This is Q5 on yesterday’s VCE Specialist Mathematics Exam 1, and I haven’t had time to go through this (or any) question, or even properly read the comments. Possibly (and possibly not) the “C” is too harsh, and maybe the “C” might better stand for “clunkiness”. I try very hard to not post unless I already have a very clear idea what is going on, which I am doing now, but there clearly seems to be enough consternation that the question warrants its own post, and I’ll try to look properly soon. For the previous discussion, readers can begin about here, and readers may also wish to consult an old PoSWW (and a very old WitCH is also somewhat relevant).
Visited 120 times, 1 visit(s) today
The student who just gets the inner product and doesn’t worry about one of the vectors being zero does better.
Wording would be clearer if in b) and c)
“collision to occur” is replaced by “this collision to occur”
Interesting that the statement “t=1” is in every question part whilst the right angled collision is not.
Don’t think overall that this question was ambiguous.
Thanks, Yaniv. What do you mean by “overall”? That context or whatever removed whatever slight ambiguity?
In VCAA questions, if the preamble is above the individual parts, then it is implied that it is relevant to all parts below.
Here, “when particles collide… velocities are at right angles” above part b and c implicitly implies that it relates to both parts.
This is not a new thing and has been present in pretty much every VCAA exam, so should not create any ambiguity within students.
For the two solutions to the perpendicular question, even though students may have been taught wrongly that zero vectors are not orthogonal, it is highly unlikely that they calculated that it was a zero vector in the exam. Even if they did, the preamble asking for two solutions removes that ambiguity.
OK, thanks, Yaniv. I haven’t looked yet.
All maths exams overall seem too easy this year, there hasn’t been a clear ‘separator’ question in any of the exams
OK, I’ve looked. I strongly disagree.
Oh well
Interesting that VCAA decided to give the hint that the velocities are perpendicular rather than letting students figure that out…
How do you figure out a (and b) without using the perpendicularity?
I meant that telling the students specifically about the “velocities”, rather than simply saying the particles are travelling perpendicular to each other, is an unnecessary hint.
They are only at “right angles” at the instant of collision. No question there is way, way, way too much handholding in these exams, but I think this one is fine.
OK. I will begin with an offering (of what is wrong with this question).
“The particles collide when
.”
However, the paths taken by the particles are not linear, so it is not reasonable to assume they ONLY collide at this one time.
“When the particles collide…” This does not specify that
for the aforementioned reason.
Yes but t=1 is stated specifically for parts b and c
I think this misses the point a bit.
The statement
comes later and does not disallow other collisions.
As the question is written, ANY OTHER collisions must happen when the velocity vectors are perpendicular to each other.
If there is a time other than
when the particles collide but the velocity vectors are not perpendicular then the question is wrong.
By that logic, any value that isn’t a=-1 or a=-3/2 would make the question ‘flawed’ because their velocities are not perpendicular at the collision at t=1. The question is SOLVING for the a and b values in which they DO collide with perpendicular velocity vectors. If we take the other collision at t=1/2, then for a=-3/4 and b=-7/4, they DO collide with perpendicular velocity vectors because I SOLVED for those a and b values.
I think this also entirely misses the spirit of what an exam question is meant to do. It is frankly preposterous and ludicrous to delve into such niche, hyper-technical interpretations of the problem. The function of an end-of-year question is to be unambiguous and assess the cohort fairly, not to invite semantic debate over contrived hypotheticals. The question as written achieves its purpose perfectly clearly, and I highly doubt any student under exam conditions would even consider the (fundamentally flawed) line of reasoning you’ve presented.
Strong language for defending a question that’s gonna screw up a lot of students.
I’m not suggesting any student would use my line of reasoning. I worked through the paper and obtained the same answers as many have reported (and I assume the answers VCAA was intending to regard as correct).
What I am suggesting is that the question (and many, many other questions from exams around the world) is making assumptions that are not valid.
Obvious or not, contrived or not, I don’t think it matters.
“Spirit” of an exam question also sounds like a fairly weak argument. Your statement that a question should be unambiguous is right on point though – even if the “purpose” of exam questions may be debatable.
Not much of a fan of Part C. Part C is rather a bit unreasonable by VCAA, if it were B ii instead of C, that would clear up any confusion that students would have to reject the solution a=-5 and b=-6 due to the statement above part b, however due to the fact part C was on the next page and only stated “when particles collide at t=1” without mentioning the collision at right angles, many students would not consider that statement in such a pressured exam. In the unlikely sense that VCAA would accept both sets of solutions, it would be interesting to see how many students failed to reject 1 set of solutions under exam pressure.
Part B however I do not think that students were affected much by the fact that they only needed to find 2 values, that the dot product of the velocity vectors was 0, I think that most if not all students would not have had the ability to worry or realise that one of the vectors was zero.
Thanks, T. I’m not a fan of A or B either. You may be correct that most students will not be affected by the clumsiness of B, but:
(i) You probably know the kids better than me but have my doubts. I’m guessing plenty of kids will be screwed up by B.
(ii) Even if it’s only a few kids, it’s a few kids too many. It’s just dumb: such line ball issues should simply not appear on the exam.
Is the consensus that VCAA really did want the zero vector included? “At right angles” seems like such a strange, deliberately-common-language choice. It strongly suggests that there is some representation where a 90 degree angle could be placed between them.
If a cue ball hits a stationary object ball flush front on, no one in their right mind would say they hit “at right angles”. And claiming that the dot product is zero would be a non sequitur.
They’ve gone out of their way to NOT say ‘orthogonal’ or even ‘perpendicular’.
Hi, Alex. I’ll get to your critique later, but in non-answer to your first question: can you see any alternative to including the zero vector? Perhaps everyone has misread the problem somehow, but I can’t see the alternative.
I definitely can’t see any way past the “two possible values”, but I was hoping someone smarter than me had a theory. Because the previous wording seemed deliberately designed to rule out one of them.
I almost feel like this question was written by two different people with two different things in mind.
Thanks, Alex. I agree with you that “right angles” is a notably geometric-vernacular term, which leans against permitting the zero vector. There would be a problem even if “perpendicular” (or “orthogonal”) had been used, because Cambridge fucked it up and because VCAA doesn’t give enough of a fuck to clarify what words mean, but “right angles” makes the issue significantly worse.
Your last sentence is well worth pondering.
I genuinely believe that the person who wrote ‘two possible values’ didn’t realise that one of the values yielded a zero vector. I believe that they were intending students to simply equate the scalar product to zero and not think about it further.
When I did the paper for myself the geometric language led me to believe they definitely wanted the balls to be moving. It is then most efficient to find the single viable solution without actually computing the scalar product as one ball is moving parallel to the j vector.
Having noted the ‘two solutions’ required by the question, I then spent a good few minutes deconstructing it and unravelling what the second desired solutions must be, even though to me it doesn’t fit the specific ‘natural language’ wording of the question.
Ultimately I wrote the ‘correct answer’ as there weren’t really any alternatives, but I was really unsatisfied by it and I think many students would have wasted valuable time in a similar way.
When you also factor in part c, I think it is a terrible, terrible question.
Thanks, tom. It is indeed a terrible, terrible question. As to what the writers were thinking, God only knows.
The below link (and others like it) may provoke further discussion:
https://math.stackexchange.com/questions/4684779/orthogonality-of-zero-vector-relative-to-non-zero-vectors
Thanks, John. There’s nothing to discuss, but your link emphasises why one wishes to use as vector-like term as possible, to stay away from “angle”.