The Fairyland Clickety-Clock

This post is again inspired by the television show The Fairies (you have been warned!).
In Fairyland, there is a peculiar procedure you have to go through in order to find out the time. The following sequence is played out several times across different Fairies episodes:

RHAPSODY: I wonder what time it is.
HARMONY: Well there’s only one way to find out the time in Fairyland…
ALL SING (with appropriate actions):
Fairyland Clickety-Clock we need to know the time.
Please show yourself in Fairyland and let us hear you chime.
Tick tock Clickety-Clock,
Tick tock Clickety-Clock,
Tick tock Clickety-Clock,
ENTER CLICKETY-CLOCK to tell everyone the time.

You can understand why they lead such a carefree existence in Fairyland: if you had to do this every time you needed the time, you wouldn’t find it easy to stick to a timetable!
Anyway, while watching the little sequence above, I was reminded of the Gramm-Schmidt Process, which the students doing Maths 1B in Summer Semester have been studying recently. It’s pretty much the only surefire way to create an orthonormal basis for a vector subspace. (If any of the words “orthonormal”, “basis”, “vector” and “subspace” are not familiar to you, then I wouldn’t worry about it too much — the point is that the Gramm-Schmidt Process is the only way to do a particular task.)
The problem with the Process is that it can take quite a lot of calculation, especially if you’re working with long vectors. This of course inspires the students to ask if there is an easier way, or at least a short-cut sometimes. It usually takes some convincing for the students to believe that in fact this is the only way and they’re going to have to get used to how long it takes. Sometimes there really is only one way to do something, and it does take a while.

RHAPSODY: I wonder what an orthonormal basis would be for this subspace.
HARMONY: Well there’s only one way to find an orthonormal basis in Fairyland…

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