Equivalence of 2 definitions of the magnitude of the cross product?

[nomadreid]

TL;DR Summary

The two definitions of the magnitude ||u x v|| of a cross product, one via coordinates, and the other with uv.sin A, are obviously equivalent, but my brute-force attempt hits a snag.

In order to prove that two of the means to calculate the magnitude of an arbitrary cross product are equivalent, I tried a direct approach, which I outline below. Any hint as to where I am going wrong would be appreciated.

So I hope this is the right section-- although this may look like HW, it isn't. It is my own fumbling.

Then

Then,

Alas, this doesn't come out. What am I doing wrong? Some silly mistake, no doubt, but I would be very grateful to have it pointed out.

 

[fresh_42]

This isn't an arbitrary cross product if the second and third coordinates are the same.

 

[nomadreid]

This isn't an arbitrary cross product if the second and third coordinates are the same.

Oops. Typo edited, thanks. Apart from that?

 

[fresh_42]

Oops. Typo edited, thanks. Apart from that?

You carried that typo throughout the calculation. There is no 2B term in the cross product. The cross product is a Lie multiplication, hence the Jacobi identity holds, hence it has to be symmetric in (x,y,z) and the pairs (x,y), (y,z), (z,x) with a cyclic permutation.

Edit: Mistake corrected.

 

[nomadreid]

Thanks very much for your reply, fresh_42. Let me start with your remark that there should be no 2B (my notation) in the result.
To find out where my mistake appears, I have broken Step 1 into four sub-steps for easy reference.

After I have figured out where my mistake(s) is (are) here, I will comb my text for the typos leading to the asymmetry you mentioned, as I don't see them at first glance. I will search again, but then I may turn to you again to specify. Thanks for your patience.

 

[fresh_42]

The essential point in your considerations is the sine of the angle between your vectors. How do you compute the sine without using the cross product? And turning it into a cosine doesn't answer the question, that's only adding confusion.

As I see it, you want to prove $\sin \measuredangle (\vec{u},\vec{v})= \dfrac{|\vec{u}\times \vec{v}|}{|\vec{u}|\cdot |\vec{v}|}$ where you have formulas for the right-hand side. So the question has to be: what is on the left? Do you use the law of cosines together with $1=\cos^2\measuredangle (\vec{u},\vec{v})+\sin^2\measuredangle (\vec{u},\vec{v})$ which is why you introduced the cosine? I guess, this can be done, even if it looks a bit like a nightmare of a calculation to me:
\begin{align*}
|(\vec{u}+\vec{v})^2|&=|\vec{u}|^2+|\vec{v}|^2-2|\vec{u}||\vec{v}|(1-\sin^2 \measuredangle (\vec{u},\vec{v}))\\
\sin^2 \measuredangle (\vec{u},\vec{v})&=1+\dfrac{|(\vec{u}+\vec{v})^2|-|\vec{u}|^2-|\vec{v}|^2}{2|\vec{u}||\vec{v}|}\\
&=\dfrac{|(\vec{u}+\vec{v})^2|-|\vec{u}|^2-|\vec{v}|^2+2|\vec{u}||\vec{v}|}{2|\vec{u}||\vec{v}|}\\
&=\dfrac{|(\vec{u}+\vec{v})^2|-(|\vec{u}|-|\vec{v}|)^2}{2|\vec{u}||\vec{v}|}\\
\sin \measuredangle (\vec{u},\vec{v})&=\sqrt{\left|\dfrac{|(\vec{u}+\vec{v})^2|-(|\vec{u}|-|\vec{v}|)^2}{2|\vec{u}||\vec{v}|}\right|}\\
\dfrac{|\vec{u}\times \vec{v}|}{|\vec{u}|\cdot |\vec{v}|}&\stackrel{!}{=}\sqrt{\left|\dfrac{|(\vec{u}+\vec{v})^2|-(|\vec{u}|-|\vec{v}|)^2}{2|\vec{u}||\vec{v}|}\right|}
\end{align*}
I didn't even use coordinates, yet, and I'm not sure whether I made a mistake or not. How do you plan to get through without any copy-and-paste errors?

If this wasn't your strategy, what else is it?
$$
$$
It doesn't answer your question, but you might be interested in this article:
https://arxiv.org/pdf/1205.5935
which has a nice introduction to the various products at its beginning.

 

[nomadreid]

Thank you, fresh_42. First, I have downloaded the ArXiv article, which looks very useful. Second, yes, your calculations are essentially the direction I was going, although I was doing it more by brute force with the coordinates to show the two methods of calculating the cross product to be equivalent -- and in the meantime I found my error (which, as I suspected, was a silly arithmetic error). To be explicit, if you keep the Step 1 in my Posts #1 and #5 and then replace the other steps in the original Post #1 by the following Steps:

one gets the desired

So, thanks again for the help.