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Re: [Phys-l] constructing perpendiculars in D dimensions



Regarding John D's challenge problem:

Hi --

Here's a cute physics-related mathematics puzzle.

I came up with this while thinking about the
recent solenoid question, but the puzzle is
more general than that.

Assume you are working in D dimensions, where D >= 2.

1) Given a set of D-1 mutually perpendicular vectors, construct a
vector that is perpendicular to each of the others.
Hint: When D=3, you can simply take the cross
product.
Hint: When D=2, you can simply rotate the
vector 90 degrees in the plane.

The objective is to find a simple and elegant method that works for
all D.

2) Extra credit if your solution works in a D-dimensional Minkowski
space, i.e. where we have D-1 spacelike dimensions and 1 timelike
dimension.


The usual jsd puzzle rules apply: This is not meant to be a trick
question or word game. The answer does not involve finding any
hidden or twisted meaning in the wording of the question.
There exists an answer that is useful and IMHO interesting. There
may be more than one way of solving the puzzle. Everything I've
said, including the hints, is AFAIK true and helpful.
On the other hand, I haven't told you everything I know about the
puzzle; for example I haven't told you the answer.

Considering John's repeated lobbying for exterior/geometric algebra
I suspect that he would be expecting an answer somewhat along the
lines of:

Take the Hodge dual of the (D-1)-form exterior product of the D-1
(covariant) basis vectors that are already mutually orthogonal.

David Bowman