We’re snowed, and we have a couple posts in the works that simply won’t behave. So, we’ll just keep the ball rolling with a WitCH, and have you commenters do the work for now. The following is the introduction to planes (a new VCE topic) in VicMaths, Nelson’s Specialist Mathematics Year 12 text.
Seriously this example was taking an overly convoluted approach.
The closer I look into this – holy-moley- the less I have fondness with these two excerpts!
1. Where is the reference point? Oh I see, origin… So the next question is: are all planes spanning out from origin?
2. Geez, setting up simultaneous equations as if they deal with linear dependence… Hold on? Didn’t they just mention vectors an and b are independent? This water pond is getting muddier…Oh wait, they just intended a vector triangle.
3. Any point P…? In the space? Which cosmos? Oh I got it, they “wish” students could infer this is a point within the plane…or a way to verify if a point lies within a plane.
Apparently appalling!
Glad we didn’t buy Nelson Spesh textbook…
Thanks, AS.
1. Yes, idiotic.
They seem to have wanted to think of a plane as being spanned by two vectors. But the way of capturing that is significantly different if the plane doesn’t contain the origin.
2. I’m not sure of your point here. I don’t see much value in the question, and the setting out is atrocious, but how else would you do it?
3. Yep. Simply not writing what they intended.
1) I believe the first sentence is grammatically incorrect (English is not my first language).
2) “A normal is a vector perpendicular to the plane”: It would be somewhat clearer to write “A normal to [or: of] the plane is …” (as the normal is defined through the plane, as is recognized two sentences further on in the screenshot by Marty).
3) “A plane may be fixed …”: What exactly is the meaning of “fix” here? “A plane may be specified …” would be better. Or, if the plane is already given through some other means, “A plane may be determined …”.
4) “Any normals of the same plane are parallel, so they are multiples of each other.” The formulation “… so they are multiples” suggests that the last mentioned property is *not* taken to be the definition of “parallel”, and instead is the geometric notion delivered by two well-aimed chalk strokes on a board. But normal vectors (in the case of school work certainly) always emanate from the origin (the zero) of the coordinate system, and they won’t any longer if shifted in the said “chalkboard” manner except in very special cases.
5) First dot point: Perhaps some students would like to see already in the preceding blurb the definition of “linear combination”, which would accommodate the case that “others” can also read “other” (see 6 below), and the case that some (or in the extreme case, all) of the scalars (real-number multipliers) in the linear combination may be zero. When looking at independence rather than dependence, one has to use a negative description as done in the dot point, which seems to me less than perfect didactically even though it saves work (to folks like those who crafted this).
6) Second dot point: “The independent vectors are in different directions.” The statement in question is really just a specialization of the preceding dot point to the case of two vectors. The student may be able to recognize this, provided they know what a linear combination is (see 5 above). Perhaps they even recognize that “different directions” should be read as excluding “opposite directions”.
I haven’t examined what is further down and the list above is possibly incomplete even for what part of the material it does cover, but it is maybe a start.
Thanks, Christian.
1. Yep, the sentence is ungrammatical.
2. Yep, the wording is appalling.
3. Yep, “fixed” is a very poor choice.
4. The sentence is as clumsy as all hell, but I’m not sure of your mathematical/definitional objection.
5. Again, the sentence is appalling, but I’m not quite sure of your specific objection. For clarification, independence and the like has already been done (badly), earlier in the text. So, these dot points would be considered a refresher, even if they’re not very refreshing.
6. Yeah, they wanted to have a dot point on independence and a dot point on spanning, but got lazy and said “Let’s just chuck stuff in”.
Thanks Marty for your responses on my points. By way of a rejoinder (if this word is neither too academic nor too humourous for this purpose – ESL speaker here), I want to add something to my 4 and 5 where you flagged issues:
4) What I had trouble with is the supposed definition of “parallel”. To me, the two normals referred to in the book passage in question (and only locally… a deep sigh to mirror what other commenters have raised) seem to be identified with the lines that contain them. If the normal vector is
then the corresponding line is the set 
. But these lines all run through the origin (the zero) of the coordinate system. This does not match the definition of parallels which says that they are straight lines that do not intersect.
5) Let me clarify as follows:
(i) In a set of two vectors as is a key situation for the book passage, whatever these vectors may be (!), none of them can be written as a linear combination of the “others”, because there is only one other vector (singular). (Maybe I should run for cover now…)
(ii) In the sentence before the colourbox “Independent vectors”, the words “… will be a linear combination of any two independent vectors in the plane” could be confusing to students who do not appreciate that in that linear combination referred to here, one or even both of the coefficients may be zero. (Afterthought: Would the word “unique” before “linear combination” be already above the intended school level? I guess so.)
Hi Christian,
Re (4), I think the normals sentence is ok, if poorly worded and irrelevant for now (since the intention is to introduce planes as linear combo things). Yes, later things are confused, because they seem to identify the vectors with lines through the origin, but I think the error is in the later stuff. But, the whole thing is worded so incredibly badly, who can be sure?
Re (5), they are thinking of the two independent vectors as spanning a 2D world, and then any (other) vector in that world can be written as a linear combination. So, again, I think it is officially mathematically ok, but the wording is so atrocious and leaves so much implicit, it’s fair enough to deem it not ok.
As for uniqueness, that’s kind of in the token linear algebra of VCE, but it’s not really relevant for the computations in the example question.
It is funny (to me, anyway) that “a position vector” … “will have position vector…”.
A lot of words for not a lot of useful information.
There is a lot going on here, but I want to start with the introduction:
After reading it, I feel I know less about planes than I did in Year 7.
So, in short, to me the main source of crap is that the whole idea of “what is a plane?” is jumbled from the start and no amount of “linearly dependent set of vectors” nonsense can un-jumble it.
Yep, the “a position vector” doesn’t belong.
Your penultimate sentence-paragraph sums it all up very well: after reading this, any student will know *less* than they did previously.
As a physicist who just happens to teach a Year 10 specialist elective, I don’t get this. Do we have to assume the origin is in the plane it’s talking about? Because, if not, we only have two points and two points don’t make a plane. Am I missing some convention here? Ah, maybe the fact that it ‘contains’ these position vectors…
Hi, Greg. I *think* that’s the assumption that has to be made. I think the text is treating the vectors as if they were lines through the origin, and the plane contains those lines. But, truth be told, I haven’t yet deciphered the thing. I read it, realised that, whatever was intended, it was screwed, and left it to the (much-appreciated) commenters. I hope to go carefully through the comments tonight or tomorrow, when I’m scheduled to breathe.
I have always (rightly or wrongly) assumed that a “position” vector begins at the origin, as opposed to a “free” vector which does not have to.
On that point, even though I still don’t think it fixes everything, I think the text may be acceptable.
As Christian has pointed out already, the writing here is sloppy, at best.
The major gripe I would have though if this were my textbook is that is completely seems to miss the main idea: If I have two vectors in a plane, I can find the equation of the plane and then very easily check if a point is in the plane or not. I do not need to test a set of vectors for linear independence and find the thought of following that process somewhere between odd and very odd.
Hi, RF. I think the purpose of the section is to introduce planes as the span of two vectors (give or take a shift from the origin), and doing this before getting into normals and equations. That’s in principle reasonable, but they’ve just done it so woefully that any reason is very difficult to locate.
OK, I accept your reasons but I now have a new question:
Why does the text not write something along the lines of: “a plane can be uniquely identified using three points and therefore two non-parallel vectors”?
If the big idea is obvious from the start, the examples are well chosen and the detail is not excessive, then the section is good.
This section fails on all three counts, in my opinion.
At least three, and more like eight.
Yes, I have other concerns as well, such as the use
as the symbol for a plane instead of 
but I feel some of these are nit-picks so I tried to focus on my main gripes.
Overall, I think the idea the text is trying to convey has been totally lost (assuming it existed).
Why give the planes a name in the first place?
Why does the name contain a subscript?
Obviously they dont want us to be confused between the plane in the prologue and the plane in the worked example. How considerate. The whole thing is gammy if you ask me.
Well played… I would have preferred a capital G since it says
not 
, but otherwise, excellent work.
Exactly. Which is why I named my daughters Eva and Eva1.
Your daughters probably mingle so it makes good sense to name them (very efficiently too, I might add) to avoid any confusion. I can’t see the planes in the prologue and worked example doing much mingling. (Said in the spirit of blithely ignoring all previous sarcasm).
Were the planes in the questions each given a different name, to avoid any danger of the plane in Question 1 being confused with the plane in Question 2 etc?
And who knows where the rain in Spain falls …
Yes, the approach is pretty ugly and inelegant and an equation for the plane makes far more sense. FWIW I *did* assume the position vectors began at the origin. I did not assume that the plane in question contained the origin… well, until I realised it had to for the question to work.
Think of it as an exercise in productive struggle.
Hi again, RF.
“position vector” is one of those school expressions that I’ve never understood. In the end, a vector is a vector is a vector. The problem is to somehow slide from “magnitude and direction” to (for high school) “represented by coordinates”. Textbooks accomplish this slide by successive sleight of hand.
I won’t try to untangle the definition and usage here. What is really important, however, and maybe what you were getting at, is that the last sentence of the introduction is both meaningless and wrong: “vectors in the plane” makes no sense; and, the position vector of a point in the plane will *not* be a combo of the given vectors (unless either the plane goes through the origin or the “position vectors” are taken relative to a point in the plane).
Pretty much.
The IB syllabi went to reasonable lengths to distinguish position vectors and direction vectors that were then used for the vector equation of a line in space.
It made sense to me as a student, and I guess I’ve never un-learned the concept.
Yes, a vector is a vector just as VCAA is VCAA.
I looked at this yesterday, but forbore to comment; thankfully everybody else has said what I would have done anyway, and of course more besides. It’s quite appalling. Not only is the mathematics quite frankly wrong; the text is ungrammatical and illiterate. How do these things get published? What sort of editors do these publishing houses use?
As to the wild and woolly first sentence: “A plane is a flat surface extends in all directions.” it’s very hard not to be reminded of Stephen Leacock’s immortal line: “Lord Ronald said nothing; he flung himself from the room, flung himself upon his horse and rode madly off in all directions.”
It is at least pleasing to be reminded of Stephen Leacock. Long time since I’ve read him.
That’s an omission which is easy to repair. Get thee to a library!
Before or after I read The Wit and Wisdom of Hannah Gadsby?
I would have thought, given the no doubt infinitesimal size of said volume, that it will take only an infinitesimal reading time … I’m working my way through “The Kindness and Humanity of Peter Dutton”, a book of basically negative length. [This is all a bit off-topic, but since we seem to be discussing blinkered stupidity wrt this WitCH, maybe not so much … ]
Euclid did not assume that a plane was necessarily infinite.
`the largest independent set’ makes no sense.
Its crappy wording, but why not?