Killing Vectors

I’ve been feeling a rant coming for some time now. Since I started teaching again three weeks ago, actually. The target of my vitriol this time is the teaching of Euclidean vectors. Not vectors themselves, of course. I like vectors. They’re great. The trouble is the way we’re forced to write them these days when we use them in introductory level physics classes.

You see, when I was a lad, I was taught to write a geometric vector in the folowing fashion:

\underline{r} =\left(\begin{array}{c} x \\ y \\ z \end{array} \right).

This is a simple column vector, where x,y,z are the components in a three-dimensional cartesian coordinate system. Other kinds of vector, such as those representing states in quantum mechanics, or anywhere else where linear algebra is used, can easily be represented in a similar fashion.

This notation is great because it’s very easy to calculate the scalar (dot) and vector (cross) products of two such objects by writing them in column form next to each other and performing a simple bit of manipulation. For example, the scalar product of the two vectors

\underline{u}=\left(\begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right) and \underline{v}=\left(\begin{array}{c} 1\\ 1 \\ -2 \end{array} \right)

can easily be found by multiplying the corresponding elements of each together and totting them up:

\underline{u}\cdot \underline{v} = (1 \times 1) + (1\times 1) + (1\times -2) =0,

showing immediately that these two vectors are orthogonal. In normalised form, these two particular vectors  appear in other contexts in physics, where they have a more abstract interpretation than simple geometry, such as in the representation of the gluon in particle physics.

Moreover, writing vectors like this makes it a lot easier to transform them via the action of a matrix, by multipying rows in the usual fashion, e.g.

\left(\begin{array}{ccc} \cos \theta & \sin\theta & 0 \\ -\sin\theta & \cos \theta & 0 \\ 0 & 0 & 1\end{array} \right) \left(\begin{array}{c} x \\ y \\ z \end{array} \right) = \left(\begin{array}{c} x\cos \theta + y\sin\theta \\ -x \sin \theta + y\cos \theta \\ z \end{array} \right)

which corresponds to a rotation of the vector in the x-y plane. Transposing a column vector into a row vector is easy too.

Well, that’s how I was taught to do it.

However, somebody, sometime, decided that, in Britain at least, this concise and computationally helpful notation had to be jettisoned and students instead must be forced to write

\underline{r} = x \underline{\hat{i}} + y \underline{\hat{j}} + z \underline{\hat{k}}

Some of you may even be used to doing it that way yourself. Why is this awful? For a start, it’s incredibly clumsy. It is less intuitive, doesn’t lend itself to easy operations on the vectors like I described above, doesn’t translate easily into the more general case of a matrix, and is generally just …well… awful.

Worse still, for the purpose of teaching inexperienced students physics, it offers the possibility of horrible notational confusion. In particular, the unit vector \underline{\hat{i}} is too easily confused with i, the square root of minus one. Introduce a plane wave with a wavevector \underline{k} and it gets even worse, especially when you want to write \exp(i\underline{k}\cdot\underline{x})!

No, give me the row and column notation any day.

I would really like to know is who decided that our schools had to teach the horrible notation, rather than the nice one, and why? I think everyone who teaches physics knows that a clear and user-friendly notation is an enormous help and a bad one is an enormous hindrance.  It doesn’t surprise me that some student struggle with even simple mathematics when its presented in such a silly way. On those grounds, I refuse to play ball, and always use the better notation.

Call me old-fashioned.

24 Responses to “Killing Vectors”

  1. I agree that the i,j,k notation is a pain in arse. I suspect it is not some sort of pedagogical backwardness but because of people writing notes in MS Word and powerpoint instead of either in LaTex or on a chalkboard. Equation editor is a very poor and non-standard add-on to Microsoft products, and maybe teachers favour it for this reason….

    Having said that I remember being taught the clumsy way and that was before powerpoint.

  2. I teach Math Methods to first years at UCL and I use both notations depending on the context. Sometimes it’s handy having things “in line”, but since I also teach matrices I gravitate towards the row/column notation over the course. From the feedback I get, I don’t think this causes the students a problem.

    Maybe using a couple of different notations is even helpful to establish the idea of a vector independent of its representation? Though they’ve never said that in feedback either…

  3. telescoper Says:

    Jon,

    I take the point that one should get used to different representations, eventually. My point was about beginners. We should help them with the notation, not saddle them with a daft one. Surely it’s better to teach the rudiments of linear algebra in a way that makes the relationship between vectors and matrices clear?

    Peter

  4. telescoper Says:

    Ed,

    I’m not sure it’s any easier to write matrices and column vectors in latex than in powerpoint. It’s certainly easier on the blackboard, but we’re not allowed blackboards any more.

    Peter

  5. Anton Garrett Says:

    Peter,

    Writing it out the way you dislike is a step toward the rephrasing of the algebra describing geometry via Clifford algebra. That is a very good thing to do – who wants to use complex analysis in one problem, vectors and tensors in another, quaternions or spinors in a further one, matrices in another case? The Clifford way unifies and extends all of these. (Did you ever wonder how to generalise the neatness of complex analysis to more than 2 dimensions…?) The trouble is that by writing it out the way you dislike but not going Clifford you have the worst of both worlds.

    Anton

  6. telescoper Says:

    Anton,

    Agreed, but I think you need to learn how to take scalar products of simple vectors and multiply matrices – and know what complex numbers are too – before you can cope with the beauty of the Clifford algebra. I was talking about learning to walk, rather than how to run like a proper athlete!

    I never got round to re-writing my Dirac notes in terms of the Clifford algebra. Maybe one day I’ll get asked to teach it again and will do so then. Sounds like a good topic for a post sometime too. I think physicists are aware of how elegant it is, but generally don’t realise how practical it is, especially for computational problems.

    Peter

  7. Anton Garrett Says:

    Peter,

    I agree that one should know how to multiply matrices (and manipulate determinants), but that’s a distinct skill from how to represent geometry algebraically. Teaching needs to be rejigged, from school upward, to inculcate the Clifford view from the start. Giving people the old way and then teaching them Clifford as a better way for ‘tough’ calculations means that the pupils have to *unlearn* something; and if you think learning is hard then unlearning is far harder. The rejigging I am calling for is a task for expert teachers who are technically competent, because you obviously can’t teach vector calculus immediately after O-level (or whatever it’s called nowadays). But I’m sure it can be done; and teaching complex analysis as 2D Clifford rather than as a brilliant dead-end might be a good start.

    Anton

    • I assume you were all just being polite in not pointing out that I rotated my vector incorrectly. I spotted it myself and have now fixed it…

  8. Anton Garrett Says:

    Vectors with attitude?

  9. I remember when I was at school reading a book about Einstein – can’t remember which one. It kept mentioning tensors, although never explained what they were. I decided to ask my maths teacher: “Sir, what’s a tensor?”.

    He replied “It’s like a vector, only more so.”

  10. Anton Garrett Says:

    Pupil: What’s a tensor?
    Teacher: Between a nine and an eleven.

    Now, what’s a hawser?

  11. My recollection from UG days is that we used both notations in different circumstances. But this was in the days of real A-levels and when matrices were part of O-level maths (I’m definitely dating myself!) so we were able to handle the different approaches. I’d hope the students could handle that now, but fear it might not be the case.

  12. I remember that when I was un undergraduate student, I first learnt the i,j,k notation, but it was not confusing.

  13. This is what you get when you ask mathematicians to teach maths. They will insist that the equation you dislike is valid (vectors can be equated to sums of vectors) but the one you like is not: a vector (coordinate-independent entity) is not identical to a coordinate-dependent column matrix. Of course this distinction is a bit high-falutin for school kids, but by the time you get to quantum state vectors it is rather crucial (at least if you’re taking the course I teach…)

    • telescoper Says:

      A vector is indeed a coordinate-independent object, but it can be represented in a given basis in column form and anyway I was talking about beginners. My point is that I think this is the best way to write vectors to learn the rudiments of vector manipulation. I agree that the deeper you go the more particular you have to be about notation, but surely you have to walk before you can run? There’s also a difference between identity and equality…

  14. Bryn Jones Says:

    Perhaps one advantage of writing vectors as sums of unit vectors (as in the i,j,k notation) is that things are more explicit when handling coordinate systems other than Cartesian. It may be helpful for polar coordinates, for example, if the unit vectors are written explicitly. So there may be some cases where using sums of unit vectors is preferable.

    Yes, students need to know and be able to use both notations. I’m not sure this is achieved on all degree courses in physics, as there is a strong preference for the i,j,k notation, and when column matrices are used, the switch is often made without a full explanation of how columns are manipulated. (In contrast, the column matrix notation seems to be preferred on mathematics degree courses, but there is nothing new in differences in culture between mathematics and physics.)

  15. I suppose it depends whether they are taught programming at the same time as vectors, as it is certainly easier in code to write them our linearly.

  16. Phil Harmsworth Says:

    Peter,

    When I used to tutor first year ‘applied mathematics’, the course used the ‘i, j, k’ notation. It has an advantage over the matrix approach when it comes to calculating cross products, IMHO. This was useful for the course that I tutored, because we covered equations for planes in 3-D, finding perpendiculars, angular momentum and so forth.

    As for the adoption of the notation, I suspect that you may have it backwards. The notation goes back to Hamilton’s quaternions: when Gibbs produced vector analysis, he kept the notation (early 20th c). I’m not sure that Cayley’s matrices were in use at that time, at least in undergraduate teaching. As far as I know, they didn’t become part of the physics syllabus until later.

    P.S. I notice that at school, you got a better response to your question about tensor analysis than I did. I was told “that’s silly” when I asked.

  17. Just waiting for a mathematician to explain that vectors aren’t triples. They are entities with a size and a direction which exists independently of co-ordinates. The ijk version (or r,theta,phi version) makes this explicit – “this is how you write me in this particular co-ordinate system” – whereas the matrix version makes it look as if the vector IS those three numbers. This is philosophically bad. Of course the matrix version is more aesthetically pleasing, and more transparent for calculation, but there is a didactic argument for starting with the ijk notation so that when uses the matrix version one remembers it is only a handy shorthand.

  18. telescoper Says:

    Andy,

    That’s essentially what Paddy said (above). My point is that the column vector is a representation which makes manipulation easier.

    I also repeat that equality is not identity. In my book, “=” is a perfectly good shorthand for “in a given basis this vector can be written as”.

    Of course, later on we have to introduce different coordinate systems and think about transforming vectors, but let’s get the students comfortable with them before doing that.

    The problem I find is that introducing vectors the ijk way erects a notational barrier that makes computation more difficult.

    Peter

  19. Teacher: Give an example of a vector.
    Student: The north star, Polaris.
    Teacher: Why is the north star a vector?
    Student: It has a magnitude, and a direction.

    Apologies to Ed Bertschinger.

  20. The notation you find so cumbersome goes all the way to the classic work on vector analysis by J.W. Gibbs, who reformulated quaternion algebra into a 3 dimensional vector algebra. His monograph can be found to this day, and is a time honored classic. Both notations have their advantages. I can move seamlessly between both notations.

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