- #1
arcnets
- 508
- 0
Hi all,
just discovered the LaTeX feature, so why not play around with it a bit.
Quaternions are (sort of) generalized complex numbers where you have one real unit and 3 imaginary units i, j, and k. The basic equation is
[tex]
i^2 = j^2 = k^2 = ijk = -1.
[/tex]
Now, if we have a 4-vector
[tex]
\vec{r} = (t, x, y, z),
[/tex]
we could write it as a quaternion
[tex]
R = t + ix + jy + kz.
[/tex]
Defining
[tex]
R_3 = ix + jy + kz,
[/tex]
and
[tex]
R_0 = t
[/tex]
we get
[tex]
R^2 = (t^2 - x^2 - y^2 - z^2) + 2R_{0}R_{3}.
[/tex]
Thus
[tex]
R^2{}_0 = S^2
[/tex]
Where S2 is a relativistic invariant.
Maybe it's just useless, I'm just playing around. Any suggestions?
just discovered the LaTeX feature, so why not play around with it a bit.
Quaternions are (sort of) generalized complex numbers where you have one real unit and 3 imaginary units i, j, and k. The basic equation is
[tex]
i^2 = j^2 = k^2 = ijk = -1.
[/tex]
Now, if we have a 4-vector
[tex]
\vec{r} = (t, x, y, z),
[/tex]
we could write it as a quaternion
[tex]
R = t + ix + jy + kz.
[/tex]
Defining
[tex]
R_3 = ix + jy + kz,
[/tex]
and
[tex]
R_0 = t
[/tex]
we get
[tex]
R^2 = (t^2 - x^2 - y^2 - z^2) + 2R_{0}R_{3}.
[/tex]
Thus
[tex]
R^2{}_0 = S^2
[/tex]
Where S2 is a relativistic invariant.
Maybe it's just useless, I'm just playing around. Any suggestions?
Last edited: