/sci/ - Science & Math

dude, what?

>>8831053
>it's a "freshman just learned about cross product and orthogonal bases in his linear algebra class" episode

>>8831076
Clearly not, or he would have known that the definition of cross product doesn't generalize to four-dimensional space.

>>8831082
I wasn't generalizing to four-dimensional space.

>>8831082
it can be roughly generalized as follows:

[eqn](u^{\mu},v^{\nu},w^{\rho}) \mapsto \varepsilon^{\alpha \beta \gamma \delta} \textbf{e}_{\alpha} u_{\beta} v_{\gamma} w_{\delta}[/eqn]

where we observe einstein summation convention, as is usual, and the e's are the basis vectors of course.

>>8831122

this generalization replicates one of the most important properties of the cross product, the product remains orthogonal to the space which the vectors span, we can demonstrate this with a quick calculation:

[eqn]v_{\mu}\textbf{e}^{\mu}(\varepsilon^{\alpha \beta \gamma \delta} \textbf{e}_{\alpha} u_{\beta} v_{\gamma} w_{\delta})=\varepsilon^{\alpha \beta \gamma \delta} v_{\mu}\textbf{e}^{\mu}(\textbf{e}_{\alpha}) u_{\beta} v_{\gamma} w_{\delta}= \delta^{\mu}_{\alpha} \varepsilon^{\alpha \beta \gamma \delta} v_{\mu} u_{\beta} v_{\gamma} w_{\delta}=\varepsilon^{\alpha \beta \gamma \delta} v_{\alpha} u_{\beta} v_{\gamma} w_{\delta}=-\varepsilon^{\alpha \beta \gamma \delta} v_{\alpha} u_{\beta} v_{\gamma} w_{\delta}= 0[/eqn]

where in the last step of the above chain of equations we use the skew symmetry of the levi civita system and the play a game of dummy index renaming.

witen...

>>8831134

of course this is a rough generalization, though the pattern should be clear. In an N dimensional vector space we can define a 'product' on (N-1) vectors which has properties similar to the cross product.

>>8831053
no it's chemtrails mate get your shit together man

>>8831134
you can write

[math] v\cross w := \star (v\wedge w)[/math]

in any dimension, with the Hodge star and the wedge product.

https://en.wikipedia.org/wiki/Hodge_dual
https://en.wikipedia.org/wiki/Exterior_algebra

It's just that 3 is a special case where the result of two 3-dim vectors is again a 3-dim vector.

>>8831134
You can write

[math] v\times w := \star \, (v\wedge w)[/math]

in any dimension, with [math] \wedge [/math] the wedge product and [math] \star [/math] the Hodge star. It's just that 3 is a special case where the result of two 3-dim vectors is again a 3-dim vector.

https://en.wikipedia.org/wiki/Hodge_dual
https://en.wikipedia.org/wiki/Exterior_algebra

>>8831171

I know the cartan formalism, but I prefer coordinates and ricci calculus, senpai.

>>8831053
Does time not exist in lower dimensions?

>>8831315
Does time exist?

>visit /sci/ after 6 years of blissful absence
>this is the top thread

time is just an idea. it's convenient.

>>8831341
W-what was it like?

>>8831053
Caltrops, OP, how do they work?

>>8831053
Look for people using their left hand on tests.

>>8831341
Did the guy in the gif die?

First of all, tell us what you think a cross product is.

>>8832353
A vector with magnitude equal to the area of the formed parallelogram given by the two vectors with direction that is perpendicular to the other two vectors.
It also preserves "handedness" whatever that means.

>>8831053

Lower dimensions are your components of a vector. They're a basis which consists of normal vectors in which the dot products of orthogonal vectors will always be zero. Therefore your theory will fail

>>8833715

>>8831336

Time has no mass, no energy - so it can be destroyed without violating conservation of entropy.

>>8832315
Yeah he did. If you reverse image it a reddit post comes up with a video that includes the landing.