  Giant resonances (GRs) provide crucial insights into nuclear physics and
astrophysics. Exciting GRs using particles like electrons is effective, yet the
angular momentum (AM) transfer of electrons, including both intrinsic spin and
orbital degrees of freedom in inelastic scattering, has never been studied.
Here, we investigate AM transfer in GRs excited by plane-wave and vortex
electrons, developing a comprehensive AM-resolved inelastic electron scattering
theory. We find that even plane-wave electrons can model-independently extract
transition strengths of higher multipolarity by selecting specific AM states of
scattered electrons. Additionally, relativistic vortex electrons with orbital
angular momentum (OAM) $\pm1$ can be efficiently generated. Vortex electrons
can also be used to extract GR transition strength as in the plane-wave case,
regardless of the position of nucleus relative to the beam axis. Furthermore,
relativistic vortex electrons with larger OAM can be generated for on-axis
nuclei due to AM conservation. Our method offers new perspectives for nuclear
structure research and paves the way for generating vortex particles.
