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I'm reading a book on linear algebra, where the author gives a method to test the handedness or chirality of a given set of 3 basis vectors.

if (v1 x v2) . v3 > 0 then it's right-handed, while if it's less than 0, it's left handed.

While what beats me is that numbers are just numbers, left or right handedness of a system depends on the viewer and how he interprets the given data.

Taking the canonical i, j, k basis vectors, in both left and right handed systems i x j = k, thereby k.k = ||k||^2 > 0 (always), then how does this test hold true?

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Orientation (handedness) is not about a set of vectors, it is about an ordered list of vectors. That is, a certain ordering, $(i,j,k)$ is agreed to as right handed. Then $(j,i,k)$ is left handed. This may or may not agree with some notion you have from physics, hard to predict.

A smooth manifold is orientable...never mind.

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  • $\begingroup$ Could you please add an example explaining in what case the condition would be false? Sorry if this is inappropriate or what I'm asking doesn't make sense. $\endgroup$ – legends2k Mar 11 '13 at 23:27
  • $\begingroup$ @legends2k, $j \times i = -k,$ so $(j \times i) \cdot k < 0,$ so the ordered basis $(j,i,k)$ has the opposite orientation compared with $(i,j,k).$ Looking at your question again, I do not know what you mean by a "left or right handed system," so I would say the use of the word "system" and the phrase "depends on the viewer" indicate thinking from some view of this that is not that of linear algebra or differential geometry texts. Perhaps computer graphics? There is no viewer in a vector space. $\endgroup$ – Will Jagy Mar 11 '13 at 23:41
  • $\begingroup$ Computer graphics it is :( However, I meant the interpreter or the person who, say, plots the values or draws the vector in a reference frame (of his/her own choice) as opposed to a hypothetical viwer in the space. For him how would this test help prove what he draw is left or right handed? $\endgroup$ – legends2k Mar 11 '13 at 23:54
  • $\begingroup$ The concept of induced orientation is probably what you want. If I stand outside a bank looking at a big glass window, I can draw a curved quarter-circle arrow with a marker on the window, indicating clockwise rotation by the location of the arrowhead. If I now walk inside the bank and look back out at the same window, the arrow seems now to indicate counterclockwise rotation. $\endgroup$ – Will Jagy Mar 12 '13 at 0:13
  • $\begingroup$ @legends2k, forgot to put at sign on previous $\endgroup$ – Will Jagy Mar 12 '13 at 0:34

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