A course that my kids are doing in conic sections insists that all positions are represented as standard 2D cartesian coordinates $(x,y)$ (i.e. row vectors), and all translations are written as 2D column vectors $\begin{pmatrix}x\\y\end{pmatrix}$. Any deviation from this is marked incorrect.


textbook image showing positions as row vectors and translations as column vectors

I believe the reasoning is that positions are not vectors and so are written in row form, while translations are vectors and therefore must be written in column form.

As far as I can recall, I have seen positions and translations in either orientations according to whatever works in context, but usually both written as row vectors or both written as column vectors. If position and translation vectors are written with different orientations, translations are unable to be obtained from subtracting position vectors, and unable to be applied to positions by addition.

Is there any basis in theory or tradition for a strict separation of positions as row vectors and translations as column vectors?

  • 3
    $\begingroup$ I have never seen such nomenclature used. Perhaps this is just for the sake of grading. Certainly it is better to think of both as rows or columns. In my courses, I use the sneaky notation that $(a,b,c)$ is a column vector whereas $[a,b,c]$ is a row vector. Or, I can write $[a,b,c]^T = (a,b,c)$. I add rows to rows and I add columns to columns. We never add rows to columns. Again, my guess, this is a grading device, and in my estimation a bad one. $\endgroup$ Sep 19 '18 at 2:13
  • $\begingroup$ @JamesS.Cook I like your notation - I might try it out myself. $\endgroup$
    – Richard
    Sep 19 '18 at 9:04
  • $\begingroup$ I don't know whether people have thought about this a lot, but in many linear algebra or MV calc texts at any rate this distinction must be preserved, because in principle a vector doesn't "start" at the origin, while coordinate points are always with reference to it. So having a notational distinction can be quite helpful. Of course, whether students are cognitively able to grasp this distinction depends partly on their age and partly on the instructor's clarity over why this is important. $\endgroup$
    – kcrisman
    Mar 11 '19 at 14:18
  • $\begingroup$ (Also, since they use parentheses and not brackets, probably the positions are not considered row "vectors", if you see what I mean.) $\endgroup$
    – kcrisman
    Mar 11 '19 at 14:18

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