Transformation to flip handedness of a 4-vector

[Gza]

Hi all,

Long time no see! Had an interesting (non-hw question lol) posed to me as to what a general transform would be to turn a right-handed system to a left handed system in 4-space. I realize that there is no analogy of a vector cross product to use for 4-vectors (which is what i'd assume you'd have to do to generate an equation with the pre- and post transformed vectors.) Thanks for any ideas, and it's great to be back!

 

[HallsofIvy]

The problem is that "right and left handedness" only makes sense in 3 dimensions. In three dimensions, flipping one axis to its negative while leaving the other alone or, fliping two axes to their negatives while leaving the third along, changes "handedness". In two dimensions, flipping one axis from positive to negative while leaving the other alone, does not really change anything, geometrically, while in dimensions 4 or higher there are several different kinds of Cartesian coordinates, not just "left and right handed".

Essentially, a diagonal matrix with some +1 entries and some -1 entries will change what you might generalize from "handedness" but while two +1 entries and one -1 entry or two -1 and one +1 will effectively give the same thing in three dimensions, that is no true of the various options in four or more dimensions.

 

[Hurkyl]

In two dimensions, flipping one axis from positive to negative while leaving the other alone, does not really change anything

It reverses orientation, exactly as it does in one dimension, three dimensions, or more.