- #1

thermobum

- 2

- 0

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter thermobum
- Start date

- #1

thermobum

- 2

- 0

- #2

HallsofIvy

Science Advisor

Homework Helper

- 43,021

- 970

The nearest you can come to the cross product in 4 dimensions is the "alternating product" [itex]z_i= \epsilon_{ijkl}u_jv_kw_l}[/itex] where [itex]\epsilon[/itex] is defined by [itex]\epsilon_{ijkl}= 1[/itex] if ijkl is an even permutation of 1234, [itex]\epsilon_{ijkl}= -1[/itex] if ijkl is an odd permutation of 1234, [itex]\epsilon_{ijkl}= 0[/itex] if ijkl is not a permutation of 1234 (i.e. at least two indices are the same). Notice that that involves the product of 3 vectors (which is what you want).

- #3

- 6,515

- 1,878

To complete your problem, you could do this...

given an orthonormal set [tex]x^a[/tex], [tex]y^a[/tex], and [tex]z^a[/tex],

choose any vector [tex]u^a[/tex] so that [tex]e_{abcd}u^a x^b y^c z^d <>0[/tex] (so that [tex]u^a[/tex] is linearly independent of the set you have).

With this [tex]u^a[/tex], subtract out all of the components parallel to the orthonormal set... [tex]t^a=u^a- (g_{bc}u^b \hat x^c) \hat x^a - (g_{bc}u^b \hat y^c) \hat y^a - (g_{bc}u^b \hat z^c) \hat z^a[/tex].

Check the signs with your metric signature conventions.

- #4

pkleinod

- 111

- 2

For an excellent and thorough formulation of the algebra, go to the link

http://modelingnts.la.asu.edu/html/STC.html [Broken]

and click on the link at the very top of the page, Spacetime Calculus, to download a pdf file.

Last edited by a moderator:

- #5

thermobum

- 2

- 0

Share:

MHB
Vector space

- Last Post

- Replies
- 4

- Views
- 735

- Replies
- 5

- Views
- 685

- Last Post

- Replies
- 7

- Views
- 484

- Last Post

- Replies
- 2

- Views
- 2K

- Replies
- 3

- Views
- 255

- Last Post

- Replies
- 5

- Views
- 786

- Last Post

- Replies
- 2

- Views
- 787

- Last Post

- Replies
- 3

- Views
- 405

- Replies
- 4

- Views
- 425

- Replies
- 38

- Views
- 4K