The (^{10}Be,^{10}B^{*}[1.74 MeV]) charge-exchange reaction at 100 AMeV is presented as a new probe for isolating the isovector (ΔT=1) nonspin-transfer (ΔS=0) response of nuclei, with ^{28}Si being the first nucleus studied. By using a secondary ^{10}Be beam produced by fast fragmentation of ^{18}O nuclei at the NSCL Coupled Cyclotron Facility, applying the dispersion-matching technique with the S800 magnetic spectrometer to determine the excitation energy in ^{28}Al, and performing high-resolution γ-ray tracking with the Gamma-Ray Energy Tracking In-beam Nuclear Array (GRETINA) to identify the 1022-keV γ ray associated with the decay from the 1.74-MeV T=1 isobaric analog state in ^{10}B, a ΔS=0 excitation-energy spectrum in ^{28}Al was extracted. Monopole and dipole contributions were determined through a multipole-decomposition analysis, and the isovector giant dipole resonance and isovector giant monopole resonance (IVGMR) were identified. The results show that this probe is a powerful tool for studying the elusive IVGMR, which is of interest for performing stringent tests of modern density functional theories at high excitation energies and for constraining the bulk properties of nuclei and nuclear matter. The extracted distributions were compared with theoretical calculations based on the normal-modes formalism and the proton-neutron relativistic time-blocking approximation. Calculated cross sections based on these strengths underestimate the data by about a factor of 2, which likely indicates deficiencies in the reaction calculations based on the distorted wave Born approximation.