In this WitCH we will again pick on the Cambridge text Specialist Mathematics VCE Units 3 & 4 (2019): see the extract below. (We’d welcome any email or comment with suggestions of other generally WitCHful texts and/or specific WitCHes.) And, a reminder that there is still plenty left to discover in WitCH 2 , WitCH 3 and Tweel’s Mathematical Puzzle still have room for comment.

Have fun.

Two issues in the first two lines…

Yes, that’s where the rot begins.

And then all of a sudden, later on, zero vectors are allowed…

This is less difficult than other WITCH, perhaps?

I don’t think so. Be detailed and precise: what exactly is the issue(s) above with zero vectors?

Moreover, though zero vectors are the source of the problem, there is also plenty more to critique. The question isn’t what is the falsity here; the question is what is the

craphere. Of course being false is one way to be crap but, alas, there are plenty of other ways.Read the excerpt. It is awful, painful to read. It is obviously crap. But why?

The existence of the crappiness is clear, but sorting out the reasons for and the nature of that crappiness takes some thought.

OK (still wanting to allow other crap-hunters their share of the fun) – you can divide the crap onto this page into two very broad categories, although there is a lot of overlap, most of the crap is dominant in one of the categories.

Category A – ideas which are just wrong. Either because they leave out key parts or are totally misleading.

Category B – ideas which might be correct but are totally useless to anyone (especially the supposed audience – students and perhaps teachers)

The whole thing is just a big over-bloated mess – a total fatberg. Imagine a student or inexperienced teacher trying to make sense of it.

Most experienced teachers cannot make sense of it.

Aren’t vectors that are are opposite in direction and different in magnitude called anti-parallel?

I don’t think you’ll find this distinction in most textbooks. An alternative definition is that two vectors are parallel if their cross product (not on the Specialist Maths course) is equal to zero.

Potii, I think John’s right. The term anti-parallel exists, but I haven’t seen it much used. More to the point, John’s suggested “alternative definition” (which can be rephrased with no reference to cross product) is quite standard, though not universal. And, very much to the point, Cambridge’s definition and John’s are not equivalent.

Two vectors are parallel if u=k*v where k is any real scalar. If u is the zero vector it is parallel to all other vectors, which can be seen by setting k=0. If k=0 is not allowed, then the zero vector is not parallel to anything.

This then means that any set of vectors which contains the zero vector will be linearly dependent. The justification is trivial, just set the coefficients of the non zero vectors to zero and the coefficient of the zero vector to anything other than zero.

This point is made in the excerpt, but by making the error about the zero earlier on, the page contradicts itself.