How to obtain these equations in 'Spinors in Physics' by J Hladik?

[grzz]

Homework Statement

Help required on how to obtain equations 1.1.11.

Relevant Equations

Please see attachments in my attempt at solution,

From the book Spinors in Physics by J Hladik:

I understood up to equations 1.1.10.
From 1.1.10 the vectors X1 and X2 depend on vector X3. But the point represented by X3 is projected stereographically to the point represented by the complex number in 1.1.4.
I understand that x3=y1z2 - y2z1 in equations 1.1.11, but I do not follow

Any help please,
Thanks.

 

[TSny]

I understand that x3=y1z2 - y2z1 in equations 1.1.11, but I do not follow
View attachment 360634

See (1.1.5) and (1.1.9)

 

[grzz]

See (1.1.5) and (1.1.9)

I want to KICK myself!
All I had to do was, just multiplying the RHS of the equations 1.1.9 by

which is equal to 1.
Thanks TSny.

 

[TSny]

All I had to do was, just multiplying the RHS of the equations 1.1.9 by View attachment 360642
which is equal to 1.

Yes. There are a lot of manipulations going on, so it is easy to overlook something simple. For example, it took me a while to see the last equality in (1.1.3) when it was only necessary to remember $x^2+y^2+z^2 = 1$.

 

[grzz]

Yes. There are a lot of manipulations going on, so it is easy to overlook something simple. For example, it took me a while to see the last equality in (1.1.3) when it was only necessary to remember $x^2+y^2+z^2 = 1$.

I want to tell you that your last reply was a comfort to me.
And now when I continued to,

I was wondering why was this particular choice made. Does one arrive at this choice by trial and error in order to obtain some desired result?
Any ideas?
Thanks.

 

[TSny]

It looks like the book pulls the choice (1.1.12) out of thin air. But you can verify that with this choice, {X₁, X₂, X₃} is a set of mutually perpendicular unit vectors with X₃ = X₁ x X₂.

 

[grzz]

Now I notice that the, apparently arbitrary, choice in 1.1.12 was made as a preparation for 1.1.24.

 

[frankieliu]

Yeah, that step from 1.1.10 to 1.1.11 is a bit dense. I think the key is treating the stereographic projection and the complex structure together — maybe try expressing the components explicitly in terms of y1,y2,z1,z2y_1, y_2, z_1, z_2y1,y2,z1,z2 and see how the cross-product forms. That helped me get some clarity when I looked at a similar setup.