently, to the abstract. Secondly, I think you, the teacher, have to be hyper-conscious when you are requiring a switch of conception, and you have to be hyper-explicit with the students: “Yeah, a vector used to be those arrow things, and they still give you intuition, however …”.
There’s plenty more to say, both about the vectors and generally, but that’s probably enough for now.
I did not like the use of the implication symbol in the Nelson book.
“did not like”. Terry, you’re hilarious.
Excellent selection of excerpts from the texts. What a beautiful POSWW. Let me add some words anyway.
Cambridge: The zero vector isn’t parallel to anything and isn’t perpendicular to anything. Zero isn’t even parallel or perpendicular to itself.
Nelson: The zero vector is parallel to everything and is perpendicular to everything. Nelson adds some unexpected spice here by going further, and saying that zero is parallel to everything in the same direction as well as in the opposite direction. Amazing.
Jacaranda: At first, zero is parallel to itself (only). But then, we are told it makes a straight angle with any other vector. That’s a bit unusual, isn’t it? I wonder what they think the difference is between making a straight angle and being parallel. Oh, but never mind, Jacaranda isn’t logically consistent because we are then told that the angle is actually a right angle. We are told that this means zero is not perpendicular. But the angle the zero vector makes with any other vector seems to be both straight and right at the same time. Oh dear.
All of this confusion!
But wait, there’s more …
The heart of the problem is that proofs have been replaced by visualizations (to put it mildly). If the authors had been forced to *define* their terms and to show, for example, that being parallel is an equivalence relation (pay attention, Jacaranda), then they might have caught most of their errors.
The heart of the heart of the problem is that no one now cares about truth.
Or textbook publishers do not sufficiently care about the accuracy of their content to actually hire a mathematician to check their page proofs…?
No one cares (± three people).
I count Peter Sullivan as -1.
Fair enough.
So if we square him, the outcome is positive? Or somehow raise him to an even power…?
Hmm. Good point. I think Sullivan to any power is negative. A whole new number system.
Someone I know asked a textbook publisher for schoolbooks in Germany why books contain so many errors nowadays, and was given the answer that one error per page is actually quite normal and nothing to worry about.
Like I said, no one cares about truth.
One per page sounds extreme, but plenty of textbooks I use for university math courses have some mistakes. It is my job to make sure that students understand what I am saying, mistakes are my responsibility.
So while it is fun to find these wrong things in high school books, IMO the “real problem” is that teachers do not teach what is correct anyhow. I guess that is difficult, if they don’t know how to fix whatever obvious mistakes are in the books (and aren’t using the same conventions etc as those setting exams).
It would be easier if those setting the exams made said conventions clear.
At a university, the lecturers set the exams (usually) and so the lectures dictate the conventions that will be used in the exams.
Putting errors in textbooks aside – teachers don’t know how to fix errors in exams because the exams do not admit there are errors.
Jesus. You think “mistakes” describes accurately what I’ve posted on school mathematics?
You’re right, these are evidence of problems well beyond mistakes, and I shouldn’t compare the odd issue here and there in a math book with these issues. I suppose I’m bemoaning that teachers don’t have the correct amount of autonomy and coordination with exam setters to make the problems with the textbooks immaterial. (If nobody purchased the books because they caused more harm than good, THEN the companies might start to care.)
It is a challenge for the teacher when the curriculum contains errors, the text books repeat the errors, and the final examination will make the same errors.
I’ve updated the post.