the reciprocal lattice vector g_hkl

If a reciprocal lattice point (hkl) is a destination, then g_hkl is the arrow that gets you there from the origin. It is the single vector that carries all the information about the (hkl) planes in one bundle: point it, measure its length, and you know both which way those planes face and how closely they are spaced. Crystallographers use it so constantly it earns its own symbol, g (some texts write it as a capital G or as H).

In formulas, g_hkl = h a* + k b* + l c* — just h steps along a*, k along b*, l along c*. Two facts make it the workhorse of diffraction. First, g_hkl is perpendicular to the (hkl) planes: it IS the plane normal. Second, its length is the reciprocal of the plane spacing: |g_hkl| = 1/d_hkl. Put those together with the plane-spacing equation and, for a cubic crystal, |g_hkl| = sqrt(h^2 + k^2 + l^2)/a, which is exactly sqrt(h^2+k^2+l^2) times a*. A longer g means more closely spaced planes and, as we will see in the diffraction field, scattering out to a wider angle.

The reason g matters beyond bookkeeping is that it is essentially the SCATTERING VECTOR of diffraction. When a wave scatters off the (hkl) planes and interferes constructively, the change in its wave-direction vector is exactly g_hkl. That is the whole content of the Laue condition, and it is why the length of g — how far out the reciprocal point sits — sets the Bragg angle. Convention note again: in the crystallographers' scheme |g| = 1/d; in the 2 pi physics scheme the same vector has length 2 pi/d and is usually called G. Same geometry, different scale factor.