We present here a new approach to generalize supersymmetric quantum mechanics to treat multiparticle and multidimensional systems. We do this by introducing a vector superpotential in an orthogonal hyperspace. In the case of N distinguishable particles in three dimensions this results in a vector superpotential with 3N orthogonal components. The original scalar Schrodinger operator can be factored using a 3N-component gradient operator and introducing vector "charge" operators: (Q) over right arrow (1) and (Q) over right arrow (dagger)(1). Using these operators, we can write the original (scalar) Hamiltonian as H-1 = (Q) over right arrow (dagger)(1)center dot(Q) over right arrow (1) + E-0((1)) where E-0((1)) is the ground-state energy. The second sector Hamiltonian is a tensor given by H-2 = (Q) over right arrow (1)(Q) over right arrow (dagger)(1) + E-0((1)) and is isospectral with H-1. The vector ground state of sector 2, (psi) over right arrow ((2))(0) can be used with the charge operator (Q) over right arrow (dagger)(1) to obtain the excited-suite wave function of the first sector. In addition, we show that 112 can also be factored in terms of a sector 2 vector superpotential with components W-2j = -(theta In psi((2))(0j)/(theta x)(j). Here psi((2))(0j) is the jth component of (psi) over right arrow ((2))(0). Then one obtains charge operators (Q) over right arrow (2) and (Q) over right arrow (dagger)(2) so that the second sector Hamiltonian can be written as (H) over right arrow (2) = (Q) over right arrow (2)(Q) over right arrow (2) + E-0((2)). This allows us to define a third sector Hamiltonian which is a scalar, H3 = (Q) over right arrow (2)* (Q) over right arrow (dagger)(2) + E-0((2)). This prescription continues with the sector Hamiltonians alternating between scalar and tensor forms, both of which can be treated by the variational method to obtain approximate solutions to both scalar and tensor sectors. We demonstrate the approach with examples of a pair of separable 1D) harmonic oscillators and the example of a nonseparable 2D anharmonic oscillator (or equivalently a pair of coupled ID oscillators). We consider both degenerate and nondegenerate cases. We also present a generalization to arbitrary curvilinear coordinate systems in the Appendix.