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(First, I am aware there is no cross product in higher dimensions. I am aware of differential forms. I am aware of the problems with the cross product. I know enough to know what I don't know. but set all that aside.)

With that, could someone help me, by properly phrasing the cross product operation?

Allow me, please, to first try to explain what I mean (and forgive me the overuse of words).

First, we choose a frame of reference that is right handed. The right-hand rules that arises from the fact that the three axes of three-dimensional space have two possible orientations and we must select one, so we select the right hand rule. I get this... no problem.

Then, we have this operation called the cross product. And if our frame is right handed, we must use a right handed rule for the cross product. WHY?

Now the cross product has a trigonometric definition: a x b = |a| |b| sin(angle-ab) n

And it has an algebraic implementation in Euclidean Space wherein the base components are found from the axes of the frame: e1 x e2 = e3, etc.

I can see how, using the algebraic implementation, that it will conform to the right handed rule of the frame.

But how can one say that the trigonometric rule also must conform?

In other words, there is a FRAME and an OPERATION and I "sort of" get the feeling that the same rule must apply to BOTH. But why? I would like someone to tell me why in better words.

Essentially (and please forgive my confusion), I see a rule to the cross product operation and I see a rule to the frame construction, but why must the operation agree with the frame? I know it must, but could someone put this in words?

With that, could someone help me, by properly phrasing the cross product operation?

Allow me, please, to first try to explain what I mean (and forgive me the overuse of words).

First, we choose a frame of reference that is right handed. The right-hand rules that arises from the fact that the three axes of three-dimensional space have two possible orientations and we must select one, so we select the right hand rule. I get this... no problem.

Then, we have this operation called the cross product. And if our frame is right handed, we must use a right handed rule for the cross product. WHY?

Now the cross product has a trigonometric definition: a x b = |a| |b| sin(angle-ab) n

And it has an algebraic implementation in Euclidean Space wherein the base components are found from the axes of the frame: e1 x e2 = e3, etc.

I can see how, using the algebraic implementation, that it will conform to the right handed rule of the frame.

But how can one say that the trigonometric rule also must conform?

In other words, there is a FRAME and an OPERATION and I "sort of" get the feeling that the same rule must apply to BOTH. But why? I would like someone to tell me why in better words.

Essentially (and please forgive my confusion), I see a rule to the cross product operation and I see a rule to the frame construction, but why must the operation agree with the frame? I know it must, but could someone put this in words?

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