Is it possible to rotate a trivector?

Question

We can rotate a vector anda bivector by applying a rotor to it. But can we rotate a trivector with rotors? If so, how?

Answer 1

In 3 dimensions, all vectors commute with trivectors. For example,

$$e_1(e_1e_2e_3)=e_1(e_1e_2)e_3=e_1(-e_2e_1)e_3=-e_1e_2(e_1e_3)=-e_1e_2(-e_3e_1)=+(e_1e_2e_3)e_1$$

and thus all multivectors commute with trivectors. (This is false in higher dimensions.) So applying a rotor to a trivector

$$RTR^{-1}=TRR^{-1}=T$$

does nothing. Geometrically, rotating 3D space in itself doesn't change it to a different 3D space, and doesn't change the volume of anything.

Answer 2

It seems to me that mr_e_man's answer applies to three dimensions, but not to four:

$e_1(e_2 e_3 e_4) = -e_2 e_1 e_3 e_4 = e_2 e_3 e_1 e_4 = -e_2 e_3 e_4 e_1$

Define $R = e_1 e_4$ and $\bar R = e_4 e_1$. Apply this rotor to the trivector $e_1 e_2 e_3$:

$$R e_1 e_2 e_3 \bar R = e_1 e_4 e_1 e_2 e_3 e_4 e_1 = - e_4 e_2 e_3 e_4 e_1 = e_4 e_2 e_4 e_3 e_1 = - e_4 e_4 e_2 e_3 e_1 = - e_2 e_3 e_1 = e_2 e_1 e_3 = - e_1 e_2 e_3$$

So, if you are working in four dimensions, you can rotate a trivector.