I'm trying to figure out why on earth the system of two vectors and their cross product must be positively oriented. Usually, this is just stated outright without further explanation.
I have however noticed that $||a x b||=||x||||y||sin \alpha $, which implies a somewhat strange solution to my problem. Since $ sin \alpha $ can be a negative value (for angles between 180 and 360 degrees) this would imply that $||a x b||$ could in fact be negative for those angles.
If by $||a x b||$ being positive we mean that it goes "up" from the plane and vice versa, this "would" in fact imply a positive orientation of the cross product and vectors a and b.
In that case the cross product vector would travel "upwards" from the plane if the angle was less than 180 degrees, making the entire triple of vectors positively oriented. It would travel "downwards" from the plane if the angle was between 180 and 360 degrees, but then we need only rotate the entire set of vectors 180 degrees and it would be plain that the system would still be positively oriented.
Am I onto something or am I grasping at straws?
The correct rule is $\Vert a\times b\Vert^2=\Vert a\Vert^2\Vert b\Vert^2\sin^2\alpha$, or equivalently $\Vert a\times b\Vert=\Vert a\Vert\Vert b\Vert|\sin\alpha|$ (note the $|\cdot|$ signs around $\sin\alpha$), because vectors cannot have negative length.