  Polycyclic Aromatic Hydrocarbons (PAHs) are organic molecules which comprise
the smallest particles of dust in the interstellar medium (ISM). Due to their
broad/complex emission profiles, obtaining kinematics is a challenge with
traditional methods, especially before the advent of the JWST. In this work we
employ Principal Component Analysis (PCA) tomography to analyse JWST/NIRSpec
IFU data of 3 nearby luminous infrared galaxies (LIRGs), namely, NGC 3256 N,
NGC 3256 S and NGC 7469. We detect the signature of rotation in the second
principal component of the 3.3 $\mu$m PAH feature in all three targets. We
construct velocity maps from the principal components for the 3.3 $\mu$m PAH
feature, Br$\beta$ (2.625$\mu$m) and molecular hydrogen, H$_2$ 1-0 S(1)
(2.12$\mu$m). We find that in each target, the PAHs qualitatively follow the
rotation of the galaxy, consistent with the rotational signature derived from
Br$\beta$ and H$_2$. There are however some differences, with the PAH rotation
in NGC 3256 N appearing at a different position angle, which suggest
differences in the motion of the dust as compared to the gas. This kind of
analysis opens a new window into this key component of the ISM.
