Why does the cross product operation adhere to the right-hand rule?

[JTC]

(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?

 

[JTC]

In other words, could one say this...

We choose a right handed frame. Then we choose an operation on vectors modeled in that frame. The cross product operation, acting on the basis vectors must follow the same rule as the frame itself. But why? I mean, I know if they don't we have inconsistency, but what inconsistency. Reflections? What does that mean? how can I say it better?

 

[Mark44]

(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.

I think there are more than two possibilities for how the axes are oriented.

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)

No, this is not correct. The left side of your equation is a vector, but the right side is a scalar. What you get on the right side is the magnitude of the cross product. IOW, the corrected version is
##|\vec u \times \vec v| = |\vec u| |\vec v| \sin(\theta)##, where ##\theta## is the angle between the two vectors.

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.

I'm not sure it has anything to do with how the axes are oriented. You could have three vectors in space, with an arbitrary arrangement of the x-, y-, and z-axes. If you take a piece of paper and draw a couple of vectors that extend from a point, with no coordinate axis system being shown, you can draw the cross product of the two vectors using the right-hand rule.

The right-hand rule is an agreed-upon convention. I don't think there's anything more to be said about it.

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?

 

[JTC]

I think there are more than two possibilities for how the axes are oriented.
No, this is not correct. The left side of your equation is a vector, but the right side is a scalar. What you get on the right side is the magnitude of the cross product. IOW, the corrected version is
##|\vec u \times \vec v| = |\vec u| |\vec v| \sin(\theta)##, where ##\theta## is the angle between the two vectors.
I'm not sure it has anything to do with how the axes are oriented. You could have three vectors in space, with an arbitrary arrangement of the x-, y-, and z-axes. If you take a piece of paper and draw a couple of vectors that extend from a point, with no coordinate axis system being shown, you can draw the cross product of the two vectors using the right-hand rule.

The right-hand rule is an agreed-upon convention. I don't think there's anything more to be said about it.

No that did not help.

Just as an aside... the issue of the cross product was a typo-error: I just forgot the vector part. that is now fixed.

But my issue remains...

The cross product is a rule applied to vectors. yes. But we also must construct a frame (your comments about many ways is a tangential issue).

Why use the same rule?

It seems we have two issues:
1. How to build the frame (right or left -- setting aside the tangent)
2. How to define the cross product.

Is there no statement that the two must be done in harmony?

 

[JTC]

Here... I am looking for something like this..

We have a cross product definition. And we apply that definition to everything in the space. We want both the frame and the cross product operation to live on th same side of the mirror. So the definition, applied to vectors, is then applied to the frame.

See... but if I do that, the vector operation comes before the construction of the right handed frame.

So I don't know if I like that, either.

 

[Mark44]

I don't understand your concern here. The usual convention for ##\mathbb R^3## has the axes labeled as in this drawing.

If you have two non-collinear vectors in ##\mathbb R^3##, their cross product will be a vector that is perpendicular to each. The convention for the right-hand rule gives you one vector; a left-hand rule gives you a vector pointing in the opposite direction, but with the same magnitude.

 

[JTC]

I don't understand your concern here. The usual convention for ##\mathbb R^3## has the axes labeled as in this drawing.
View attachment 226252
If you have two non-collinear vectors in ##\mathbb R^3##, their cross product will be a vector that is perpendicular to each. The convention for the right-hand rule gives you one vector; a left-hand rule gives you a vector pointing in the opposite direction, but with the same magnitude.

All that is, as I explained, obvious to me.

I am looking for a statement like this...

"We define a cross product such that the output vector follows the same orientation as the frame itself, BECAUSE if we don't..."

In other words... we have two THINGS
1. the cross product operation
2. the orientation of the frame.

What came first? Why must the other one adhere to it?

 

[Mark44]

I am looking for a statement like this...

"We define a cross product such that the output vector follows the same orientation as the frame itself, BECAUSE if we don't..."

... you get a vector that points in the opposite direction.

What came first?

I imagine that orientation of the coordinate axis system (either in two dimensions or in three) preceded notions of vectors and in particular the cross product.

Why must the other one adhere to it?

Why do we adhere to the convention that ##\sqrt 4## represents the positive square root of 4, or 2? Because it's the generally agreed-to convention.

From post #1:

First, I am aware there is no cross product in higher dimensions.

There's a cross product in ##\mathbb R^7##. See https://en.wikipedia.org/wiki/Seven-dimensional_cross_product.

 

[JTC]

... you get a vector that points in the opposite direction.

I imagine that orientation of the coordinate axis system (either in two dimensions or in three) preceded notions of vectors and in particular the cross product.Why do we adhere to the convention that ##\sqrt 4## represents the positive square root of 4, or 2? Because it's the generally agreed-to convention.

From post #1: There's a cross product in ##\mathbb R^7##. See https://en.wikipedia.org/wiki/Seven-dimensional_cross_product.

So as trivial as this may appear, you have helped me a lot. THANK YOU.

Now I will risk this and try to reword things and I ask for your comment on this...

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First, we establish a coordinate system. To rise above ambiguity, we resolve on a right handed coordinate system. Done.

NEXT, we establish a useful operation: the cross product as a GEOMETRIC definition: a x b = |a| |b| sin(theta_ab) n

We like this GEOMETRIC DEFINITION because it maintains information about the plane of the vectors, the magnitude of the vectors and the angle between them. Indeed, there are problems with this operation (lack of adherence to associativity, difficulty in generalizing to higher dimensions, etc.) but for an undergraduate mechanical engineering, we accept this operation.

The problem with the GEOMETRICAL definition, however, two-fold. First,the direction of the resulting vector is not defined. Second, it is not always an easy operation to implement.

So we infuse the definition with the SAME handedness (right) of our coordinate system in order to ensure an unambiguous direction to the resulting vector. Then, as a RESULT of this, we find that: e1 x e2 = e3, e2 x e1 = -e3, e2 x e3 = e1, etc.

And as a RESULT of that, along with the Euclidean nature of space, we find a simpler ALGEBRAIC instantiation of our GEOMETRIC definition. And that is:
a x b = (a2 b3 - a3 b2) e1 + (a3 b1 - a1 b3) e2 + (a1 b2 - a2 b1) e3
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Is this a correct statement?