Writing Vectors

Once again it’s time to introduce first-year Mathematical Physics students to the joy of vectors, or specifically Euclidean vectors. Some of my students have seen them before, but probably aren’t aware of how much we use them theoretical physics. Obviously we introduce the idea of a vector in the simplest way possible, as a directed line segment. It’s only later on, in the second year, that we explain how there’s much more to vectors than that and explain their relationship to matrices and tensors.

Although I enjoy teaching this subject I always have to grit my teeth when I write them in the form that seems obligatory these days.

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

\vec{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

\vec{u}=\left(\begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right) and \vec{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:

\vec{u} \cdot \vec{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 a vector laboriously in terms of base vectors:

\vec{r} = x\hat{\imath} + y \hat{\jmath} + z \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. The only amusing thing about this is that you get to tell students not to put a dot on the “i” or the “j” – it always gets a laugh when you point out that these little dots are called “tittles“.

Worse still, for the purpose of teaching inexperienced students physics, it offers the possibility of horrible notational confusion. In particular, the unit vector \hat{\imath} is too easily confused with i, the square root of minus one. Introduce a plane wave with a wavevector \vec{k} and it gets even worse, especially when you want to write \exp(i\vec{k}\cdot \vec{x}), and if you want the answer to be the current density \vec{j} then you’re in big trouble!

Call me old-fashioned, but I’ll take the row and column notation any day!

(Actually it’s better still just to use the index notation, a_i which generalises easily to a_{ij} and, for that matter, a^{i}.)

Or perhaps being here in Ireland we should, in honour of Hamilton, do everything in quaternions.

4 Responses to “Writing Vectors”

  1. A long time ago but pretty sure I was taught using r = x i + y j etc. Can’t have been a British/Irish thing as I was taught by Irish christian brothers. However had the line under the vector i, j etc, rather than an arrow on top.

    I also seem to recall that we learned about vectors before matrices, which may explain it.

    • telescoper Says:

      I used underline rather than the arrow (and double underline for a matrix) until I started to get them confused with printers’ instructions for italic and bold face espectively…

    • I was taught with rows and columns. The summation notation is not intuitive: we wouldn’t write coordinates on earth that way (or RA and Dec, for that matter). I was wondering whether it came about from typesetting, as it keeps everything on the line?

  2. Anton Garrett Says:

    The version you are complaining about is actually optimal when you get into higher dimensions and numbers of suffixes. Use Clifford Algebra and you never have to write out those Dirac and Pauli matrices.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: